# Convergence of the Distributed SG Algorithm Under Cooperative Excitation
Condition

Die Gan and Zhixin Liu Corresponding author: Zhixin Liu.This work was
supported by the National Key R$\&$D Program of China under Grant
2018YFA0703800. Natural Science Foundation of China, NSFC 11688101.D. Gan and
Z. X. Liu are with the Key Laboratory of Systems and Control, Academy of
Mathematics and Systems Science, Chinese Academy of Sciences, and School of
Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, P.
R. China. Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, a distributed stochastic gradient (SG) algorithm is proposed
where the estimators are aimed to collectively estimate an unknown time-
invariant parameter from a set of noisy measurements obtained by distributed
sensors. The proposed distributed SG algorithm combines the consensus strategy
of the estimation of neighbors with the diffusion of regression vectors. A
cooperative excitation condition is introduced, under which the convergence of
the distributed SG algorithm can be obtained without relying on the
independency and stationarity assumptions of regression vectors which are
commonly used in existing literature. Furthermore, the convergence rate of the
algorithm can be established. Finally, we show that all sensors can cooperate
to fulfill the estimation task even though any individual sensor can not by a
simulation example.

###### Index Terms:

Stochastic gradient algorithm, cooperative excitation condition, distributed
estimation, stochastic dynamic system, convergence.

##  I Introduction

Parameter estimation or filtering is one of the important issues in diverse
fields including statistical learning, signal processing, system
identification and adaptive control. With the development of computer science
and communication, sensor networks are widely applied due to the advantages of
flexibility, fault tolerance, and ease of deployment. The sensor networks
bring more and more data, and how to apply these data to design proper
estimation algorithms is a promising research direction.

Generally speaking, there are three manners to process the information from
the sensors: centralized, distributed and a combination of both (cf., [1]).
For the centralized method, the information measured by the sensors are
transmitted to a fusion center which uses all information to estimate the
unknown signals or parameters. Compared with the distributed algorithms, the
centralized ones are lack of robustness in addition to the burden brought by a
large amount of computation and communication. In distributed algorithms, the
sensors can accomplish complicated tasks in a cooperative manner even though
each sensor can only receive local information. A amount of theoretical
results on distributed estimation or learning algorithms (e.g., [2][3][4])
arise because of comprehensive practical applications in engineering systems,
such as target localization, collaborative spectral sensing, see e.g., [5][6].

In the investigation of distributed estimation algorithms, how to use the
local information to design the algorithms is important for the property of
the algorithms. Three types of strategies are often adopted in the current
literature: incremental strategy (cf., [7]), consensus strategy (cf., [8]),
and diffusion strategy (cf., [9][10]). Based on these three strategies, many
different distributed adaptive estimation algorithms are proposed, such as the
diffusion least mean squares (LMS), the consensus-based Kalman filter, the
diffusion least squares. Correspondingly, the stability and the convergence
analysis of the distributed estimation algorithms are also investigated under
some signal conditions (cf.,[11]-[20]). Schizas et al. in [16] established the
stability results for a distributed LMS-type adaptive algorithm with the
strictly stationary ergodic regressor vectors. Takahashi et al. in [17]
investigated the mean-transient and mean-square performance analysis of the
diffusion LMS algorithm with the independent and identically distributed
regressors. Cattivelli and Sayed in [18] provided the steady-state mean and
mean-square analysis of the diffusion LMS algorithm with the independent
Gaussian regressors. Arablouei et al. in [19] presented the convergence
analysis of a partial-diffusion recursive least squares algorithm for the
independent and ergodic input vectors. Lei and Chen in [20] studied the
convergence of the distributed stochastic approximation algorithm with ergodic
signals. So far, most results require that the regression signals satisfy some
stringent conditions, such as independency, stationarity and ergodicity
assumptions, which makes it hard or even impossible to apply these theoretical
results to practical feedback control systems. We remark that a preliminary
attempt towards the relaxation of the independency and stationarity
assumptions was made by Chen, Liu and Guo (cf., [21]), where they provided a
cooperative excitation condition to guarantee the stability of the diffusion
LMS algorithm, and some elegant results for the distributed LMS algorithm were
further established by Xie and Guo in [9] and [22] under a general cooperative
information condition.

It is well-known that the standard SG algorithm has the advantages of simple
expression and easy computation. The stochastic gradient algorithm is widely
applied in the area of adaptive control, and also has deep connections with
stochastic gradient descent algorithm and its variants which are widely used
to deal with optimization problems in machine learning [23]. With the
development of sensor networks, the distributed implementations of stochastic
gradient algorithms have attracted much attention of researchers (cf.,
[24]-[29]). For example, George et al. in [28] proposed a distributed
stochastic gradient descent algorithm for solving non-convex optimization
problems with applications in distributed supervised learning. Cavalcante and
Stanczak in [29] devised and analyzed a novel online distributed algorithm for
dynamic optimization problems.

In this paper, we consider a network of sensors which are aimed to
collectively estimate an unknown time-invariant parameter. We propose a
distributed SG algorithm based on the combined diffusion and consensus
strategies and study the convergence properties of the proposed algorithm for
a dynamic system. The main contributions of this paper are summarized as
follows.

($i$) We propose a novel distributed SG algorithm where each sensor is only
allowed to communicate with its neighbors. The information of the regression
vectors are first diffused through the sensor networks, and then the
estimation of the unknown parameters is obtained by using the consensus-based
strategy.

($ii$) We introduce a cooperative excitation condition on the regressor
signals, under which the strong consistency of the distributed SG algorithm
can be established. By the cooperative excitation condition, we see that the
estimation task can be still fulfilled by the cooperation of multiple sensors
even if any of them can not. The properties of the product of non-independent
and non-stationary random matrices are investigated, which is crucial for our
analysis. Our results can be degenerated to the convergence results on the
standard stochastic gradient algorithm (cf., [30][31]).

($iii$) We finally establish the convergence rate of the distributed
stochastic gradient algorithm under the cooperative excitation condition. We
remark that our convergence results and the convergence rate of the proposed
algorithm are obtained without relying on the assumptions of the independency
and stationarity of the regression signals, which makes it possible for
applications to the stochastic feedback systems.

The rest of this paper is organized as follows. In Section II, we first
propose the distributed SG algorithm, and introduce the cooperative excitation
condition. A necessary and sufficient condition for the strong consistency of
the proposed algorithm, and the conditions of the regressors for the
convergence of the algorithm are given in Section III. The convergence rate of
the distributed SG algorithm is given in Section IV. A simulation example is
given in Section V to illustrate our theoretical results. The concluding
remarks are made in the last section.

##  II Problem Formulation

###  II-A Some Preliminaries

In this paper, we use $\bm{A}\in\mathbb{R}^{m\times n}$ to denote an $m\times
n$-dimensional matrix. For a matrix $\bm{A}$, $\|\bm{A}\|$ denotes its
Euclidean norm, i.e.,
$\|\bm{A}\|\triangleq(\lambda_{\max}(\bm{A}\bm{A}^{T}))^{\frac{1}{2}}$, where
the notation $T$ denotes the transpose operator and $\lambda_{\max}(\cdot)$
denotes the largest eigenvalue of the matrix. The notations $\det(\cdot)$ and
$tr(\cdot)$ are used to denote the determinant and trace of the corresponding
matrix respectively. If all elements of a matrix are nonnegative, then it is a
nonnegative matrix, and furthermore if $\sum^{n}_{j=1}a_{ij}=1$ for all $i$,
then it is called a stochastic matrix. The Kronecker product
$\bm{A}\otimes\bm{B}$ of two matrices $\bm{A}=(a_{ij})\in\mathbb{R}^{m\times
n}$ and $\bm{B}\in\mathbb{R}^{p\times q}$ is defined as

$\displaystyle\bm{A}\otimes\bm{B}=\left(\begin{matrix}a_{11}\bm{B}&\cdots&a_{1n}\bm{B}\\\
\vdots&\ddots&\vdots\\\ a_{n1}\bm{B}&\cdots&a_{nn}\bm{B}\\\
\end{matrix}\right)\in\mathbb{R}^{mp\times nq}.$

Our purpose is to propose a distributed estimation algorithm based on the
information from a set of sensors and investigate the convergence properties
of the proposed algorithm. The sensors in sensor networks are modeled as
nodes, and the relationship between sensors are presented as an undirected
weighted graph $\mathscr{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$, where
$\mathcal{V}=\\{1,2,3,\cdots,n\\}$ is the set of sensors (i.e., nodes), the
edge set $\mathcal{E}\in\mathcal{V}\times\mathcal{V}$ denotes the
communication between sensors, and $\mathcal{A}=(a_{ij})$ is the weighted
matrix. The elements of the matrix $\mathcal{A}$ satisfy: $a_{ij}>0$ if
$(i,j)\in\mathcal{E}$ and $a_{ij}=0$ otherwise. The neighbor set of the sensor
$i$ is denoted as $N^{i}=\\{j\in\mathcal{V},(i,j)\in\mathcal{E}\\}$, and the
sensor $i$ is also included in this set. Each sensor can only exchange
information with its neighbors. A path of length $\ell$ is a sequence of nodes
$\\{i_{1},...i_{\ell}\\}$ satisfying $(i_{j},i_{j+1})\in\mathcal{E}$ for all
$1\leq j\leq\ell-1$. The graph $\mathscr{G}$ is called connected if for any
two sensors $i$ and $j$, there is a path connecting them. The diameter
$D(\mathscr{G})$ of the graph $\mathscr{G}$ is defined as the maximum length
of paths between any two sensors. For simplicity of analysis, the properties
of the distributed algorithm are considered under the condition that the
weighted matrix $\mathcal{A}$ is symmetric and stochastic. Hence, the
Laplacian matrix $\bm{L}$ of the graph $\mathscr{G}$ can be written as
$\bm{L}=\bm{I}-\mathcal{A}$ with $\bm{I}$ being the identity matrix.

A classical result for the Laplacian matrix $\bm{L}$ can be stated as follows.

###### Lemma II.1

[32] The Laplacian matrix $\bm{L}$ has at least one zero eigenvalue, with
other eigenvalues positive and less than or equal to 2. Moreover, if the graph
$\mathscr{G}$ is connected, then $\bm{L}$ has only one zero eigenvalue.

The following lemma is often used in our analysis and we list it as follows.

###### Lemma II.2

[33] Let $D_{t}\triangleq 1+\sum^{t}_{j=1}d_{j},~{}d_{j}\geq 0$, then

$\displaystyle\sum^{\infty}_{j=1}\frac{d_{j}}{D^{\alpha}_{j}}<\infty,~{}~{}\forall~{}\alpha>1,$
$\displaystyle\sum^{\infty}_{j=1}\frac{d_{j}}{D_{j}}=\infty~{}~{}{\rm
iff}~{}~{}\lim_{j\rightarrow\infty}D_{j}=\infty.$

###  II-B Distributed SG Algorithm

In this paper, we consider a network consisting of $n$ sensors. The signal
model of each sensor $i\in\\{1,...,n\\}$ is assumed to obey the following
time-invariant regression stochastic model,

$\displaystyle
y^{i}_{k+1}=\bm{\theta}^{T}{\bm{\varphi}^{i}_{k}}+\varepsilon^{i}_{k+1},~{}~{}k\geq
0,$ (1)

where $y^{i}_{k}$ is the scalar observation of the sensor $i$ at the time
instant $k$, $\bm{\varphi}^{i}_{k}\in\mathbb{R}^{m}$ is the random regression
vector which may be the function of the current and past inputs and outputs,
$\bm{\theta}\in\mathbb{R}^{m}$ is an unknown parameter to be estimated, and
$\\{\varepsilon^{i}_{k}\\}$ is a noise process.

We aim at designing a distributed adaptive estimation algorithm where all
sensors cooperatively estimate the unknown parameter $\bm{\theta}$ of the
stochastic dynamical system (1) by using local information
$\\{y^{j}_{k+1},\bm{\varphi}^{j}_{k}\\}_{j\in N^{i}}$, and further
establishing the (almost sure) convergence property and the convergence rate
of the proposed distributed algorithm.

The standard SG algorithm is commonly used in the area of adaptive control and
system identification (e.g.,[31][34]). Inspired by [2] and based on the
standard SG algorithm in the context of stochastic adaptive control, we
propose the following distributed SG algorithm to cooperatively estimate the
unknown parameter $\bm{\theta}$. The detailed algorithm can be found in
Algorithm 1.

Algorithm 1 Distributed SG Algorithm

Given any initial estimate $\bm{\hat{\theta}}^{i}_{0}$ of each sensor $i$, the
distributed SG algorithm is given as follows:

Step 1. Set the initial value at each time instant $k$ for each sensor
$i\in\\{1,...,n\\}$ as

$\displaystyle
x^{i}_{k}(0)=\frac{\|\bm{\varphi}^{i}_{k}\|^{2}}{r^{i}_{k}},~{}~{}r^{i}_{k}\triangleq
1+\sum_{j=1}^{k}{\left\|\bm{\varphi}^{i}_{j}\right\|}^{2}.$

Step 2. Perform the following diffusion process for $q=0,1,2,\cdots,Q$ with
$Q\geq D(\mathscr{G})$,

$\displaystyle x^{i}_{k}(q+1)=\sum_{j\in N^{i}}a_{ij}x^{j}_{k}(q).$

Step 3. Update the estimate $\bm{\hat{\theta}}^{i}_{k}$ of the unknown
parameter for each sensor $i$ as follows,

$\displaystyle\bm{z}^{i}_{k}(\bm{\hat{\theta}}^{i}_{k})$ $\displaystyle=$
$\displaystyle x^{i}_{k}(Q)\sum_{l\in
N^{i}}a_{li}(\bm{\hat{\theta}}^{i}_{k}-\bm{\hat{\theta}}^{l}_{k}),$
$\displaystyle\bm{\hat{\theta}}^{i}_{k+1}$ $\displaystyle=$
$\displaystyle\underbrace{\bm{\hat{\theta}}^{i}_{k}+\mu\frac{\bm{\varphi}^{i}_{k}}{r^{i}_{k}}(y^{i}_{k+1}-({\bm{\varphi}^{i}_{k}})^{T}\bm{\hat{\theta}}^{i}_{k})}_{\rm{Standard~{}SG}}$
(2) $\displaystyle\underbrace{-\mu\nu\sum_{j\in
N^{i}}a_{ij}\left(\bm{z}^{i}_{k}(\bm{\hat{\theta}}^{i}_{k})-\bm{z}^{j}_{k}(\bm{\hat{\theta}}^{j}_{k})\right)}_{\rm{Consensus-
based~{}item}},$

where $\mu$ and $\nu$ are two step sizes lying in $(0,1)$, $r^{i}_{k}$ is
defined in Step $1$.

###### Remark II.1

Algorithm 1 designed by combining the consensus strategy of the estimation of
neighbors with the diffusion strategy of regression vectors is online and
updated from time to time by using new measurement data. We see that the right
hand side of (2) in Algorithm 1 consists of two parts: the first part is the
standard SG algorithm which tries to minimize the prediction error by using
the innovation, while the second part can be regarded as the result of
minimizing the weighted distance between the estimates of the sensor $i$ and
its neighbors.

###### Remark II.2

In Step 2, the multi-step diffusion strategy of regression vectors is used,
which was widely employed in the design of the distributed algorithms
(e.g.,[35]-[37]). The diffusion step $Q$ plays an important role in
establishing the contraction property of the product of random matrices
$\prod^{k}_{p=j}(\bm{I}_{mn}-\mu\bm{G}_{p})$ (see the proof of Theorem IV.1 in
Appendix D). Moreover, by following the proof of Lemma III.4, we see that if
the condition number of $\bm{\Phi}^{T}_{k}\bm{\Phi}_{k}$ is bounded, then the
multi-step diffusion step can be removed.

For convenience of analysis, we introduce the following notations (see Table
I). In Table I, $col(\cdot,\cdots,\cdot)$ denotes the vector stacked by the
specified vectors, and $diag(\cdot,\cdots,\cdot)$ denotes the block matrix
formed in a diagonal manner of the corresponding vectors or matrices.

TABLE I: Some notations Notation | Definition | Dimension
---|---|---
$\bm{Y}_{k}$ | $\\{y^{1}_{k},\cdots,y^{n}_{k}\\}$ | $1\times n$
$\bm{\Phi}_{k}$ | $diag\\{\bm{\varphi}^{1}_{k},\cdots,\bm{\varphi}^{n}_{k}\\}$ | $mn\times n$
$\bm{\Xi}_{k}$ | $\\{\varepsilon^{1}_{k},\cdots,\varepsilon^{n}_{k}\\}$ | $1\times n$
$\bm{\Theta}$ | $col\\{\underbrace{\bm{\theta},\cdots,\bm{\theta}}_{n}\\}$ | $mn\times 1$
$\hat{\bm{\Theta}}_{k}$ | $col\\{\hat{\bm{\theta}}^{1}_{k},\cdots,\hat{\bm{\theta}}^{n}_{k}\\}$ | $mn\times 1$
$\widetilde{\bm{\Theta}}_{k}$ |  $col\\{\widetilde{\bm{\theta}}^{1}_{k},\cdots,\widetilde{\bm{\theta}}^{n}_{k}\\}$, $\widetilde{\bm{\theta}}^{i}_{k}=\bm{\theta}-\hat{\bm{\theta}}^{i}_{k}$ | $mn\times 1$
$\bm{R}_{k}$ | $diag\\{r^{1}_{k},\cdots,r^{n}_{k}\\}$ | $n\times n$
$\mathscr{L}$ |  $\bm{L}\otimes\bm{I}_{m}$, $\bm{L}$ is the Laplacian matrix | $mn\times mn$
$\bm{A}_{k}$ | ${\bm{\Phi}_{k}}{{\bm{R}}^{-1}_{k}}{{\bm{\Phi}}^{T}_{k}}$ | $mn\times mn$
$\bm{X}_{k}(Q)$ | $diag\\{x^{1}_{k}(Q),\cdots,x^{n}_{k}(Q)\\}$ | $n\times n$
$\bm{G}_{k}$ | $\bm{A}_{k}+\nu\mathscr{L}(\bm{X}_{k}(Q)\otimes\bm{I}_{m})\mathscr{L}$ | $mn\times mn$

By the notations in Table I, we can rewrite (1) and (2) into the following
matrix form,

$\displaystyle\bm{Y}_{k+1}$ $\displaystyle=$
$\displaystyle\bm{\Theta}^{T}{{\bm{\Phi}}_{k}}+\bm{\Xi}_{k+1},$
$\displaystyle\hat{\bm{\Theta}}_{k+1}$ $\displaystyle=$
$\displaystyle\hat{\bm{\Theta}}_{k}+\mu\bm{\Phi}_{k}{\bm{R}}^{-1}_{k}(\bm{Y}^{T}_{k+1}-{\bm{\Phi}^{T}_{k}}\hat{\bm{\Theta}}_{k})$
$\displaystyle-\mu\nu\mathscr{L}(\bm{X}_{k}(Q)\otimes\bm{I}_{m})\mathscr{L}\hat{\bm{\Theta}}_{k}.$

Let $\widetilde{\bm{\Theta}}_{k}=\bm{\Theta}-\hat{\bm{\Theta}}_{k}$. It is
clear that $\mathscr{L}\bm{\Theta}=0$, then we have the following error
equation,

$\displaystyle\widetilde{\bm{\Theta}}_{k+1}=(\bm{I}_{mn}-\mu\bm{G}_{k})\widetilde{\bm{\Theta}}_{k}-\mu{\bm{\Phi}_{k}}{{\bm{R}}^{-1}_{k}}\bm{\Xi}^{T}_{k+1}.$
(3)

In order to proceed with our analysis, we introduce some assumptions
concerning the graph, regression vectors and the system noise.

###### Assumption II.1

The undirected graph $\mathscr{G}$ is connected.

###### Assumption II.2

(Cooperative Excitation Condition) There exist two positive constants $N$ and
$K_{0}$ such that for $k\geq K_{0}$, the following inequality is satisfied,

$\displaystyle\frac{\lambda^{(k)}_{\max}}{\lambda^{(k)}_{\min}}\leq
N\left(log(\|\bm{R}_{k}\|)\right)^{\frac{1}{3}},{~{}~{}~{}\rm a.s.}$ (4)

where $\lambda^{(k)}_{\max}$, $\lambda^{(k)}_{\min}$ represent the maximum and
minimum eigenvalues of the matrix
$\frac{n}{m}\bm{I}_{m}+\sum^{n}_{i=1}\sum^{k}_{j=1}\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T},$
and $\|\bm{R}_{k}\|\rightarrow\infty$ as $k\rightarrow\infty$.

###### Remark II.3

We give an intuitive explanation for the necessity of the above cooperative
excitation condition. Consider an extreme case where all regressor vectors
$\bm{\varphi}^{i}_{j}$ are equal to zero. It is clear that Assumption II.2 is
not satisfied, and the unknown parameter $\bm{\theta}$ can not be identified
since the observations do not contain any information about the unknown
parameter. In order to estimate $\bm{\theta}$, we should impose some nonzero
information (or excitation) conditions on the regressor vectors
$\bm{\varphi}^{i}_{j}$. The following persistent excitation (PE) condition for
the single sensor case is commonly used in the literature (see
e.g.,[38]-[39]).

$\mathbf{(PE):}$ $r^{i}_{k}\xrightarrow[]{k\rightarrow\infty}\infty$ and for
all $k\geq 0$, there exits a positive constant $c_{1}$ such that

$\displaystyle\frac{\lambda_{\max}\left(\sum^{k}_{j=1}\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}\right)}{\lambda_{\min}\left(\sum^{k}_{j=1}\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}\right)}\leq{c_{1}}<\infty,~{}{\rm{a.s..}}$
(5)

Therefore, a similar excitation condition is needed to guarantee the
convergence of the distributed SG algorithm.

###### Remark II.4

It can be verified that the independent and identically distributed (i.i.d.)
signals (by the strong law of large numbers) and the stationary ergodic
signals (by the ergodic theorem) have the following property,

$\mathbf{(Ergodicity~{}Property):}$ For any $i\in\\{1,...,n\\}$, the regressor
vectors $\bm{\varphi}^{i}_{j}$ satisfy the ergodicity property, i.e., there
exists a matrix $\bm{H}_{i}$ such that

$\displaystyle\frac{1}{k}\sum^{k}_{j=1}\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}~{}~{}\underrightarrow{k\rightarrow\infty}~{}~{}\bm{H}_{i},~{}{\rm
a.s.}.$

Furthermore, if $\sum^{n}_{i=1}\bm{H}_{i}$ is positive definite (cf., [20]),
then the ergodicity property implies the PE condition in the multiple sensor
case, i.e., ${\lambda^{(k)}_{\max}}/{\lambda^{(k)}_{\min}}\leq{c_{2}}$ with
${c_{2}}$ being a positive constant, which means that Assumption II.2 is
satisfied.

###### Remark II.5

Guo in [30] proved that the convergence of the standard SG algorithm under the
following excitation condition,

$\displaystyle\frac{\lambda_{\max}\Big{(}\sum^{k}_{j=1}\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}\Big{)}}{\lambda_{\min}\Big{(}\sum^{k}_{j=1}\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}\Big{)}}\leq\tilde{N}\Big{(}\log
r^{i}_{k}\Big{)}^{\frac{1}{3}},~{}{\rm a.s.},$ (6)

where $r^{i}_{k}\rightarrow\infty$ as $k\rightarrow\infty.$ It is clear that
the condition (6) is weaker than the PE condition (5) for the single sensor
case, and Assumption II.2 can be degenerated to the condition (6) when $n=1$.

###### Remark II.6

The cooperative excitation condition (i.e., Assumption II.2) reflects the
cooperative effect of multiple sensors. By Example V.1 in Section V, we can
show that the estimation task can be still fulfilled by the cooperation of
multiple sensors even if any of them can not.

###### Assumption II.3

We assume that the system noise $\\{\varepsilon^{i}_{k},i=1,...n,k\geq 1\\}$
is a martingale difference sequence, that is,
$E(\bm{\Xi}_{k+1}|\mathscr{F}_{k})=0$ with
$\mathscr{F}_{k}=\sigma\\{\bm{\varphi}^{i}_{j},\varepsilon^{i}_{j},i=1,...n,j\leq
k\\}$ and $E(\cdot|\cdot)$ being the conditional mathematical expectation, and
there exist constants $c_{0}>0$ and $\varepsilon\in[0,1)$ (which may depend on
$\omega$) such that $E(\|\bm{\Xi}_{k+1}\|^{2}|\mathscr{F}_{k})\leq
c_{0}\|\bm{R}_{k}\|^{\varepsilon}$ a.s..

##  III Convergence of Distributed SG Algorithm

In this section, we will provide the convergence analysis of the proposed
distributed SG algorithm.

Let the state transition matrix ${\bm{\Psi}}(k,j)$ be recursively defined by

$\displaystyle{\bm{\Psi}}(k+1,j)=(\bm{I}_{mn}-\mu\bm{G}_{k}){\bm{\Psi}}(k,j),~{}{\bm{\Psi}}(j,j)=\bm{I}_{mn}.$
(7)

From the error equation (3), we can see that the analysis of the error
$\widetilde{\bm{\Theta}}_{k+1}$ can be divided into two key steps:

(i) Analyzing properties of the product of random matrices
${\bm{\Psi}}(k,j)=\prod_{p=j}^{k-1}(\bm{I}_{mn}-\mu\bm{G}_{p})$.

(ii) Analyzing the cumulative effect of noises.

In order to establish the properties of ${\bm{\Psi}}(k,j)$, we first analyze
the properties of $\bm{G}_{k}$.

###### Lemma III.1

Suppose that Assumption II.1 is satisfied. If $\mu(1+4\nu)\leq 1$, then we
have

$\displaystyle 0\leq\mu\bm{G}_{k}\leq\bm{I}_{mn}.$

Proof. By Step 2 in Algorithm 1, we have

$\displaystyle
x^{i}_{k}(Q)=\sum^{n}_{j=1}a^{(Q)}_{ij}\frac{\|\bm{\varphi}^{j}_{k}\|^{2}}{r^{j}_{k}},$
(8)

where $a^{(Q)}_{ij}$ is the $i$-th row, $j$-th column element of
$\mathcal{A}^{Q}$. The matrix $\mathcal{A}$ is stochastic, so is the matrix
$\mathcal{A}^{Q}$ for $Q\geq 1$. Hence we have for $i\in\\{1,\cdots,n\\}$,

$\displaystyle x^{i}_{k}(Q)\leq\Big{(}\max_{1\leq j\leq
n}\frac{\|\bm{\varphi}^{j}_{k}\|^{2}}{r^{j}_{k}}\Big{)}\sum^{n}_{j=1}a^{(Q)}_{ij}=\max_{1\leq
j\leq n}\frac{\|\bm{\varphi}^{j}_{k}\|^{2}}{r^{j}_{k}}\leq\|\bm{A}_{k}\|.$

By the definition $\bm{X}_{k}(Q)$ in Table I, we have
$\|\bm{X}_{k}(Q)\otimes\bm{I}_{m}\|\leq\|\bm{A}_{k}\|$. By Lemma II.1, it
follows that

$\displaystyle\|\mu\bm{G}_{k}\|\leq\mu(\|\bm{A}_{k}\|+4\nu\|\bm{A}_{k}\|)\leq\mu(1+4\nu)\leq
1.$ (9)

This completes the proof of the lemma. $\blacksquare$

By the definition of $\bm{G}_{k}$ in Table I, we see that $\bm{G}_{k}$ is a
nonnegative definite matrix. Thus, there exists a matrix sequence
$\\{\bm{B}_{k},k\geq 0\\}$, such that for all $k$ we have

$\displaystyle\bm{B}^{2}_{k}=\mu\bm{G}_{k}.$ (10)

In the following, we will analyze the properties of ${\bm{\Psi}}(k,j)$.

###### Lemma III.2

Assume that the step sizes $\mu$ and $\nu$ satisfy $\mu(1+4\nu)\leq 1$. Then
for any $k\geq 0$ the following inequality holds,

$\displaystyle\sum^{k-1}_{j=0}\|\bm{\Psi}(k,j+1)\bm{B}_{j}\|^{2}\leq mn.$

Proof. By the definition of the state transition matrix in (7), we have
$\bm{\Psi}(k,k)=\bm{I}_{mn}$ and

$\displaystyle{\bm{\Psi}}(k,j+1){\bm{\Psi}}(j+1,j)={\bm{\Psi}}(k,j).$

Then,

$\displaystyle mn$ $\displaystyle=$ $\displaystyle
tr\Big{(}{\bm{\Psi}}(k,k)\bm{\Psi}^{T}(k,k)\Big{)}$ $\displaystyle\geq$
$\displaystyle
tr\Bigg{(}\sum^{k-1}_{j=0}\Big{[}{\bm{\Psi}}(k,j+1){\bm{\Psi}^{T}}(k,j+1)$
$\displaystyle-{\bm{\Psi}}(k,j){\bm{\Psi}^{T}}(k,j)\Big{]}\Bigg{)}$
$\displaystyle=$ $\displaystyle
tr\Bigg{(}\sum^{k-1}_{j=0}{\bm{\Psi}}(k,j+1)\Big{[}\bm{I}_{mn}-{\bm{\Psi}}(j+1,j)$
$\displaystyle\cdot{\bm{\Psi}^{T}}(j+1,j)\Big{]}{\bm{\Psi}^{T}}(k,j+1)\Bigg{)}.$

Furthermore, by ${\bm{\Psi}}(j+1,j)=\bm{I}_{mn}-\mu\bm{G}_{j}$, we have

$\displaystyle mn$ $\displaystyle\geq$ $\displaystyle
tr\Bigg{(}\sum^{k-1}_{j=0}{\bm{\Psi}}(k,j+1)\Big{[}\mu\bm{G}_{j}+\mu\bm{G}_{j}(\bm{I}_{mn}-\mu\bm{G}_{j})\Big{]}$
$\displaystyle\cdot{\bm{\Psi}^{T}}(k,j+1)\Bigg{)}$ $\displaystyle\geq$
$\displaystyle
tr\Bigg{(}\sum^{k-1}_{j=0}{\bm{\Psi}}(k,j+1)\mu\bm{G}_{j}{\bm{\Psi}^{T}}(k,j+1)\Bigg{)}$
$\displaystyle=$ $\displaystyle
tr\Bigg{(}\sum^{k-1}_{j=0}{\bm{\Psi}}(k,j+1)\bm{B}^{2}_{j}{\bm{\Psi}^{T}}(k,j+1)\Bigg{)}$
$\displaystyle\geq$
$\displaystyle\sum^{k-1}_{j=0}\|{\bm{\Psi}}(k,j+1)\bm{B}_{j}\|^{2},$

which completes the proof of the lemma. $\blacksquare$

How to deal with the noise effect of the distributed SG algorithm is a crucial
step for the convergence analysis of the algorithm. The following lemma
provides an upper bound of the cumulative summation of the noises.

###### Lemma III.3

Suppose that Assumption II.3 is satisfied, and the condition number of
$\bm{R}_{k}$ is bounded (i.e., there exists a positive constant $\gamma$ which
may depend on the sample $\omega$ such that $\max\limits_{1\leq i\leq
n}{r^{i}_{k}}\Big{/}{\min\limits_{1\leq i\leq n}{r^{i}_{k}}}$ $\leq\gamma$),
then $\bm{S}_{k}$ tends to a finite limit $\bm{S}$ as $k\rightarrow\infty$,
where
$\bm{S}_{k}\triangleq\sum^{k}_{j=0}{\bm{\Phi}_{j}}{{\bm{R}}^{-1}_{j}}\bm{\Xi}^{T}_{j+1}$.
Furthermore, there exists a positive constant $c$ which may depend upon the
sample $\omega$ such that

$\displaystyle\left\|\widetilde{\bm{S}}_{k-1}\right\|\leq
c\|\bm{R}_{k}\|^{-\delta},$ (12)

where $\widetilde{\bm{S}}_{k-1}\triangleq\bm{S}-\bm{S}_{k-1}$ and
$\delta\in(0,\frac{1-\varepsilon}{2})$ with $\varepsilon$ defined in
Assumption II.3.

The proof is put in Appendix A.

Now, we present a necessary and sufficient condition for the strong
consistency of the distributed SG algorithm.

###### Theorem III.1

Suppose that the condition number of $\bm{R}_{k}$ is bounded, and
$\mu(1+4\nu)<1$. Then under Assumptions II.1 and II.3, the estimate
$\hat{\bm{\Theta}}_{k}$ defined in Table I converges to the true parameter
${\bm{\Theta}}$ almost surely (a.s.) for any initial value
$\hat{\bm{\Theta}}_{0}$ if and only if $\bm{\Psi}(k,0)\rightarrow 0$ almost
surely as $k\rightarrow\infty$.

Proof. By $(\ref{10})$ and $(\ref{11})$, we have the following expression,

$\displaystyle\widetilde{\bm{\Theta}}_{k+1}$ $\displaystyle=$
$\displaystyle{\bm{\Psi}}(k+1,0)\widetilde{\bm{\Theta}}_{0}$ (13)
$\displaystyle-\mu\sum^{k}_{j=0}{\bm{\Psi}}(k+1,j+1)\bm{\Phi}_{j}\bm{R}^{-1}_{j}\bm{\Xi}^{T}_{j+1}.$

We notice that the second term on the right-hand side of $(\ref{46})$ is
independent of $\widetilde{\bm{\Theta}}_{0}$. Thus,
$\widetilde{\bm{\Theta}}_{k+1}\rightarrow 0$ for any
$\widetilde{\bm{\Theta}}_{0}$ implies
${\bm{\Psi}}(k+1,0)\widetilde{\bm{\Theta}}_{0}\rightarrow 0$ as
$k\rightarrow\infty$, which means that ${\bm{\Psi}}(k+1,0)\rightarrow 0$ as
$k\rightarrow\infty$. This completes the proof of the necessity part of the
theorem.

Now, let us move on to the sufficiency part. It is clear that in order to
prove the convergence of the algorithm, we just need to prove

$\displaystyle\sum^{k}_{j=0}{\bm{\Psi}}(k+1,j+1)\bm{\Phi}_{j}\bm{R}^{-1}_{j}\bm{\Xi}^{T}_{j+1}\rightarrow
0\ {\rm a.s.},\ \hbox{as}~{}k\rightarrow\infty.$ (14)

Set $\bm{S}_{-1}=0$. By the definition of ${\bm{\Psi}}(\cdot,\cdot)$ in (7),
we have

$\displaystyle\bigg{\|}\sum^{k}_{j=0}{\bm{\Psi}}(k+1,j+1)\bm{\Phi}_{j}\bm{R}^{-1}_{j}\bm{\Xi}^{T}_{j+1}\bigg{\|}$
(15) $\displaystyle=$
$\displaystyle\bigg{\|}\sum^{k}_{j=0}{\bm{\Psi}}(k+1,j+1)(\bm{S}_{j}-\bm{S}_{j-1})\bigg{\|}$
$\displaystyle=$
$\displaystyle\bigg{\|}\bm{S}_{k}-\sum^{k}_{j=0}[{\bm{\Psi}}(k+1,j+1)-{\bm{\Psi}}(k+1,j)]\bm{S}_{j-1}\bigg{\|}$
$\displaystyle=$
$\displaystyle\bigg{\|}\bm{S}_{k}-\sum^{k}_{j=0}[{\bm{\Psi}}(k+1,j+1)-{\bm{\Psi}}(k+1,j)]\bm{S}$
$\displaystyle+\sum^{k}_{j=0}[{\bm{\Psi}}(k+1,j+1)-{\bm{\Psi}}(k+1,j)]\widetilde{\bm{S}}_{j-1}\bigg{\|}$
$\displaystyle=$
$\displaystyle\bigg{\|}\bm{S}_{k}-\bm{S}+{\bm{\Psi}}(k+1,0)\bm{S}+\sum^{k}_{j=0}{\bm{\Psi}}(k+1,j+1)$
$\displaystyle\cdot[\bm{I}_{mn}-{\bm{\Psi}}(j+1,j)]\widetilde{\bm{S}}_{j-1}\bigg{\|},$

where $\bm{S},\bm{S}_{k},\widetilde{\bm{S}}_{k-1}$ are defined in Lemma III.3.

By Lemma III.3 we have $\bm{S}_{k}-\bm{S}\rightarrow 0$. By the condition that
${\bm{\Psi}}(k+1,0)\rightarrow 0$ as $k\rightarrow\infty$, we have
${\bm{\Psi}}(k+1,0)\bm{S}\rightarrow 0$ as $k\rightarrow\infty$. By Lemma
III.3 and Hölder inequality, we have

$\displaystyle\bigg{\|}\sum^{k}_{j=0}{\bm{\Psi}}(k+1,j+1)[\bm{I}_{mn}-{\bm{\Psi}}(j+1,j)]\widetilde{\bm{S}}_{j-1}\bigg{\|}$
(16) $\displaystyle=$
$\displaystyle\bigg{\|}\sum^{k}_{j=0}{\bm{\Psi}}(k+1,j+1)\mu\bm{G}_{j}\widetilde{\bm{S}}_{j-1}\bigg{\|}$
$\displaystyle\leq$ $\displaystyle
c\sum^{M}_{j=0}\|{\bm{\Psi}}(k+1,j+1)\bm{B}_{j}\|\frac{\|\bm{B}_{j}\|}{\|\bm{R}_{j}\|^{\delta}}$
$\displaystyle+c\Big{(}\sum^{k}_{j=M+1}\|{\bm{\Psi}}(k+1,j+1)\bm{B}_{j}\|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle~{}~{}~{}~{}\cdot\Big{(}\sum^{k}_{j=M+1}\frac{\|\bm{B}_{j}\|^{2}}{\|\bm{R}_{j}\|^{2\delta}}\Big{)}^{\frac{1}{2}}.$

Furthermore, by Lemma II.2 and Lemma III.1 we have

$\displaystyle\sum^{\infty}_{j=1}\frac{\|\bm{B}_{j}\|^{2}}{\|\bm{R}_{j}\|^{2\delta}}=\sum^{\infty}_{j=1}\frac{\|\mu\bm{G}_{j}\|}{\|\bm{R}_{j}\|^{2\delta}}$
(17) $\displaystyle\leq$
$\displaystyle\mu(1+4\nu)\sum^{\infty}_{j=1}\frac{\|\bm{A}_{j}\|}{\|\bm{R}_{j}\|^{2\delta}}$
$\displaystyle\leq$
$\displaystyle\sum^{\infty}_{j=1}\frac{\|\bm{\Phi}_{j}\|^{2}\|\bm{R}^{-1}_{j}\|}{\|\bm{R}_{j}\|^{2\delta}}\leq\gamma\sum^{\infty}_{j=1}\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+2\delta}}<\infty.$

According to $(\ref{55})$ and Lemma III.2, we see that the two terms on the
right-hand-side of (16) tend to zero if we first let $k\rightarrow\infty,$ and
then let $M\rightarrow\infty$. Hence (14) holds. This completes the proof of
the theorem. $\blacksquare$

A key problem still remains unresolved: what conditions on the regression
signals $\\{\bm{\varphi}^{i}_{k}\\}$ can guarantee that
$\bm{\Psi}(k,0)\rightarrow 0$ as $k\rightarrow\infty$? In the following, we
will prove that under the cooperative excitation condition (i.e., Assumption
II.2) the convergence results for the distributed algorithm can be
established.

Before stating the main theorem of this section, we first give a lemma which
provides a key step for the convergence of the matrix ${\bm{\Psi}}(k,0)$.

###### Lemma III.4

Suppose that Assumption II.1 is satisfied. Then there exists a positive
constant $\sigma$, such that the following inequality holds for all $t\geq 0$
and all $u\geq 0$,

$\displaystyle\lambda_{\min}\left(\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\bm{G}_{k}\right)\geq\sigma\lambda_{\min}\left(\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\sum^{n}_{i=1}\bm{A}^{i}_{k}\right),$
(18)

where
$\bm{A}^{i}_{k}\triangleq\frac{\bm{\varphi}^{i}_{k}(\bm{\varphi}^{i}_{k})^{T}}{r^{i}_{k}}$,
$\hat{g}(t)\triangleq\max\\{k:d_{k}\leq t\\}$, and
$d_{k}\triangleq\sum^{n}_{i=1}\sum^{k-1}_{j=K_{0}}\frac{\|\bm{\varphi}^{i}_{j}\|^{2}}{tr(\bm{R}_{j})(\log
tr(\bm{R}_{j-1}))^{\frac{1}{3}}}$, $K_{0}$ is defined in Assumption II.2.

Proof. Set

$\displaystyle\bm{H}^{i}_{t}=\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\bm{A}^{i}_{k},\
\ \ \bm{H}_{t}=diag\\{\bm{H}^{1}_{t},...\bm{H}^{n}_{t}\\},$
$\displaystyle\bm{\Delta}_{t}=\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\bm{G}_{k},\
\ \ \bm{\Gamma}_{t}=\sum^{n}_{i=1}\bm{H}^{i}_{t}.$

By the definition of $\bm{A}_{k},\bm{G}_{k}$ in Table I, we have

$\displaystyle\bm{\Delta}_{t}$
$\displaystyle=\bm{H}_{t}+\nu\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\mathscr{L}(\bm{X}_{k}(Q)\otimes\bm{I}_{m})\mathscr{L},$
$\displaystyle\bm{\Gamma}_{t}$
$\displaystyle=\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\sum^{n}_{i=1}\bm{A}^{i}_{k}.$

The eigenvalues of $\mathscr{L}$ are denoted in a non-decreasing order as
$l_{1},\cdots,l_{m},l_{m+1},\cdots,l_{mn}$, and the corresponding unit
orthogonal eigenvectors are denoted as
$\xi_{1},\cdots,\xi_{m},\xi_{m+1},\cdots,\xi_{mn}$. By Assumption II.1, we see
that $l_{1}=\cdots=l_{m}=0$ and

$\displaystyle\bm{\xi}_{1}=\frac{1}{\sqrt{n}}\bm{1}_{n}\otimes\bm{e}_{1},...,\bm{\xi}_{m}=\frac{1}{\sqrt{n}}\bm{1}_{n}\otimes\bm{e}_{m},$
(19)

where $\bm{1}_{n}$ denotes the $n-$diamensional vector with all entries equal
to 1, $\bm{e}_{j}~{}(j=1,\cdots,m)$ is the $j$th column of the identity matrix
$\bm{I}_{m}$.

Hence for any unit vector $\bm{\eta}\in\mathbb{R}^{mn}$, we have the following
expression,

$\displaystyle\bm{\eta}=\sum^{m}_{j=1}{\kappa_{j}}\bm{\xi}_{j}+\sum^{mn}_{j=m+1}{\kappa_{j}}\bm{\xi}_{j}\triangleq\bm{\eta}_{1}+\bm{\eta}_{2},$

where $\sum^{m}_{j=1}{\kappa^{2}_{j}}+\sum^{mn}_{j=m+1}{\kappa^{2}_{j}}=1$.
Then we have

$\displaystyle\bm{\eta}^{T}\bm{\Delta}_{t}\bm{\eta}$ (20) $\displaystyle=$
$\displaystyle(\bm{\eta}_{1}+\bm{\eta}_{2})^{T}\Big{(}\bm{H}_{t}+\nu\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\mathscr{L}(\bm{X}_{k}(Q)\otimes\bm{I}_{m})\mathscr{L}\Big{)}$
$\displaystyle\cdot(\bm{\eta}_{1}+\bm{\eta}_{2})$ $\displaystyle=$
$\displaystyle\bm{\eta}^{T}_{1}\bm{H}_{t}\bm{\eta}_{1}+\bm{\eta}^{T}_{2}\bm{H}_{t}\bm{\eta}_{2}+2\bm{\eta}^{T}_{1}\bm{H}_{t}\bm{\eta}_{2}$
$\displaystyle+\nu\bm{\eta}^{T}_{1}\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\mathscr{L}(\bm{X}_{k}(Q)\otimes\bm{I}_{m})\mathscr{L}\bm{\eta}_{1}$
$\displaystyle+\nu\bm{\eta}^{T}_{2}\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\mathscr{L}(\bm{X}_{k}(Q)\otimes\bm{I}_{m})\mathscr{L}\bm{\eta}_{2}$
$\displaystyle+2\nu\bm{\eta}^{T}_{1}\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\mathscr{L}(\bm{X}_{k}(Q)\otimes\bm{I}_{m})\mathscr{L}\bm{\eta}_{2}$
$\displaystyle\triangleq$ $\displaystyle s_{1}+s_{2}+s_{3}+s_{4}+s_{5}+s_{6}.$

In the following, we will estimate $s_{i}(i=1,2,\cdots,6)$. By the definition
of $\bm{\eta}_{1}$, we see that $\bm{\eta}_{1}$ is the eigenvector
corresponding to zero eigenvalue of $\mathscr{L}$. Hence we have
$s_{4}=s_{6}=0$.

We note that $\bm{H}_{t}$ is a non-negative definite matrix. Thus we can
decompose it as $\bm{H}_{t}=\bm{H}^{\frac{1}{2}}_{t}\bm{H}^{\frac{1}{2}}_{t}$,
then

$\displaystyle s_{3}$ $\displaystyle=$ $\displaystyle
2\bm{\eta}^{T}_{1}\bm{H}_{t}\bm{\eta}_{2}\geq-\zeta\bm{\eta}^{T}_{1}\bm{H}_{t}\bm{\eta}_{1}-\frac{1}{\zeta}\bm{\eta}^{T}_{2}\bm{H}_{t}\bm{\eta}_{2}$
$\displaystyle=$ $\displaystyle-\zeta s_{1}-\frac{1}{\zeta}s_{2},$

where the inequality
$2\bm{M}^{T}_{1}\bm{M}_{2}\leq\zeta\bm{M}^{T}_{1}\bm{M}_{1}+\frac{1}{\zeta}\bm{M}^{T}_{2}\bm{M}_{2}$
is used, with $\zeta$ being a positive constant and $\bm{M}_{1}$ and
$\bm{M}_{2}$ being two matrices with appropriate dimensions.

Let $y=\sum^{m}_{j=1}{\kappa^{2}_{j}}$. Then by (8) and the definition of
$\bm{X}_{k}(Q)$, we have

$\displaystyle s_{5}$
$\displaystyle=\nu\bm{\eta}^{T}_{2}\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\mathscr{L}(\bm{X}_{k}(Q)\otimes\bm{I}_{m})\mathscr{L}\bm{\eta}_{2}$
$\displaystyle\geq\nu\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}\lambda_{\min}(\bm{X}_{k}(Q)\otimes\bm{I}_{m})\sum^{mn}_{j=m+1}l^{2}_{j}{\kappa^{2}_{j}}$
$\displaystyle\geq
a\nu\sum^{\hat{g}(t+u)-1}_{k=\hat{g}(t)}tr(\bm{A}_{k})l^{2}_{m+1}(1-y)$
$\displaystyle=a\nu tr(\bm{H}_{t})l^{2}_{m+1}(1-y),$ (21)

where $a\triangleq\min\limits_{i,j\in\\{1,\cdots,n\\}}a^{(Q)}_{ij}$ is a
positive constant for $Q\geq D(\mathscr{G})$ (cf., [40]).

In the following, we will estimate $s_{1}$.

$\displaystyle s_{1}$ $\displaystyle=$
$\displaystyle\bm{\eta}^{T}_{1}\bm{H}_{t}\bm{\eta}_{1}=(\sum^{m}_{j=1}{\kappa_{j}}\bm{\xi}_{j})^{T}\bm{H}_{t}(\sum^{m}_{j=1}{\kappa_{j}}\bm{\xi}_{j})$
(22) $\displaystyle=$
$\displaystyle{\bm{K}}^{T}\bm{\Xi}^{T}\bm{H}_{t}\bm{\Xi}{\bm{K}},$

where ${\bm{K}}=({\kappa_{1}},...,{\kappa_{j}})^{T}$ and
$\bm{\Xi}=(\bm{\xi}_{1},...,\bm{\xi}_{m})$. By the definition of
$\bm{\xi}_{i}$ in (19), we have

$\displaystyle\bm{\Xi}$
$\displaystyle=\frac{1}{\sqrt{n}}\left(\begin{matrix}\bm{e}_{1}&\bm{e}_{2}&\cdots&\bm{e}_{m}\\\
\bm{e}_{1}&\bm{e}_{2}&\cdots&\bm{e}_{m}\\\ \vdots&\vdots&\ddots&\vdots\\\
\bm{e}_{1}&\bm{e}_{2}&\cdots&\bm{e}_{m}\\\ \end{matrix}\right),$
$\displaystyle\bm{H}_{t}\bm{\Xi}$
$\displaystyle=\frac{1}{\sqrt{n}}\left(\begin{matrix}\bm{H}^{1}_{t}\bm{e}_{1}&\bm{H}^{1}_{t}\bm{e}_{2}&\cdots&\bm{H}^{1}_{t}\bm{e}_{m}\\\
\bm{H}^{2}_{t}\bm{e}_{1}&\bm{H}^{2}_{t}\bm{e}_{2}&\cdots&\bm{H}^{2}_{t}\bm{e}_{m}\\\
\vdots&\vdots&\ddots&\vdots\\\
\bm{H}^{n}_{t}\bm{e}_{1}&\bm{H}^{n}_{t}\bm{e}_{2}&\cdots&\bm{H}^{n}_{t}\bm{e}_{m}\\\
\end{matrix}\right),$
$\displaystyle=\frac{1}{\sqrt{n}}(\bm{H}^{1}_{t}~{}\bm{H}^{2}_{t}\ldots\bm{H}^{n}_{t})^{T},$
$\displaystyle\bm{\Xi}^{T}\bm{H}_{t}\bm{\Xi}$
$\displaystyle=\frac{1}{n}\left(\begin{matrix}\bm{e}^{T}_{1}\bm{H}^{1}_{t}+e^{T}_{1}\bm{H}^{2}_{t}+\cdots+e^{T}_{1}\bm{H}^{n}_{t}\\\
\bm{e}^{T}_{2}\bm{H}^{1}_{t}+e^{T}_{2}\bm{H}^{2}_{t}+\cdots+e^{T}_{2}\bm{H}^{n}_{t}\\\
\vdots\\\
\bm{e}^{T}_{m}\bm{H}^{1}_{t}+e^{T}_{m}\bm{H}^{2}_{t}+\cdots+e^{T}_{m}\bm{H}^{n}_{t}\\\
\end{matrix}\right).$

Hence we have
$\bm{\Xi}^{T}\bm{H}_{t}\bm{\Xi}=\frac{1}{n}\sum^{n}_{i=1}\bm{H}^{i}_{t}=\frac{1}{n}\bm{\Gamma}_{t}.$
By this and (22) we have

$\displaystyle
s_{1}=\frac{1}{n}{\bm{K}}^{T}\bm{\Gamma}_{t}{\bm{K}}\geq\frac{\lambda_{\min}(\bm{\Gamma}_{t})}{n}y.$
(23)

By the definition of $s_{2}$, we have

$\displaystyle s_{2}=\bm{\eta}^{T}_{2}\bm{H}_{t}\bm{\eta}_{2}\leq
tr(\bm{H}_{t})(1-y).$ (24)

Substitute (III)-(24) into (20), we have for $\zeta\in(0,1)$,

$\displaystyle\bm{\eta}^{T}\bm{\Delta}_{t}\bm{\eta}$ $\displaystyle\geq$
$\displaystyle(1-\zeta)s_{1}+(1-\frac{1}{\zeta})s_{2}+s_{5}$ (25)
$\displaystyle\geq$
$\displaystyle\frac{(1-\zeta)\lambda_{\min}(\bm{\Gamma}_{t})}{n}y+(1-\frac{1}{\zeta})tr(\bm{H}_{t})(1-y)$
$\displaystyle+a\nu tr(\bm{H}_{t})l^{2}_{m+1}(1-y).$

Thus, we have

$\displaystyle\lambda_{\min}(\bm{\Delta}_{t})$ $\displaystyle\geq$
$\displaystyle\Big{[}\frac{(1-\zeta)\lambda_{\min}(\bm{\Gamma}_{t})}{n}-\Big{(}a\nu
tr(\bm{H}_{t})l^{2}_{m+1}$
$\displaystyle+tr(\bm{H}_{t})-\frac{tr(\bm{H}_{t})}{\zeta}\Big{)}\Big{]}y$
$\displaystyle+a\nu
tr(\bm{H}_{t})l^{2}_{m+1}+tr(\bm{H}_{t})-\frac{tr(\bm{H}_{t})}{\zeta}.$

Taking $\zeta=\frac{1}{1+0.5l^{2}_{m+1}a\nu}\in(0,1)$, then we can obtain the
following inequality,

$\displaystyle\lambda_{\min}(\bm{\Delta}_{t})$ $\displaystyle\geq$
$\displaystyle\Big{[}\sigma\lambda_{\min}(\bm{\Gamma}_{t})-0.5a\nu
l^{2}_{m+1}tr(\bm{H}_{t})\Big{]}y$ $\displaystyle+0.5a\nu
l^{2}_{m+1}tr(\bm{H}_{t}),$

where $\sigma\triangleq\frac{l^{2}_{m+1}a\nu}{2n+l^{2}_{m+1}a\nu n}\in(0,1).$
Hence by $0\leq\lambda_{\min}(\bm{\Gamma}_{t})\leq tr(\bm{H}_{t})$ and
$y\in(0,1)$, it is easy to obtain

$\displaystyle\lambda_{\min}(\bm{\Delta}_{t})-\sigma\lambda_{\min}(\bm{\Gamma}_{t})$
$\displaystyle\geq$
$\displaystyle[\sigma\lambda_{\min}(\bm{\Gamma}_{t})-0.5a\nu
l^{2}_{m+1}tr(\bm{H}_{t})](y-1)>0.$

This completes the proof of the lemma. $\blacksquare$

###### Remark III.1

From Theorem III.1, we can see that the properties of the product of the
matrices $(I_{mn}-\mu\bm{G}_{j}),~{}j\geq 0$ is crucial for the strong
consistency of the distributed SG algorithm. Lemma III.4 establishes a
connection between the eigenvalues of $\\{\bm{A}^{i}_{k}\\}$ and
$\\{\bm{G}_{k}\\}$, and thus builds a bridge between the standard SG algorithm
and the distributed SG algorithm.

By using Lemma III.4, we have the following theorem.

###### Theorem III.2

Let $\mu(1+4\nu)<1$. Suppose that there exists $i_{1}\in\\{1,\cdots,n\\}$ such
that
$\limsup_{k\rightarrow\infty}\frac{r^{i_{1}}_{k}}{r^{i_{1}}_{k-1}}\triangleq
r^{*}<\infty$, and the condition number of $\bm{R}_{k}$ is bounded. Under
Assumptions II.1 and II.2, we have $\bm{\Psi}(k,0)\rightarrow 0$ as
$k\rightarrow\infty$ a.s..

The proof of Theorem III.2 is complicated, and we put it in Appendix B.

###### Remark III.2

For some typical cases such as the bounded sequence
$\\{\bm{\varphi}^{i}_{k}\\}~{}(i.e.,~{}c^{\prime}\leq\|\bm{\varphi}^{i}_{k}\|\leq
c^{\prime\prime})$ and the i.i.d. sequence $\\{\bm{\varphi}^{i}_{k}\\}$, the
condition
$\limsup_{k\rightarrow\infty}\frac{r^{i_{1}}_{k}}{r^{i_{1}}_{k-1}}\triangleq
r^{*}<\infty$ of Theorem III.2 can be easily verified.

By Theorems III.1 and III.2, we have the following corollary.

###### Corollary III.1

Under the conditions of Theorem III.2, if Assumption II.3 is further
satisfied, the convergence of the distributed SG algorithm designed in
Algorithm 1 can be obtained.

Proof. By Theorem III.2, we have $\bm{\Psi}(k,0)\rightarrow 0,$ as
$k\rightarrow\infty$. Hence from Theorem III.1, the estimate
$\hat{\bm{\Theta}}_{k}$ defined in Table I converges to the true parameter
${\bm{\Theta}}$ a.s., which completes the proof of the corollary.
$\blacksquare$

###### Remark III.3

The cooperative excitation condition proposed in Assumption II.2 is reasonable
in a sense that if there exists only one sensor which satisfies the excitation
condition (6), then all sensors in the network can satisfy Assumption II.2. We
show this point under a mild condition (the condition number of $\bm{R}_{k}$
is bounded) as follows,

$\displaystyle\frac{{\lambda^{(k)}_{\max}}}{{\lambda^{(k)}_{\min}}}\leq
mn\gamma\cdot\frac{\lambda_{\max}\big{\\{}\sum^{k}_{j=1}\bm{\varphi}^{i_{2}}_{j}{\bm{\varphi}^{i_{2}}_{j}}^{T}\big{\\}}}{\lambda_{\min}\big{\\{}\sum^{k}_{j=1}\bm{\varphi}^{i_{2}}_{j}{\bm{\varphi}^{i_{2}}_{j}}^{T}\big{\\}}}$
$\displaystyle\leq$ $\displaystyle mn\gamma\tilde{N}\Big{(}\log
r^{i_{2}}_{k}\Big{)}^{\frac{1}{3}}\leq
mn\gamma\tilde{N}\left(\log(\|\bm{R}_{k}\|)\right)^{\frac{1}{3}}.$

where ${i_{2}}$ denotes the index of the sensor satisfying (6), $\gamma$ is
defined in Lemma III.3, ${\lambda^{(k)}_{\max}}$ and ${\lambda^{(k)}_{\min}}$
are defined in Assumption II.2.

Combining Remark III.3 with Corollary III.1, we see that if only one sensor
can fulfill the estimate task, then all sensors in our proposed algorithm can
fulfill it.

###### Remark III.4

Different form most results in the literature, our results are obtained
without using the independency and stationarity assumptions on the regression
signals, which makes it possible to apply the distributed algorithm to
practical feedback systems.

##  IV Convergence Rate of the Distributed SG Algorithm

In this section, we will consider the convergence rate of the distributed SG
algorithm. In order to prove the theorems of this section, we first introduce
the following two lemmas.

###### Lemma IV.1

If $\mu(1+4\nu)<1$, then there exists a constant $\tau_{1}\geq 1$ such that
for any $k\geq 0$, we have

$\displaystyle
det(\bm{I}_{mn}-\mu\bm{G}_{k})\geq[det(\bm{I}_{mn}-\bm{A}_{k})]^{\tau_{1}}~{}.$

Proof. By the definition of $\bm{G}_{k}$ and (9), we have

$\displaystyle det(\bm{I}_{mn}-\mu\bm{G}_{k})$
$\displaystyle\geq(\lambda_{\min}(\bm{I}_{mn}-\mu\bm{G}_{k}))^{mn}$
$\displaystyle=(1-\lambda_{\max}(\mu\bm{G}_{k}))^{mn}$
$\displaystyle\geq[1-\lambda_{\max}({\mu(1+4\nu)}\bm{A}_{k})]^{mn}$
$\displaystyle=[\lambda_{\min}(\bm{I}-{\mu(1+4\nu)}\bm{A}_{k})]^{mn}$
$\displaystyle\geq\left[det(\bm{I}-{\mu(1+4\nu)}\bm{A}_{k})\right]^{mn}$
$\displaystyle\geq\left[det(\bm{I}-\bm{A}_{k})\right]^{mn}.$

The lemma can be proved by taking $\tau_{1}=mn$. $\blacksquare$

###### Lemma IV.2

If ${\mu(1+4\nu)<1}$ , then we have the following inequalities111 Let
$\\{\bm{A}_{k}\\}$ be a matrix sequence and $\\{b_{k}\\}$ be a positive scalar
sequence. Then by $\bm{A}_{k}=O(b_{k})$ we mean that there exists a constant
$M>0$ such that $\|\bm{A}_{k}\|\leq Mb_{k},~{}\forall k\geq 0$.,

$\displaystyle(i)$ $\displaystyle\|\bm{\Psi}(k,j)\|\leq 1,~{}~{}~{}~{}0\leq
j\leq k,~{}k\geq 0;~{}$ $\displaystyle(ii)$
$\displaystyle\frac{1}{\|\bm{R}_{k}\|^{\tau_{1}}}=O(\|{\bm{\Psi}}(k+1,0)\|^{m})~{},~{}~{}~{}k\geq
1~{};$ $\displaystyle(iii)$
$\displaystyle\|{\bm{\Psi}}(k,j+1)\|=O(\|{\bm{\Psi}}(k,0)\|\|\bm{R}_{j}\|^{n\tau_{1}});$
$\displaystyle(iv)$
$\displaystyle\sum^{\infty}_{j={M}+1}\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+{\varsigma}}}\leq\frac{n^{1+{\varsigma}}}{{\varsigma}}\frac{1}{\|\bm{R}_{M}\|^{\varsigma}}~{},{\varsigma}>0.$

where $\tau_{1}$ is defined in Lemma IV.1.

The proof of the above lemma is given in Appendix C.

In the following, we establish the specific relationship between the error
$\widetilde{\bm{\Theta}}_{k}$ and ${\bm{\Psi}}(k,0)$.

###### Lemma IV.3

Under the conditions of Theorem III.1, if
$\lim_{k\rightarrow\infty}\bm{\Psi}(k,0)=0$, then we have the following
inequality

$\displaystyle\|\hat{\bm{\Theta}}_{k}-{\bm{\Theta}}\|=O\left(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{n\tau_{1}(1+\delta)}}\right)~{}~{}{\rm
a.s.},$

where $\delta$ is defined in Lemma III.3 and $\tau_{1}$ can be taken as $mn$.

Proof. Let

$\displaystyle\beta(k)=\max\\{j:\|\bm{R}_{j}\|^{n\tau_{1}}\leq
k\\},~{}~{}~{}~{}k\geq 0,$
$\displaystyle\bar{\Delta}(k)=\beta(\|{\bm{\Psi}}(k,0)\|^{-\frac{1}{1+\delta}}),~{}~{}~{}k\geq
0.$

By the definition of $\beta(k)$ and $\bar{\Delta}(k)$, we have

$\displaystyle\begin{aligned}
&\|\bm{R}_{\bar{\Delta}(k)}\|^{n\tau_{1}}\leq\|{\bm{\Psi}}(k,0)\|^{-\frac{1}{1+\delta}},\\\
&\|\bm{R}_{\bar{\Delta}(k)+1}\|^{n\tau_{1}}>\|{\bm{\Psi}}(k,0)\|^{-\frac{1}{1+\delta}}.\end{aligned}$
(26)

According to Lemma $\ref{l9}\ (iii)$, we have

$\displaystyle\|{\bm{\Psi}}(k,\bar{\Delta}(k)+1)\|$
$\displaystyle=O(\|{\bm{\Psi}}(k,0)\|\cdot\|\bm{R}_{\bar{\Delta}(k)}\|^{n\tau_{1}})$
$\displaystyle=O(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{1+\delta}}).$ (27)

We prove the following inequality by contradiction,

$\displaystyle\bar{\Delta}(k)<k-1,\ \ \hbox{for large}\ \ k.$ (28)

Suppose that there exists a large constant $k_{0}$ such that
$\bar{\Delta}(k_{0})\geq k_{0}-1$. Then by (26), we have

$\displaystyle\|\bm{R}_{k_{0}-1}\|^{n\tau_{1}}\leq\|{\bm{\Psi}}(k_{0},0)\|^{-\frac{1}{1+\delta}}.$

By Lemma IV.2 $(ii)$, we see that there exists a positive constant $\hat{c}$
such that

$\displaystyle\|\bm{R}_{k_{0}-1}\|^{n\tau_{1}}\geq\hat{c}\|{\bm{\Psi}}(k_{0},0)\|^{-mn}.$

Thus we have

$\displaystyle\hat{c}\leq\|{\bm{\Psi}}(k_{0},0)\|^{mn-\frac{1}{1+\delta}},$

which is contradictory with ${\bm{\Psi}}(k,0)\rightarrow 0$ as
$k\rightarrow\infty$.

Note that by (9), we have
$\|\mu\bm{G}_{j}\|\leq\mu\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|}$. Hence
from Lemma III.3 and (15) in the proof of Theorem III.1, we have the following
estimation for the noise term of the system,

$\displaystyle\left\|\sum^{k-1}_{j=0}{\bm{\Psi}}(k,j+1)\bm{\Phi}_{j}\bm{R}^{-1}_{j}\bm{\Xi}^{T}_{j+1}\right\|$
$\displaystyle\leq$
$\displaystyle\|\widetilde{\bm{S}}_{k-1}\|+\|{\bm{\Psi}}(k,0)\bm{S}\|$
$\displaystyle+\sum^{k-1}_{j=0}\|{\bm{\Psi}}(k,j+1)\|\cdot\|\mu\bm{G}_{j}\|\cdot\|\widetilde{\bm{S}}_{j-1}\|$
$\displaystyle=$ $\displaystyle
O(\|\bm{R}_{k}\|^{-\delta})+O(\|{\bm{\Psi}}(k,0)\|)$
$\displaystyle+c\mu(1+4\nu)\gamma\sum^{k-1}_{j=0}\|{\bm{\Psi}}(k,j+1)\|\cdot\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+\delta}}.$

Combining this with (28), we have

$\displaystyle\left\|\sum^{k-1}_{j=0}{\bm{\Psi}}(k,j+1)\bm{\Phi}_{j}\bm{R}^{-1}_{j}\bm{\Xi}^{T}_{j+1}\right\|$
(29) $\displaystyle=$ $\displaystyle
O(\|\bm{R}_{\bar{\Delta}(k)+1}\|^{-\delta})+O(\|{\bm{\Psi}}(k,0)\|)$
$\displaystyle+O\Big{(}\sum^{k-1}_{j=0}\|{\bm{\Psi}}(k,j+1)\|\cdot\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+\delta}}\Big{)}.$

In the following, we estimate the last term on the right hand of (29). By
Lemma IV.2 $(i)$ and (27), we have

$\displaystyle\sum^{k-1}_{j=0}\|{\bm{\Psi}}(k,j+1)\|\cdot\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+\delta}}$
(30) $\displaystyle\leq$
$\displaystyle\sum^{\bar{\Delta}(k)}_{j=0}\|{\bm{\Psi}}(k,\bar{\Delta}(k)+1)\|\cdot\|{\bm{\Psi}}(\bar{\Delta}(k)+1,j+1)\|$
$\displaystyle\cdot\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+\delta}}+\sum^{k-1}_{j=\bar{\Delta}(k)+1}\|{\bm{\Psi}}(k,j+1)\|\cdot\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+\delta}}$
$\displaystyle=$ $\displaystyle
O(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{1+\delta}})\sum^{\infty}_{j=0}\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+\delta}}$
$\displaystyle+\sum^{k-1}_{j=\bar{\Delta}(k)+1}\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+\delta}}.$

Furthermore, by Lemma II.2 and Lemma IV.2 $(iv)$, we have

$\displaystyle\sum^{k-1}_{j=0}\|{\bm{\Psi}}(k,j+1)\|\cdot\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+\delta}}$
$\displaystyle=$ $\displaystyle
O(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{1+\delta}})+\frac{\|\bm{\Phi}_{\bar{\Delta}(k)+1}\|^{2}}{\|\bm{R}_{\bar{\Delta}(k)+1}\|^{1+\delta}}$
$\displaystyle+\sum^{\infty}_{j=\bar{\Delta}(k)+2}\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+\delta}}$
$\displaystyle=$ $\displaystyle
O(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{1+\delta}})+O(\|\bm{R}_{\bar{\Delta}(k)+1}\|^{-\delta}).$

Combining the above inequality with (26) and (29), then we have

$\displaystyle\Bigg{\|}\sum^{k-1}_{j=0}{\bm{\Psi}}(k,j+1)\bm{\Phi}_{j}\bm{R}^{-1}_{j}\bm{\Xi}^{T}_{j+1}\Bigg{\|}$
(31) $\displaystyle=$ $\displaystyle
O(\|\bm{R}_{\bar{\Delta}(k)+1}\|^{-\delta})+O(\|{\bm{\Psi}}(k,0)\|)$
$\displaystyle+O(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{1+\delta}})$
$\displaystyle=$ $\displaystyle
O(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{n\tau_{1}(1+\delta)}})+O(\|{\bm{\Psi}}(k,0)\|)$
$\displaystyle+O(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{1+\delta}})$
$\displaystyle=$ $\displaystyle
O(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{n\tau_{1}(1+\delta)}}),$

where $\|{\bm{\Psi}}(k,0)\|\leq 1$ is used in the above inequality. Hence by
(13) and (31), we have

$\displaystyle\|\widetilde{\bm{\Theta}}_{k}\|=O(\|{\bm{\Psi}}(k,0)\|^{\frac{\delta}{n\tau_{1}(1+\delta)}})~{}~{}{\rm
a.s.}$ (32)

This completes the proof of the theorem. $\blacksquare$

We further establish the convergence rate under the cooperative excitation
condition (Assumption II.2). We see that the rate can be expressed via simply
characterizable quantities.

###### Theorem IV.1

Under the conditions of Theorem III.2 and Assumption II.3, we have

$\displaystyle\|\widetilde{\bm{\Theta}}_{k}\|=O\Big{(}(\log\|\bm{R}_{k}\|)^{-\delta_{1}}\Big{)}{~{}~{}\rm
a.s.},$

with $\delta_{1}$ being a positive constant.

The proof of the above theorem is given in Appendix D.

###### Remark IV.1

By Theorem IV.1, the convergence rate of the proposed distributed SG algorithm
is mainly determined by the parameter $\delta_{1}$ and $\|\bm{R}_{k}\|$. After
a careful calculation by following the proof of Theorem IV.1 in Appendix D, we
can obtain that the parameter $\delta_{1}$ is positively related to $\mu,\nu$
and $l_{m+1}$, where $l_{m+1}$ is the $(m+1)$-th eigenvalue of the Laplacian
matrix $\mathscr{L}=\bm{L}\otimes\bm{I}_{m}$ in a non-decreasing order and can
be regarded as a measure of the connectivity of the graph $\mathscr{G}$.

##  V  A Simulation Example

In this section, we provide an example of non-independent regression signals
to illustrate the cooperative effect in this paper.

###### Example V.1

The network is composed of $n=28$ sensors whose dynamics obey the equation (1)
with $m=10$. The system noises $\varepsilon^{i}_{k},(i=1,\cdots,n,~{}k\geq 1)$
in (1) are independent and identically distributed with
$\varepsilon^{i}_{k}\sim\mathcal{N}(0,1.2^{2})$ (Gaussian distribution with
zero mean and variance $1.2^{2}$). Set the unknown parameter as
$\bm{\theta}=[1,2,\cdots,10]^{T}$. Let
$\bm{\varphi}^{i}_{k}\in\mathbb{R}^{10}~{}(i=1,\cdots,n,~{}k\geq 1)$ be
generated by a state space model,

$\displaystyle\bm{u}^{i}_{k}$ $\displaystyle=$
$\displaystyle\bm{A}_{i}\bm{u}^{i}_{k-1}+\bm{B}_{i}\xi^{i}_{k},$
$\displaystyle\bm{\varphi}^{i}_{k}$ $\displaystyle=$
$\displaystyle\bm{C}_{i}\bm{u}^{i}_{k}.$ (33)

For convenience, we take

$\displaystyle\bm{A}_{i}$ $\displaystyle=$ $\displaystyle
diag\\{1.2,\cdots,1.2\\}\in\mathbb{R}^{10\times 10},$
$\displaystyle\bm{B}_{i}$ $\displaystyle=$
$\displaystyle\bm{e_{j}}\in\mathbb{R}^{10},$ $\displaystyle\bm{C}_{i}$
$\displaystyle=$ $\displaystyle
col\\{0,\cdots,0,\underset{j^{th}}{\bm{e_{j}}},0,\cdots,0\\}\in\mathbb{R}^{10\times
10},$

where $j=\mod(i,m)$ and $\bm{e}_{j}~{}(j=1,\cdots,m)$ is the $j$th column of
the identity matrix $\bm{I}_{m}$ $(m=10)$.

and let $\\{\xi^{i}_{k},k\geq 1,i=1,\cdots,n\\}$ be independent and
identically distributed with
$\xi^{i}_{k}\in\mathbb{R}\sim\mathcal{N}(0,0.3^{2})$.

The network structure is shown in Fig. 1. Here we use the Metropolis rule in
[41] to construct the weights.

Figure 1: Network topology

We can verify that for each sensor $i(i=1,\cdots,28)$, the regression signals
$\bm{\varphi}_{k}^{i}($ generated by (33) can not satisfy the excitation
condition (6), but they can cooperate to satisfy Assumption II.2. We repeat
the simulation for $s=500$ times with the same initial states. For fixed step
sizes $\mu=0.25$ and $\nu=0.7$, the simulation results are shown in Fig. 2. We
see that if all the sensors use the non-cooperative SG algorithm to estimate
$\bm{\theta}$, the mean square error (MSE) of each sensor cannot converge to
zero, while the MSE of each sensor in the distributed SG algorithm converges
to zero. It is clear that the estimation task can be fulfilled through
exchanging information between sensors even though any individual sensor can
not.

Figure 2:  The maximal and minimal MSEs of sensors by using non-cooperative SG
algorithm and cooperative SG algorithm.

##  VI Concluding Remarks

This paper proposed a distributed SG algorithm based on the consensus strategy
and the diffusion of the regression vectors to cooperatively estimate an
unknown time-invariant parameter. We introduced a cooperative excitation
condition, under which the almost sure convergence of the proposed algorithm
can be guaranteed, and the convergence rate of the algorithm can be
established. Compared with the existing results concerning the distributed
estimation in the literature, our results are obtained without relying on the
independency and stationarity assumptions, which makes it possible to apply
our results to the feedback control systems. Furthermore, we found that the
sensors can cooperate to finish the estimation task even though any individual
can not. Many interesting problems deserve to be further investigated, for
example, the convergence of the distributed SG algorithm with correlated
noise, the analysis of other distributed algorithms such as the distributed
Kalman filter, and the combination of the distributed adaptive estimation with
the distributed control.

##  Appendix A Proof of Lemma III.3

Proof. By $\varepsilon<1$ we see that $2-\varepsilon-2\delta>1$. Since
$\bm{R}_{k}=\bm{R}_{k-1}+\bm{\Phi}^{T}_{k}\bm{\Phi}_{k}$, we obtain

$\displaystyle
tr(\bm{R}_{k})=tr(\bm{R}_{k-1})+tr(\bm{\Phi}^{T}_{k}\bm{\Phi}_{k}).$ (34)

Then by Lemma II.2, we have the following inequality,

$\displaystyle\sum^{\infty}_{k=1}\frac{\|\bm{\Phi}_{k}\|^{2}}{\|\bm{R}_{k}\|^{2-2\delta-\varepsilon}}\leq{n^{2-2\delta-\varepsilon}}\sum^{\infty}_{k=1}\frac{tr(\bm{\Phi}^{T}_{k}\bm{\Phi}_{k})}{(tr(\bm{R}_{k}))^{2-2\delta-\varepsilon}}<\infty.$

By the boundedness of the condition number of $\bm{R}_{k}$, we have
$\|\bm{R}_{k}\|\|\bm{R}^{-1}_{k}\|\leq\gamma$ for all $k$. Using this and the
above inequality, we have,

$\displaystyle\sum^{\infty}_{k=1}E(\|\bm{\Phi}_{k}\bm{R}^{\delta-1}_{k}\bm{\Xi}^{T}_{k+1}\|^{2}|\mathscr{F}_{k})$
$\displaystyle\leq$ $\displaystyle
c_{0}\sum^{\infty}_{k=1}\|\bm{\Phi}_{k}\|^{2}\|\bm{R}^{\delta-1}_{k}\|^{2}\|\bm{R}_{k}\|^{\varepsilon}$
$\displaystyle=$ $\displaystyle
c_{0}\sum^{\infty}_{k=1}\|\bm{\Phi}_{k}\|^{2}\|\bm{R}^{-1}_{k}\|^{2-2\delta}\|\bm{R}_{k}\|^{\varepsilon}$
$\displaystyle\leq$ $\displaystyle
c_{0}\gamma^{2-2\delta}\sum^{\infty}_{k=1}\frac{\|\bm{\Phi}_{k}\|^{2}}{\|\bm{R}_{k}\|^{2-2\delta-\varepsilon}}<\infty,~{}~{}{\rm
a.s.}$

where Assumption II.3 is used in the first inequality. By the martingale
convergence theorem it follows that as $k\rightarrow\infty$,
$\sum^{k}_{j=1}\bm{\Phi}_{j}\bm{R}^{\delta-1}_{j}\bm{\Xi}^{T}_{j+1}$ converges
a.s.. Hence for any $\eta>0$, if $k$ is large enough, then we have
$\|\widetilde{\bm{S}}_{k-1,\delta}\|<\eta$, where
$\widetilde{\bm{S}}_{k-1,\delta}\triangleq\sum^{\infty}_{j=k}\bm{\Phi}_{j}\bm{R}^{\delta-1}_{j}\bm{\Xi}^{T}_{j+1}$.

By the definition of $\bm{\Phi}_{j}$ and $\bm{R}_{j}$ in Table I, we see that
$\bm{\Phi}_{j}\bm{R}^{-\delta}_{j}=(\bm{R}^{-\delta}_{j}\otimes\bm{I}_{m})\bm{\Phi}_{j}$
. Then summation by parts yields the following result,

$\displaystyle\Big{\|}(\bm{R}^{\delta}_{k}\otimes\bm{I}_{m})\widetilde{\bm{S}}_{k-1}\Big{\|}$
(35) $\displaystyle=$
$\displaystyle\Big{\|}(\bm{R}^{\delta}_{k}\otimes\bm{I}_{m})\sum^{\infty}_{j=k}(\bm{R}^{-\delta}_{j}\otimes\bm{I}_{m})\bm{\Phi}_{j}\bm{R}^{\delta-1}_{j}\bm{\Xi}^{T}_{j+1}\Big{\|}$
$\displaystyle=$
$\displaystyle\Big{\|}(\bm{R}^{\delta}_{k}\otimes\bm{I}_{m})\sum^{\infty}_{j=k}(\bm{R}^{-\delta}_{j}\otimes\bm{I}_{m})({\widetilde{\bm{S}}_{j-1,\delta}}-{\widetilde{\bm{S}}_{j,\delta}})\Big{\|}$
$\displaystyle=$
$\displaystyle\Big{\|}{\widetilde{\bm{S}}_{k-1,\delta}}-(\bm{R}^{\delta}_{k}\otimes\bm{I}_{m})\sum^{\infty}_{j=k}((\bm{R}^{-\delta}_{j}-\bm{R}^{-\delta}_{j+1})\otimes\bm{I}_{m}){\widetilde{\bm{S}}_{j,\delta}}\Big{\|}$
$\displaystyle\leq$
$\displaystyle\eta+\eta\Big{\|}(\bm{R}^{\delta}_{k}\otimes\bm{I}_{m})\sum^{\infty}_{j=k}\left((\bm{R}^{-\delta}_{j}-\bm{R}^{-\delta}_{j+1})\otimes\bm{I}_{m}\right)\Big{\|}$
$\displaystyle\leq$ $\displaystyle 2\eta.$

Furthermore, by (35) we have

$\displaystyle\|\widetilde{\bm{S}}_{k-1}\|$
$\displaystyle=\|(\bm{R}^{-\delta}_{k}\otimes\bm{I}_{m})(\bm{R}^{\delta}_{k}\otimes\bm{I}_{m})\widetilde{\bm{S}}_{k-1}\|$
$\displaystyle\leq\|(\bm{R}^{-\delta}_{k}\otimes\bm{I}_{m})\|\|(\bm{R}^{\delta}_{k}\otimes\bm{I}_{m})\widetilde{\bm{S}}_{k-1}\|$
$\displaystyle\leq
2\eta\|(\bm{R}^{-1}_{k}\otimes\bm{I}_{m})\|^{\delta}=2\eta\|\bm{R}^{-1}_{k}\|^{\delta}\leq\frac{2\eta\gamma^{\delta}}{\|\bm{R}_{k}\|^{\delta}},$

where $\|\bm{R}^{-1}_{k}\otimes\bm{I}_{m}\|=\|\bm{R}^{-1}_{k}\|$ is used. This
completes the proof of the lemma. $\blacksquare$

##  Appendix B Proof of Theorem III.2

Before proving Theorem III.2, we first introduce two lemmas, whose proof can
be found in [30].

###### Lemma B.1

[30] Suppose that $0\leq\bm{C}_{k}\leq\bm{I}_{p},~{}k\geq 0$ with
$\bm{C}_{k}\in\mathbb{R}^{p\times p}$. Set
$\bm{\Pi}(k+1,j)=(\bm{I}-\bm{C}_{k})\bm{\Pi}(k,j),~{}\bm{\Pi}(j,j)=\bm{I},~{}\forall~{}k\geq
j,$ then we have

$\displaystyle\|\bm{\Pi}(M,k)\|\leq\left(1-\frac{\lambda_{\min}(\bm{F}_{kM})}{2(1+r^{2}_{kM})}\right)^{\frac{1}{2}},\
\ \ M>k,$ (36)

where $\bm{F}_{kM}\triangleq\sum^{M-1}_{j=k}\bm{C}_{j}$,
$r_{kM}\triangleq\sum^{M-1}_{j=k}\|\bm{C}_{j}\|$, and
$\lambda_{\min}(\bm{F}_{kM})$ represents the minimum eigenvalue of
$\bm{F}_{kM}$.

###### Lemma B.2

[30] Suppose that $0\leq\bm{C}_{k}\leq\bm{I}_{p},~{}k\geq 0$ with
$\bm{C}_{k}\in\mathbb{R}^{p\times p}$. Then $\bm{\Pi}(k,0)$ defined in Lemma
B.1 converges to 0 as $k\rightarrow\infty$, if there exists a sequence of
monotonically increasing positive integers $\\{t_{k}\\}$ with
$t_{k}\rightarrow\infty$ as $k\rightarrow\infty$, such that

$\displaystyle\sum^{\infty}_{k=1}\frac{\lambda_{\min}(\bm{F}_{k})}{1+\lambda^{2}_{\max}(\bm{F}_{k})}=\infty,$
(37)

where $\bm{F}_{k}\triangleq\sum^{t_{k}-1}_{j=t_{k-1}}\bm{C}_{j}$.

Proof of Theorem III.2.

Proof. By Lemma B.2, we just need to show that there exists an integer
sequence $\\{t_{k}\\}$ with $t_{k}\rightarrow\infty$ as $k\rightarrow\infty$,
such that

$\displaystyle\sum^{\infty}_{k=1}\frac{\lambda_{\min}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)}{1+\lambda^{2}_{\max}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)}=\infty.$
(38)

In the following, we will show this by three steps.

Step 1: Construction of the integer sequence $\\{t_{k}\\}$.

Without loss of generality, the constant $K_{0}$ in Assumption II.2 can be
taken to satisfy the inequality $\log tr(\bm{R}_{K_{0}})\geq 1$ since
$\|\bm{R}_{k}\|\rightarrow\infty$ as $k\rightarrow\infty$. Thus, by the
definition of $d_{k}$ and $\hat{g}(t)$ in Lemma III.4, we have for $t\geq 0$,

$\displaystyle t\leq d_{\hat{g}(t)+1}\leq d_{\hat{g}(t)}+1\leq t+1.$ (39)

We first prove the following result,

$\displaystyle\hat{g}(t)\rightarrow\infty,~{}t\rightarrow\infty.$ (40)

By the boundedness of the condition number of $\bm{R}_{k}$, we have for large
$k$

$\displaystyle\frac{\|\bm{R}_{k}\|}{\|\bm{R}_{k-1}\|}$
$\displaystyle=\frac{\max_{i}r^{i}_{k}}{\max_{i}r^{i}_{k-1}}=\frac{\max_{i}r^{i}_{k}}{\min_{i}r^{i}_{k}}\cdot\frac{\min_{i}r^{i}_{k}}{\max_{i}r^{i}_{k-1}}$
$\displaystyle\leq\frac{\max_{i}r^{i}_{k}}{\min_{i}r^{i}_{k}}\cdot\frac{r^{i_{1}}_{k}}{r^{i_{1}}_{k-1}}\leq\gamma
r^{*}.$

Hence we have

$\displaystyle\frac{tr\bm{R}_{k}}{tr\bm{R}_{k-1}}\leq\frac{n\|\bm{R}_{k}\|}{\|\bm{R}_{k-1}\|}\leq
n\gamma r^{*}.$ (41)

Then by (34) and (41), we have for any $k\geq K_{0}$,

$\displaystyle d_{k}$
$\displaystyle=\sum^{k-1}_{j=K_{0}}\frac{tr\bm{R}_{j}-tr\bm{R}_{j-1}}{tr(\bm{R}_{j})(\log
tr(\bm{R}_{j-1}))^{\frac{1}{3}}}$ $\displaystyle\geq\frac{1}{n\gamma
r^{*}}\sum^{k-1}_{j=K_{0}}\frac{tr\bm{R}_{j}-tr\bm{R}_{j-1}}{tr(\bm{R}_{j-1})(\log
tr(\bm{R}_{j-1}))^{\frac{1}{3}}}$ $\displaystyle\geq\frac{1}{n\gamma
r^{*}}\sum^{k-1}_{j=K_{0}}\int^{tr\bm{R}_{j}}_{tr\bm{R}_{j-1}}\frac{dx}{x(\log
x)^{\frac{1}{3}}}$ $\displaystyle=\frac{3}{2n\gamma
r^{*}}\left(\log^{\frac{2}{3}}tr\bm{R}_{k-1}-\log^{\frac{2}{3}}tr\bm{R}_{K_{0}-1}\right).$
(42)

Since $\|\bm{R}_{k}\|\xrightarrow[]{k\rightarrow\infty}\infty$, then we have
$d_{k}\rightarrow\infty$ as $k\rightarrow\infty$. By the definition of
$\hat{g}(t)$, we further have $\hat{g}(t)<\infty$ for all $t>0$ and
$\hat{g}(t)\rightarrow\infty$ as $t\rightarrow\infty.$

By (40) and $\|\bm{R}_{j}\|\xrightarrow[]{j\rightarrow\infty}\infty$, we see
that there exists sufficiently large $N_{1}$ such that for
$j\geq\hat{g}(N_{1})$,

$\displaystyle\frac{n(\log
tr\bm{R}_{j})^{\frac{1}{3}}}{tr\bm{R}_{j}}\leq\frac{1}{2N},$ (43)

where $N$ is defined in Assumption II.2. The integer sequence $\\{t_{k}\\}$
can be taken as

$\displaystyle t_{k}=\hat{g}(N_{1}+k\alpha),\ \ \alpha=2Nm(n+2)+1.$ (44)

Then by (40), we have $t_{k}\rightarrow\infty$ as $k\rightarrow\infty$.

Step 2: Estimation of
$\lambda_{\min}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)$.

By the properties of $r_{j}^{i}$, we have

$\displaystyle\sum^{n}_{i=1}\sum^{t_{k}-1}_{j=t_{k-1}}\frac{\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}}{r^{i}_{j}}\geq\sum^{n}_{i=1}\sum^{t_{k}}_{j=t_{k-1}}\frac{\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}}{r^{i}_{j}}-n\bm{I}_{m}$
(45) $\displaystyle=$
$\displaystyle\sum^{t_{k}}_{j=t_{k-1}}\sum^{n}_{i=1}\frac{1}{r^{i}_{j}}\left(\sum^{j}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}-\sum^{j-1}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}\right)-n\bm{I}_{m}$
$\displaystyle\geq$
$\displaystyle\sum^{t_{k}}_{j=t_{k-1}}\frac{1}{tr\bm{R}_{j}}\sum^{n}_{i=1}\left(\sum^{j}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}-\sum^{j-1}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}\right)-n\bm{I}_{m}$
$\displaystyle=$
$\displaystyle\sum^{t_{k}+1}_{j=t_{k-1}+1}\frac{1}{tr\bm{R}_{j-1}}\sum^{n}_{i=1}\sum^{j-1}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}-n\bm{I}_{m}$
$\displaystyle-\sum^{t_{k}}_{j=t_{k-1}}\frac{1}{tr\bm{R}_{j}}\sum^{n}_{i=1}\sum^{j-1}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}.$

By the definition of $\lambda^{(k)}_{\min}$ in Assumption II.2 and (45), we
obtain

$\displaystyle\sum^{n}_{i=1}\sum^{t_{k}-1}_{j=t_{k-1}}\frac{\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}}{r^{i}_{j}}$
(46) $\displaystyle\geq$
$\displaystyle\sum^{t_{k}}_{j=t_{k-1}+1}\left(\frac{1}{tr\bm{R}_{j-1}}-\frac{1}{tr\bm{R}_{j}}\right)\sum^{n}_{i=1}\sum^{j-1}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}-n\bm{I}_{m}$
$\displaystyle+\frac{1}{tr\bm{R}_{t_{k}}}\sum^{n}_{i=1}\sum^{t_{k}}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}-\frac{1}{tr\bm{R}_{t_{k-1}}}\sum^{n}_{i=1}\sum^{t_{k-1}}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}$
$\displaystyle\geq$
$\displaystyle\sum^{t_{k}}_{j=t_{k-1}+1}\left(\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{tr\bm{R}_{j}tr\bm{R}_{j-1}}\right)\sum^{n}_{i=1}\sum^{j-1}_{l=1}\bm{\varphi}^{i}_{l}{\bm{\varphi}^{i}_{l}}^{T}-(n+1)\bm{I}_{m}$
$\displaystyle\geq$
$\displaystyle\sum^{t_{k}}_{j=t_{k-1}+1}\left(\lambda^{(j-1)}_{\min}-\frac{n}{m}\right)\cdot\left(\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{tr\bm{R}_{j}tr\bm{R}_{j-1}}\right)\bm{I}_{m}$
$\displaystyle\hskip 128.0374pt-(n+1)\bm{I}_{m}.$

Combining this with Assumption II.2, we have

$\displaystyle\sum^{n}_{i=1}\sum^{t_{k}-1}_{j=t_{k-1}}\frac{\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}}{r^{i}_{j}}+(n+1)\bm{I}_{m}$
$\displaystyle\geq$
$\displaystyle\sum^{t_{k}}_{j=t_{k-1}+1}\Big{(}\frac{\lambda^{(j-1)}_{\max}}{N(\log
tr\bm{R}_{j-1})^{\frac{1}{3}}}-\frac{n}{m}\Big{)}\Big{(}\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{tr\bm{R}_{j}tr\bm{R}_{j-1}}\Big{)}\bm{I}_{m}$
$\displaystyle\geq$
$\displaystyle\frac{1}{m}\sum^{t_{k}}_{j=t_{k-1}+1}\Big{(}\frac{tr\bm{R}_{j-1}}{N(\log
tr\bm{R}_{j-1})^{\frac{1}{3}}}-n\Big{)}\Big{(}\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{tr\bm{R}_{j}tr\bm{R}_{j-1}}\Big{)}\bm{I}_{m}$
$\displaystyle=$
$\displaystyle\frac{1}{m}\sum^{t_{k}}_{j=t_{k-1}+1}\left(\frac{1}{N}-\frac{n(\log
tr\bm{R}_{j-1})^{\frac{1}{3}}}{tr\bm{R}_{j-1}}\right)$ $\displaystyle\hskip
85.35826pt\cdot\left(\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{tr\bm{R}_{j}(\log
tr\bm{R}_{j-1})^{\frac{1}{3}}}\right)\bm{I}_{m}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2Nm}\sum^{t_{k}}_{j=t_{k-1}+1}\left(\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{tr\bm{R}_{j}(\log
tr\bm{R}_{j-1})^{\frac{1}{3}}}\right)\bm{I}_{m},$

where (43) is used in the last inequality. Hence by the definition of $d_{k}$
in Lemma III.4, we can obtain

$\displaystyle\sum^{n}_{i=1}\sum^{t_{k}-1}_{j=t_{k-1}}\frac{\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}}{r^{i}_{j}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2Nm}(d_{t_{k}+1}-d_{t_{k-1}+1})\bm{I}_{m}-(n+1)\bm{I}_{m}.$

Furthermore, by Lemma III.4, (39) and (44), we have

$\displaystyle\lambda_{\min}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\bm{G}_{j}\right)\geq\sigma\lambda_{\min}\left(\sum^{n}_{i=1}\sum^{t_{k}-1}_{j=t_{k-1}}\frac{\bm{\varphi}^{i}_{j}{\bm{\varphi}^{i}_{j}}^{T}}{r^{i}_{j}}\right)$
(47) $\displaystyle\geq$
$\displaystyle\frac{\sigma}{2Nm}(d_{\hat{g}(N_{1}+k\alpha)+1}-d_{\hat{g}(N_{1}+(k-1)\alpha+1)})-\sigma(n+1)$
$\displaystyle\geq$
$\displaystyle\frac{\sigma}{2Nm}(N_{1}+k\alpha-(N_{1}+(k-1)\alpha+1))-\sigma(n+1)$
$\displaystyle=$
$\displaystyle\sigma\left(\frac{\alpha-1}{2Nm}-(n+1)\right)=\sigma.$

Step 3: Estimation of
$\lambda_{\max}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)$.

By the basic properties of the trace and the Euclidean norm of the matrix, we
have

$\displaystyle\sum^{t_{k}-1}_{j=t_{k-1}}\|\bm{A}_{j}\|$ (48)
$\displaystyle\leq$
$\displaystyle\sum^{t_{k}-1}_{j=t_{k-1}}tr(\bm{R}^{-1}_{j}\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})\leq\sum^{t_{k}-1}_{j=t_{k-1}}tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})\|\bm{R}^{-1}_{j}\|$
$\displaystyle\leq$
$\displaystyle\gamma\sum^{t_{k}-1}_{j=t_{k-1}}\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{\|\bm{R}_{j}\|}\leq
n\gamma\sum^{t_{k}-1}_{j=t_{k-1}}\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{tr\bm{R}_{j}}$
$\displaystyle\leq$ $\displaystyle n\gamma(\log
tr\bm{R}_{t_{k}-1})^{\frac{1}{3}}\sum^{t_{k}-1}_{j=t_{k-1}}\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{tr\bm{R}_{j}(\log
tr\bm{R}_{j-1})^{\frac{1}{3}}}.$

By the definition of $d_{\hat{g}(t)}$, (39), and (48), we have

$\displaystyle\sum^{t_{k}-1}_{j=t_{k-1}}\|\bm{A}_{j}\|\leq n\gamma(\log
tr\bm{R}_{t_{k}-1})^{\frac{1}{3}}(d_{t_{k}}-d_{t_{k-1}})$ $\displaystyle=$
$\displaystyle n\gamma(\log
tr\bm{R}_{t_{k}-1})^{\frac{1}{3}}(d_{\hat{g}(N_{1}+k\alpha)}-d_{\hat{g}(N_{1}+(k-1)\alpha)})$
$\displaystyle\leq$ $\displaystyle n\gamma(\alpha+1)(\log
tr\bm{R}_{t_{k}-1})^{\frac{1}{3}}.$

Then combining this with (9), we have

$\displaystyle\lambda_{\max}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\bm{G}_{j}\right)$
(49) $\displaystyle\leq$
$\displaystyle\sum^{t_{k}-1}_{j=t_{k-1}}\|\bm{G}_{j}\|\leq(1+4\nu)\sum^{t_{k}-1}_{j=t_{k-1}}\|\bm{A}_{j}\|$
$\displaystyle\leq$ $\displaystyle n\gamma(1+4\nu)(\alpha+1)(\log
tr\bm{R}_{t_{k}-1})^{\frac{1}{3}}.$

According to (39), (42) and (44), we can see that

$\displaystyle N_{1}+k\alpha\geq d_{\hat{g}(N_{1}+k\alpha)}$ (50)
$\displaystyle\geq$ $\displaystyle\frac{3}{2n\gamma
r^{*}}\left(\log^{\frac{2}{3}}tr\bm{R}_{t_{k}-1}-\log^{\frac{2}{3}}tr\bm{R}_{K_{0}-1}\right).$

By (49) and (50), we have

$\displaystyle\lambda^{2}_{\max}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\bm{G}_{j}\right)$
(51) $\displaystyle\leq$ $\displaystyle
n^{2}\gamma^{2}(1+4\nu)^{2}(\alpha+1)^{2}$
$\displaystyle\cdot\left(\frac{2n\gamma
r^{*}}{3}(N_{1}+k\alpha)+\log^{\frac{2}{3}}tr\bm{R}_{K_{0}-1}\right)$
$\displaystyle\triangleq$ $\displaystyle p_{k}=O(k).$

Combining this with $(\ref{89})$ yields

$\displaystyle\sum^{\infty}_{k=1}\frac{\lambda_{\min}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)}{1+\lambda^{2}_{\max}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)}$
(52) $\displaystyle\geq$
$\displaystyle\sum^{\infty}_{k=1}\frac{\mu\sigma}{1+\mu^{2}p_{k}}=\infty.$

According to Lemma B.2, we can see that ${\bm{\Psi}}(k,0)\rightarrow 0$ as
$k\rightarrow\infty$. This completes the proof of the theorem. $\blacksquare$

##  Appendix C Proof of Lemma IV.2

Proof. By the definition of ${\bm{\Psi}}(k,j)$ in (7), it is clear that we
have $(i)$.

By Lemma IV.1, we have

$\displaystyle det{\bm{\Psi}}(k+1,0)$ (53) $\displaystyle=$
$\displaystyle\prod^{k}_{j=0}det(\bm{I}_{mn}-\mu\bm{G}_{j})\geq\prod^{k}_{j=0}(det(\bm{I}_{mn}-\bm{A}_{j}))^{\tau_{1}}$
$\displaystyle=$
$\displaystyle(det(\bm{I}_{mn}-\bm{A}_{0}))^{\tau_{1}}\left(\prod^{n}_{i=1}\prod^{k}_{j=1}\frac{r^{i}_{j-1}}{r^{i}_{j}}\right)^{\tau_{1}}$
$\displaystyle=$
$\displaystyle(det(\bm{I}_{mn}-\bm{A}_{0}))^{\tau_{1}}\prod^{n}_{i=1}\frac{1}{(r^{i}_{k})^{\tau_{1}}}$
$\displaystyle=$
$\displaystyle\frac{1}{det(\bm{R}^{\tau_{1}}_{k})}\prod^{n}_{i=1}(1-\|\bm{\varphi}^{i}_{0}\|^{2})^{\tau_{1}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{\|\bm{R}_{k}\|^{n\tau_{1}}}\prod^{n}_{i=1}(1-\|\bm{\varphi}^{i}_{0}\|^{2})^{\tau_{1}}.$

Therefore, we have

$\displaystyle\left(\frac{1}{\|\bm{R}_{k}\|^{n\tau_{1}}}\prod^{n}_{i=1}(1-\|\bm{\varphi}^{i}_{0}\|^{2})^{\tau_{1}}\right)^{2}$
$\displaystyle\leq$ $\displaystyle
det({\bm{\Psi}}(k+1,0){\bm{\Psi}^{T}}(k+1,0))\leq\|{\bm{\Psi}}(k+1,0)\|^{2mn}.$

Since the initial value $\bm{\varphi}^{i}_{0}$ can be arbitrarily selected,
without loss of generality we suppose $\|\bm{\varphi}^{i}_{0}\|\neq 1$ for all
$i\in\\{1,...,n\\}$. It is clear that $(ii)$ of the lemma holds.

By Lemma IV.1, we have

$\displaystyle\|{\bm{\Psi}}(k,j+1)\|\leq\|{\bm{\Psi}}(k,0)\|\|\bm{\Psi}^{-1}(j+1,0)\|$
(54) $\displaystyle\leq$
$\displaystyle\|{\bm{\Psi}}(k,0)\|\prod^{j+1}_{p=1}\|(\bm{I}_{mn}-\mu\bm{G}_{p-1})^{-1}\|$
$\displaystyle\leq$
$\displaystyle\|{\bm{\Psi}}(k,0)\|\prod^{j+1}_{p=1}det\left((\bm{I}_{mn}-\mu\bm{G}_{p-1})^{-1}\right)$
$\displaystyle\leq$
$\displaystyle\|{\bm{\Psi}}(k,0)\|\prod^{j+1}_{p=1}\frac{1}{\left(det(\bm{I}_{mn}-\bm{A}_{p-1})\right)^{\tau_{1}}}$
$\displaystyle=$
$\displaystyle\|{\bm{\Psi}}(k,0)\|\left(\frac{1}{det(\bm{I}_{mn}-\bm{A}_{0})}\right)^{\tau_{1}}\prod^{n}_{i=1}\prod^{j+1}_{p=2}\left(\frac{r^{i}_{p-1}}{r^{i}_{p-2}}\right)^{\tau_{1}}$
$\displaystyle=$
$\displaystyle\|{\bm{\Psi}}(k,0)\|\cdot\left(\frac{1}{det(\bm{I}_{mn}-\bm{A}_{0})}\right)^{\tau_{1}}\left(\prod^{n}_{i=1}r^{i}_{j}\right)^{\tau_{1}}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{det(\bm{I}_{mn}-\bm{A}_{0})}\right)^{\tau_{1}}\cdot\|{\bm{\Psi}}(k,0)\|\cdot\|\bm{R}_{j}\|^{n\tau_{1}}.$

Hence $(iii)$ of the lemma can be proved.

Now we will prove $(iv)$.

$\displaystyle\sum^{\infty}_{j=M+1}\frac{\|\bm{\Phi}_{j}\|^{2}}{\|\bm{R}_{j}\|^{1+{\varsigma}}}\leq\sum^{\infty}_{j=M+1}\frac{tr(\bm{\Phi}^{T}_{j}\bm{\Phi}_{j})}{\frac{1}{n^{1+{\varsigma}}}(tr\bm{R}_{j})^{1+{\varsigma}}}$
$\displaystyle=$ $\displaystyle
n^{1+{\varsigma}}\sum^{\infty}_{j=M+1}\int^{tr\bm{R}_{j}}_{tr\bm{R}_{j-1}}\frac{1}{(tr\bm{R}_{j})^{1+{\varsigma}}}~{}dt$
$\displaystyle\leq$ $\displaystyle
n^{1+{\varsigma}}\sum^{\infty}_{j=M+1}\int^{tr\bm{R}_{j}}_{tr\bm{R}_{j-1}}\frac{1}{t^{1+{\varsigma}}}~{}dt$
$\displaystyle\leq$ $\displaystyle
n^{1+{\varsigma}}\int^{\infty}_{tr\bm{R}_{M}}\frac{1}{t^{1+{\varsigma}}}~{}dt$
$\displaystyle=$
$\displaystyle\frac{n^{1+{\varsigma}}}{{\varsigma}}\frac{1}{(tr\bm{R}_{M})^{{\varsigma}}}\leq\frac{n^{1+{\varsigma}}}{{\varsigma}}\frac{1}{\|\bm{R}_{M}\|^{{\varsigma}}}.$

This completes the proof of the lemma. $\blacksquare$

##  Appendix D Proof of Theorem IV.1

Proof. By (47) and (51) in Appendix B, there exists a constant $c^{*}>0$ such
that

$\displaystyle\frac{\lambda_{\min}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)}{1+\lambda^{2}_{\max}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)}\geq\frac{c^{*}}{1+k},$

where $t_{k}$ is defined in (44) of Appendix B. In fact, since
$\mu(1+4\nu)<1$, we can take

$\displaystyle c^{*}$ $\displaystyle=$
$\displaystyle\frac{\mu\sigma}{1+[n\gamma(\alpha+1)]^{2}\max\\{\frac{2n\gamma
r^{*}\alpha}{3},\frac{2n\gamma
r^{*}N_{1}}{3}+\log^{\frac{2}{3}}tr\bm{R}_{K_{0}-1}\\}}$

with $\alpha=2Nm(n+2)+1$, $N_{1}\propto n$, where
$\sigma=\frac{l^{2}_{m+1}a\nu}{2n+l^{2}_{m+1}a\nu n}$, $\gamma$ and $r^{*}$
are respectively defined in Lemma III.3 and Theorem III.2.

By Lemma B.1, we have

$\displaystyle\|{\bm{\Psi}}(t_{k},t_{k-1})\|$ (55) $\displaystyle\leq$
$\displaystyle\left(1-\frac{\lambda_{\min}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)}{2\left(1+\left(\sum^{t_{k}-1}_{j=t_{k-1}}\|\mu\bm{G}_{j}\|\right)^{2}\right)}\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\left(1-\frac{\lambda_{\min}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)}{2\left(1+m^{2}n^{2}\lambda^{2}_{\max}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)\right)}\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\left(1-\frac{\lambda_{\min}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)}{2m^{2}n^{2}\left(1+\lambda^{2}_{\max}\left(\sum^{t_{k}-1}_{j=t_{k-1}}\mu\bm{G}_{j}\right)\right)}\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\left(1-\frac{\bar{c}}{1+k}\right)^{\frac{1}{2}}~{}~{}~{}~{}~{}\left(\bar{c}=\frac{c^{*}}{2m^{2}n^{2}}\right).$

By the definition of $t_{k}$ in (44), we have the following estimate about
$\|{\bm{\Psi}}(\hat{g}(N_{1}+j\alpha),0)\|$ with $N_{1}$ and $\alpha$ defined
in Appendix B.

$\displaystyle\|{\bm{\Psi}}(\hat{g}(N_{1}+j\alpha),0)\|$ (56)
$\displaystyle\leq$
$\displaystyle\prod^{j}_{l=1}\|{\bm{\Psi}}(\hat{g}(N_{1}+l\alpha),\hat{g}(N_{1}+(l-1)\alpha)\|$
$\displaystyle\cdot\|{\bm{\Psi}}(\hat{g}(N_{1}),0)\|$ $\displaystyle\leq$
$\displaystyle\prod^{j}_{l=1}\left(1-\frac{\bar{c}}{1+l}\right)^{\frac{1}{2}}\leq\prod^{j}_{l=1}e^{-\frac{\bar{c}}{2(1+l)}}$
$\displaystyle=$ $\displaystyle e^{-\sum^{j}_{l=1}\frac{\bar{c}}{2(1+l)}}\leq
e^{-\frac{\bar{c}}{2}\cdot\log({\frac{j+2}{2}})}=\left(\frac{j+2}{2}\right)^{-\frac{\bar{c}}{2}},$

where the inequalities $1-x\leq e^{-x}$ for all $x\geq 0$ and
$\sum^{k}_{j=1}\frac{1}{1+j}\geq\log^{\frac{k+2}{2}}$ for all $~{}k\geq 1$ are
used.

Since $\hat{g}(t)\rightarrow\infty$ as $t\rightarrow\infty$ in (40), then for
any $k\geq\hat{g}(N_{1}+\alpha)$, there exists $j\geq 1$ such that

$\displaystyle\hat{g}(N_{1}+j\alpha)\leq k\leq\hat{g}(N_{1}+(j+1)\alpha).$

By the monotonicity of $d_{k}$ and (39), we have

$\displaystyle d_{k}\leq d_{\hat{g}(N_{1}+(j+1)\alpha)}\leq
N_{1}+(j+1)\alpha.$

Thus, $j\geq\frac{d_{k}-N_{1}-\alpha}{\alpha}$. According to $(\ref{138})$, we
obtain

$\displaystyle\|{\bm{\Psi}}(k,0)\|$ (57) $\displaystyle\leq$
$\displaystyle\|{\bm{\Psi}}(k,\hat{g}(N_{1}+j\alpha))\|\cdot\|{\bm{\Psi}}(\hat{g}{(N_{1}+j\alpha)},0)\|$
$\displaystyle\leq$
$\displaystyle\|{\bm{\Psi}}(\hat{g}(N_{1}+j\alpha),0)\|\leq\left(\frac{d_{k}-N_{1}+\alpha}{2\alpha}\right)^{-\frac{\bar{c}}{2}}.$

Combining (42) with (57), we have for large $k$

$\displaystyle\|{\bm{\Psi}}(k,0)\|$ $\displaystyle=$ $\displaystyle O(\log
tr\bm{R}_{k-1})^{-\frac{\bar{c}}{3}}$ $\displaystyle=$ $\displaystyle O(\log
tr\bm{R}_{k})^{-\frac{\bar{c}}{3}}.$

Hence by Lemma IV.3, we have
$\|\widetilde{\bm{\Theta}}_{k}\|=O((\log\|\bm{R}_{k}\|)^{-\delta_{1}})$ with
$\delta_{1}=\frac{\delta\bar{c}}{3n\tau_{1}(1+\delta)}=\frac{\delta
c^{*}}{6m^{3}n^{4}(1+\delta)}.$ $\blacksquare$

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