# Reducing sequential change detection to sequential estimation

Shubhanshu Shekhar  Department of Statistics and Data Science, Carnegie Mellon
University  Aaditya Ramdas  Department of Statistics and Data Science,
Carnegie Mellon University  Machine Learning Department, Carnegie Mellon
University

###### Abstract

We consider the problem of sequential change detection, where the goal is to
design a scheme for detecting any changes in a parameter or functional
$\theta$ of the data stream distribution that has small detection delay, but
guarantees control on the frequency of false alarms in the absence of changes.
In this paper, we describe a simple reduction from sequential change detection
to sequential estimation using confidence sequences: we begin a new
$(1-\alpha)$-confidence sequence at each time step, and proclaim a change when
the intersection of all active confidence sequences becomes empty. We prove
that the average run length is at least $1/\alpha$, resulting in a change
detection scheme with minimal structural assumptions (thus allowing for
possibly dependent observations, and nonparametric distribution classes), but
strong guarantees. Our approach bears an interesting parallel with the
reduction from change detection to sequential testing of Lorden (1971)and the
e-detector of Shin et al. (2022).

##  1 Introduction

We consider the following problem of _sequential change detection_ (SCD): for
a general space $\mathcal{X}$, given a stream of $\mathcal{X}$-valued
observations $X_{1},X_{2},\ldots$, our goal is to design a method to detect
any changes in a prespecified parameter or functional $\theta$ (possibly
infinite dimensional) associated with the source generating this stream. Let
$\mathcal{P}$ denote a class of probability distributions on the infinite
product space $\Omega=\mathcal{X}^{\infty}$, and let
$\theta:\mathcal{P}\to\Theta$, denote a mapping from probability distributions
to some (possibly infinite dimensional) parameter space $\Theta$. The data
distribution satisfies the following for some
$T\in\mathbb{N}\cup\\{\infty\\}$:

  * •

For $n\leq T$, the observations are the first $T$ elements of a trajectory of
$P_{0}\in\mathcal{P}$, with $\theta(P_{0})=\theta_{0}$.

  * •

For $n>T$, the observations are drawn from another distribution
$P_{1}\in\mathcal{P}$, such that $\theta(P_{1})=\theta_{1}\neq\theta_{0}$.

In particular, $P_{0},P_{1}$ are distributions over an infinite sequence of
observations, and the data is not assumed to be iiid (independent and
identically distributed), meaning that we do not assume that $P_{0},P_{1}$
take the form $p^{\infty}$ for some distribution $p$ over $\mathcal{X}$.
Further, $P_{1}$ (and hence $\theta_{1}$) is allowed to depend on
$X_{1},\dots,X_{T}$.

Under the above model, our goal is to design a data-driven method for
detecting any change in the value of $\theta$, without any knowledge of
$T,P_{0},P_{1}$. In technical terms, our objective is to define a stopping
time $\tau$, at which we stop collecting more data, and declare a changepoint
has previously occurred.

There are two problem settings to consider: non-partitioned and partitioned.
By default, define the filtration $\mathcal{F}\equiv(\mathcal{F}_{n})_{n\geq
0}$ as given by $\mathcal{F}_{n}\coloneqq\sigma(X_{1},\ldots,X_{n})$ and
$\mathcal{F}_{0}=\\{\emptyset,\Omega\\}$.

######  Problem 1.1 (Non-partitioned SCD).

For some unknown triple $(T,P_{0},P_{1})$, suppose $X_{1},X_{2},\ldots$ denote
a data stream, such that $(X_{n})_{n\leq T}$ are drawn from $P_{0}$, and
$(X_{n})_{n>T}$ are drawn from $P_{1}$, such that
$\theta(P_{1})=\theta_{1}\neq\theta_{0}=\theta(P_{0})$. We must define a
stopping time $\tau$, adapted to $\mathcal{F}$ satisfying:

  * •

If $T=\infty$, and there is no changepoint, we require that
$\mathbb{E}_{\infty}[\tau]\geq 1/\alpha$, for a prespecified $\alpha\in(0,1)$.
The term $\mathbb{E}_{\infty}[\tau]$ is called the average run length (ARL),
and represents the frequency of false alarms.

  * •

If $T<\infty$, and there is a changepoint at which the distribution changes
from $P_{0}$ to $P_{1}$, we desire the detection delay,
$\mathbb{E}_{T}[(\tau-T)^{+}]$, to be as small as possible.

The non-partitioned SCD problem stated above, does not require any pre-
specified partitioning of $\mathcal{P}$ into pre- and post change distribution
classes. That is, it assumes that the data generating distribution changes
from one unknown distribution ($P_{0}$) in $\mathcal{P}$ to another
distribution ($P_{1}$). This is in contrast to another formulation of the SCD
problem, where it is assumed that we know some partition of $\mathcal{P}$ into
$\mathcal{P}_{0},\mathcal{P}_{1}$ such that $P_{0},P_{1}$ are known to
respectively lie in $\mathcal{P}_{0},\mathcal{P}_{1}$ (in other words, we are
not interested in changes within $\mathcal{P}_{0}$, only changes from
$\mathcal{P}_{0}$ to $\mathcal{P}_{1}$). This additional knowledge about the
two partitioned distribution classes, that is not available in the first
formulation, is often critical in designing optimal SCD schemes, especially in
parametric problems. The strategy we develop in this paper is also applicable
to an intermediate variant of the SCD problem, that only assumes the knowledge
of the pre-change parameter class, $\Theta_{0}$.

######  Problem 1.2 (Partitioned SCD).

For some unknown triple $(T,P_{0},P_{1})$, suppose $X_{1},X_{2},\ldots$ denote
a stream of $\mathcal{X}$-valued observations, satisfying the following
assumptions: The observations $(X_{n})_{n\leq T}$ are drawn according to a
process $P_{0}\in\mathcal{P}$ with parameter
$\theta_{0}\in\Theta_{0}\subset\Theta$, and this parameter set $\Theta_{0}$ is
assumed to be known. The observations $(X_{n})_{n>T}$ are drawn from $P_{1}$
with parameter $\theta_{1}$, such that $\theta_{1}\not\in\Theta_{0}$. The
parameter $\theta_{1}$ is not assumed to be known. Our objective is to design
a stopping time $\tau$, satisfying the two bulleted criteria stated in 1.1.

Sequential changepoint detection is a very well-studied problem in the
sequential analysis literature, going back to the early works by Shewhart
(1925, 1930); Page (1954); Shiryaev (1963). These initial papers developed
computationally efficient likelihood-based schemes for known pre- and post-
change distributions, which have since then been extended to more general
cases of composite, but parametric, pre- and post-change distribution classes.
We refer the reader to the book by Tartakovsky et al. (2014) for a detailed
discussion of the parametric SCD problem. Unlike the parametric case, there
exist very few general principles (analogous to the likelihood and generalized
likelihood based schemes) for designing SCD methods with nonparametric
distribution classes. Two recent exceptions to this trend include the paper on
e-detectors by Shin et al. (2022) for 1.2 (partitioned), and the BCS-Detector
scheme proposed by Shekhar and Ramdas (2023) for 1.1 (non-partitioned), which
is the subject of this paper.

The BCS-Detector scheme of Shekhar and Ramdas (2023) is also a reduction
changepoint detection to estimation, but the current paper proposes a
different and even simpler reduction. To elaborate, BCS-Detector uses a single
confidence sequence (CS) in the forward direction, but with every new
observation, it constructs a new CS in the backward direction (the so-called
“backward CS” or BCS, constructed using observations with their time indices
reversed). The scheme stops and declares a detection, as soon as the
intersection of the CS and the all active BCSs becomes empty. Since it is
critical to our simplified scheme as well, we recall the definition of CSs
below.

######  Definition 1.3 (Confidence Sequences).

Given observations $X_{1},X_{2},\ldots$ drawn from a distribution $P$ with
associated parameter $\theta\equiv\theta(P)$, a level-$(1-\alpha)$ CS for
$\theta$, is a sequence of sets $(C_{n})_{n\geq 1}$, satisfying:

  * •

For every $n\geq 1$, the set $C_{n}\subset\Theta$ is
$\mathcal{F}_{n}=\sigma(X_{1},\ldots,X_{n})$-measurable. In words, the set
$C_{n}$ can be constructed using the information contained in the first $n$
observations.

  * •

The sets satisfy a uniform coverage guarantee: $\mathbb{P}\left(\forall
n\in\mathbb{N}:\theta(P)\in C_{n}\right)\geq 1-\alpha$. Equivalently, for any
$\mathcal{F}$-stopping time $\tau$, $\mathbb{P}\left(\theta(P)\in
C_{\tau}\right)\geq 1-\alpha$.

######  Remark 1.4.

Due to the uniform coverage guarantee, if $(C_{n})$ is a CS, then so is
$(\cap_{m\leq n}C_{m})$. Thus, we can assume without loss of generality that
the sets involved in a CS are nested; that is $C_{n}\subset C_{n^{\prime}}$
for all $n^{\prime}<n$.

The BCS-Detector scheme of Shekhar and Ramdas (2023) satisfies several
favorable properties: it can be instantiated for a large class of parametric
and nonparametric problems, it provides non-asymptotic control over the ARL,
and has strong guarantees over the detection delay. However, a closer look at
their scheme reveals that it implicitly makes a “bidirectional” assumption
about the data generating process: at any $n\geq 1$, the BCS-Detector assumes
the ability to construct a CS in the forward direction (based on
$X_{1},\ldots,X_{n}$), as well as in the backward direction (using
$Y_{1}=X_{n},Y_{2}=X_{n-1},\ldots,Y_{n}=X_{1}$). Most methods for constructing
CSs proceed by designing martingales or supermartingales adapted to the
natural (forward) filtration of the observations. Hence, the BCS-Detector
implicitly involves constructing martingales (or supermartingales) in both,
the forward and reverse directions; this in turn is typically only possible if
the observations are independent. This restriction limits the applicability of
the BCS-Detector scheme, and this paper’s simplified scheme addresses this
weakness.

#### Contributions.

Our main contribution in this paper is a new SCD scheme, that we refer to as
the repeated-FCS-Detector. This scheme proceeds by constructing a new forward
CS with each observation (Definition 2.1), and stops as soon the intersection
of all the active CSs becomes empty. Since it relies only on forward CSs, it
eliminates the aforementioned weakness of the BCS-Detector of Shekhar and
Ramdas (2023). Further, as we show in Theorem 2.2, our scheme achieves a
tighter bound (by a factor $2$) on the ARL, as compared to the BCS-Detector,
while matching its detection delay guarantees. Finally, we note in Section 3,
that our reduction from change detection to constructing CSs is related to
(but different from) a famous reduction by Lorden (1971) from parametric
change detection to one-sided sequential tests.

##  2 Our proposed reduction

We now describe our new SCD scheme that proceeds by starting a new CS in the
forward direction with each new observation.

######  Definition 2.1 (repeated-FCS-Detector).

Suppose we are given a stream of observations $X_{1},X_{2},\ldots$, and a
functional $\theta$ associated with the source. For 1.1 (non-partitioned),
define $C^{(0)}_{n}=\Theta$ for all $n\geq 1$, while for 1.2 (partitioned) set
$C^{(0)}_{n}=\Theta_{0}$, for all $n\geq 1$. Proceed as follows for
$n=1,2,\ldots:$

  * •

Initialize a new level-$(1-\alpha)$ forward CS,
$C^{(n)}\equiv(C^{(n)}_{t})_{t\geq n}$, which will be formed using data
$X_{n},X_{n+1},\dots$.

  * •

Observe the next data-point $X_{n}$.

  * •

Update all CSs initialized so far using $X_{n}$, meaning that we form the sets
$\\{C_{n}^{(m)}:1\leq m\leq n\\}$.

  * •

If the intersection of all initialized CSs becomes empty, meaning
$\cap_{m=0}^{n}C_{n}^{(m)}=\emptyset$, then set $\tau\leftarrow n$, and
declare a detection.

(In the last step, we have implicitly used the nestedness discussed Remark
1.4, but if the CSs are not nested, we can use the stopping criterion
$\cap_{m=0}^{n}\cap_{i=m}^{n}C_{i}^{(m)}=\emptyset$.)

Compared to the BCS-Detector of Shekhar and Ramdas (2023), the main change in
the above scheme is that it creates a new forward CS with each new
observation, instead of a new backward CS. We now show that our new SCD scheme
defined above admits a nonasymptotic lower bound on the ARL.

######  Theorem 2.2 (ARL control).

The changepoint detection scheme described in Definition 2.1 controls the
average run length (ARL) at level $1/\alpha$. That is, when $T=\infty$, our
proposed stopping time $\tau$ satisfies

$\displaystyle\mathbb{E}_{\infty}[\tau]\geq\frac{1}{\alpha}.$ (1)

The proof of this result is in Section 4.1. Note that for the BCS-Detector,
used with level-$(1-\alpha)$ CSs, Shekhar and Ramdas (2023) obtained a lower
bound of $1/2\alpha-3/2$ on the ARL. Thus, our repeated-FCS-Detector achieves
an improved (by a factor of $2$) lower bound, while matching the detection
delay guarantees of BCS-Detector, as we show in the next section.

###  2.1 Analyzing the detection delay

We now state an assumption under which we will analyze the detection delay of
our SCD scheme.

######  Assumption 2.3.

Letting $d$ denote a metric on $\Theta$, $X^{n}$ be shorthand for
$(X_{1},\dots,X_{n})$, and $(C_{n})_{n\geq 0}$ be a given confidence sequence,
we assume that the width of the set $C_{n}\equiv C_{n}(X^{n},\alpha)$ has a
deterministic upper bound

$\displaystyle\sup_{\theta^{\prime},\theta^{\prime\prime}\in
C_{n}}d(\theta^{\prime},\theta^{\prime\prime})\stackrel{{\scriptstyle
a.s.}}{{\leq}}w(n,P,\alpha),$ (2)

such that $\lim_{n\to\infty}w(n,P,\alpha)=0$, for all $P\in\mathcal{P}$ and
$\alpha\in(0,1]$.

The above assumption requires the existence of a deterministic envelope
function for the diameter of the confidence sequence, which converges to zero
pointwise for every $(P,\alpha)$, as $n$ increases. This is a very weak
assumption, and essentially all known CSs satisfy it.

We now analyze the detection delay of our SCD scheme for 1.1 under 2.3.

######  Theorem 2.4.

Consider the SCD problem with observations $X_{1},X_{2},\ldots$ such that
$(X_{n})_{n\leq T}$ are drawn from a distribution $P_{0}$ (with parameter
$\theta_{0}$), while $(X_{n})_{n>T}$ are drawn from a product distribution
$P_{1}$ (with parameter $\theta_{1}\neq\theta_{0}$), and are independent of
the pre-change observations. Suppose the repeated-FCS-Detector from Definition
2.1 is applied to this problem, with the CSs satisfying 2.3. Let
$\mathcal{E}:=\\{\theta_{0}\in\cap_{n=1}^{T}C^{(1)}_{n}\\}$ denote the “good
event” (having at least $(1-\alpha)$ probability) that the CS up to the
changepoint covers the true parameter. For 1.1 (non-partitioned), if $T$ is
large enough to ensure that $w(T,P_{0},\alpha)<d(\theta_{0},\theta_{1})$, then
the detection delay of our proposed scheme satisfies

$\displaystyle\mathbb{E}_{T}[(\tau-T)^{+}|\mathcal{E}]\leq\frac{3}{1-\alpha}u(\theta_{0},\theta_{1},T),$
(3) where $\displaystyle u(\theta_{0},\theta_{1},T)\coloneqq\min\\{n\geq
1:w(T,P_{0},\alpha)+w(n,P_{1},\alpha)<d(\theta_{0},\theta_{1})\\}.$ (4)

For 1.2 (partitioned), with a known $\Theta_{0}$, the repeated-FCS-Detector
satisfies

$\displaystyle\mathbb{E}_{T}[(\tau-T)^{+}]=\frac{3}{1-\alpha}u(\Theta_{0},\theta_{1}),$
(5) where $\displaystyle u(\Theta_{0},\theta_{1})\coloneqq\min\\{n\geq
1:w(n,P_{1},\alpha)<\inf_{\theta^{\prime}\in\Theta_{0}}d(\theta^{\prime},\theta_{1})\\}.$
(6)

for all values of $T<\infty$.

The proof of this result adapts the arguments developed by Shekhar and Ramdas
(2023) for analyzing the BCS-Detector, and we present the details in Section
4.2.

######  Remark 2.5.

The above detection delay bound _exactly_ matches that obtained by the BCS-
Detector of Shekhar and Ramdas (2023), which had an ARL guarantee of
$\text{ARL}\geq 1/(2\alpha)-3/2$. Recalling Theorem 2.2, our new scheme
achieves a significantly improved (by a factor of $2$) nonasymptotic lower
bound on the ARL as compared to the BCS-Detector, while matching their
detection delay performance.

This result provides a general detection delay bound applicable to a large
class of problems. However, this guarantee can be sharpened when the repeated-
FCS-Detector is applied to problems with additional structure. We demonstrate
this next, through an application of our scheme to the problem of detecting
changes in mean of bounded random variables.

###  2.2 A nonparametric example: change in mean for bounded random variables

We now analyze the performance of our changepoint detection scheme the problem
of detecting changes in the mean of bounded real-valued random variables
supported on $\mathcal{X}=[0,1]$. Note that despite the simple observation
space, the class of distributions on this $\mathcal{X}$ is highly composite
and nonparametric. In particular, there does not exist a common dominating
measure for all distributions in this class, which renders likelihood based
techniques inapplicable to this problem.

Formally, we consider the instances of 1.1 and 1.2 with $\mathcal{X}=[0,1]$,
and the parameter space $\Theta=[0,1]$ with metric
$d(\theta,\theta^{\prime})=|\theta-\theta^{\prime}|$. For 1.2, we assume that
the pre-change mean $\theta_{0}$ lies in a known set
$\Theta_{0}\subset\Theta$. For an unknown value
$T\in\mathbb{N}\cup\\{\infty\\}$, the distribution generating the observations
changes from $P_{0}$, with mean $\theta_{0}$, to another distribution $P_{1}$,
with mean $\theta_{1}\neq\theta_{0}$. The (unknown) change magnitude is
denoted by $d(\theta_{0},\theta_{1})=\Delta=|\theta_{1}-\theta_{0}|$.

For the problem setting discussed above, we instantiate our repeated-FCS-
Detector strategy using the betting-based construction of CSs (details in
Section 4.3) for the means of bounded random variables proposed by Waudby-
Smith and Ramdas (2023). Our next result analyzes its performance.

######  Proposition 2.6.

Consider the problem of detecting changes in mean with bounded observations
discussed above, under the following additional conditions: (i) the post-
change observations are independent of the pre-change observations, and (ii)
both, the pre- and post-change observations are i.i.d. (that is $P_{0},P_{1}$
are infinite products of some distributions $p_{0},p_{1}$ on $\mathcal{X}$).
For 1.1 (non-partitioned), if $T\geq 64\log(64/\Delta^{2}\alpha)/\Delta^{2}$,
the repeated-FCS-Detector instantiated with the betting CS of Waudby-Smith and
Ramdas (2023) satisfies:

$\displaystyle\mathbb{E}_{\infty}[\tau]\geq\frac{1}{\alpha},\quad\text{and}\quad\mathbb{E}_{T}[(\tau-T)^{+}|\mathcal{E}]=\mathcal{O}\left(\frac{\log(1/\alpha
K_{1})}{K_{1}}\right),$ (7) where $\displaystyle
K_{1}=K_{1}(P_{1},\theta_{0})\coloneqq\inf_{P_{\theta}:|\theta-\theta_{0}|\leq\Delta/2}d_{\text{KL}}(p_{1}\parallel
p_{\theta}).$ (8)

In the display above, $\mathcal{E}$ is the “good event” in Theorem 2.4, having
probability at least $1-\alpha$.

For 1.2 (partitioned), the repeated-FCS-Detector satisfies the following:

$\displaystyle\mathbb{E}_{\infty}[\tau]\geq\frac{1}{\alpha},\quad\text{and}\quad\mathbb{E}_{T}[(\tau-T)^{+}]=\mathcal{O}\left(\frac{\log(1/\alpha
K_{2})}{K_{2}}\right),$ (9) where $\displaystyle K_{2}\equiv
K_{2}(P_{1},\Theta_{0})=\inf_{P_{\theta}:\theta\in\Theta_{0}}d_{\text{KL}}(p_{1}\parallel
p_{\theta}).$ (10)

In the statements above, $P_{\theta}=p_{\theta}^{\infty}$ denotes any product
distribution on $\mathcal{X}^{\infty}$ with mean $\theta$.

The proof of this result is in Section 4.3, and relies on a careful analysis
of the behavior of betting CSs. If the pre-change distribution $P_{0}$ is also
i.i.d. (say $P_{0}=p_{0}^{\infty}$), and is known, then $K_{2}$ in (9) reduces
to $d_{\text{KL}}(p_{1}\parallel p_{0})$. The resulting detection delay is
order optimal, according to Lorden (1971)[Theorem 3], and furthermore, this
optimality is achieved for an unknown $P_{1}$ lying in a nonparametric
distribution class.

######  Remark 2.7.

By an application of Pinsker’s inequality, we know that both $K_{1}$ and
$K_{2}$ are $\Omega(\Delta^{2})$, which gives us the weaker upper bound on the
detection delay, $\mathcal{O}\left(\log(1/\alpha\Delta)/\Delta^{2}\right)$.
This is the upper bound on the detection delay derived by Shekhar and Ramdas
(2023) for the change of mean detection problem, using the empirical-Bernstein
CS of Waudby-Smith and Ramdas (2023), and a direct application of the general
delay bound of Theorem 2.4.

##  3 Connection to Lorden’s reduction from SCD to testing

Using the duality between confidence sequences and sequential hypothesis
tests, we now show that our repeated-FCS-Detector strategy is a generalization
of a well-known result of Lorden (1971), that reduces the problem of SCD (with
separated distribution classes) to that of repeated sequential tests.

Lorden’s work built upon the interpretation of CuSum algorithm as repeated
sequential probability ratio tests (SPRT) for known pre- and post-change
distributions by Page (1954). In particular, Lorden (1971) considered a
parametric SCD problem with a known pre-change distribution $P_{0}$, and a
parametric composite class of post-change distributions
$\\{P_{\theta_{1}}:\theta_{1}\in\Theta_{1}\\}$. Then, given a sequential test,
or equivalently, extended stopping time, $\\{N(\alpha):\alpha\in(0,1)\\}$,
satisfying $\mathbb{E}_{P_{0}}(N(\alpha)<\infty)\leq\alpha$, Lorden (1971)
proposed the following SCD strategy:

  * •

For every $m\geq 1$, define $N^{(m)}(\alpha)$ as the stopping time $N(\alpha)$
applied to the observations $X_{m},X_{m+1},\ldots$.

  * •

Using these, declare the changepoint at the time $N^{*}(\alpha)$, defined as

$\displaystyle N^{*}(\alpha)=\min_{m\geq 1}\;\\{N^{(m)}(\alpha)+m-1\\}.$ (11)

In words, this scheme can be summarized as: _initiate a new sequential test
with every new observation, and stop and declare a detection as soon as one of
the active tests rejects the null_. For this scheme, Lorden (1971) established
the ARL control; that is, $\mathbb{E}_{P_{0}}[N^{*}(\alpha)]\geq 1/\alpha$,
for the specified $\alpha\in(0,1)$. Furthermore, under certain assumptions on
the expected stopping time of the test $N(\alpha)$ under the alternative,
Lorden (1971) also established the minimax optimality of the scheme in the
regime of $\alpha\to 0$.

Due to the duality between confidence sequences and sequential tests, it is
easy to verify that the SCD scheme defined in (11) is a special instance of
our general strategy. In particular, let us consider the problem studied by
Lorden (1971), with a known pre-change distribution $P_{0}$ with parameter
$\theta_{0}$ (which could, for instance, be the distribution function of
$P_{0}$), and a class of post-change distributions
$\\{P_{\theta_{1}}:\theta_{1}\in\Theta_{1}\\}$. Given a method for
constructing confidence sequences for $\theta$, denoted by $\mathcal{C}$, we
can define a sequential test

$\displaystyle N(\alpha)=\inf\\{n\geq 1:\theta_{0}\not\in
C_{n}\\},\quad\text{where}\quad C_{n}=\mathcal{C}(X_{1},\ldots,X_{n};\alpha).$
(12)

By the uniform coverage guarantee of confidence sequences, we have
$\mathbb{P}_{P_{0}}\left(N(\alpha)<\infty\right)=\mathbb{P}_{P_{0}}\left(\exists
n\in\mathbb{N}:\theta_{0}\not\in C_{n}\right)\leq\alpha$. Thus, $N(\alpha)$ is
a valid level-$\alpha$ sequential test for the simple null $\\{P_{0}\\}$.
Similarly, $N^{(m)}$ for $m\geq 1$, can be defined as the stopping time
$N(\alpha)$ constructed using observations $(X_{n})_{n\geq m}$ starting at
time $m$. More specifically, we have

$\displaystyle N^{(m)}(\alpha)=\inf\\{n-m+1:\theta_{0}\not\in
C^{(m)}_{n},\;n\geq m\\},\quad\text{where}\quad
C_{n}^{(m)}=\mathcal{C}\left(X_{m},X_{m+1}\ldots,X_{n};\alpha\right).$ (13)

Using this, we can define $N^{*}(\alpha)=\inf{m\geq
1}\\{N^{(m)}(\alpha)+m-1\\}$. We now observe, that this $N^{*}(\alpha)$ is
exactly the same as our proposed stopping time, $\tau$, in Definition 2.1. In
particular, for all $n\geq 1$, we have

$\displaystyle\\{N^{*}(\alpha)\leq n\\}$ $\displaystyle=\\{\exists
n^{\prime}\leq n,\exists m\leq n^{\prime}:N^{*}(m)(\alpha)=n^{\prime}-m+1\\}$
(14) $\displaystyle=\\{\exists n^{\prime}\leq n,\exists m\leq
n^{\prime}:\theta_{0}\not\in C^{(m)}_{n^{\prime}}\\}=\\{\exists n^{\prime}\leq
n:\cap_{m=1}^{n^{\prime}}C^{(m)}_{n^{\prime}}=\emptyset\\}$ (15)
$\displaystyle=\\{\cap_{n^{\prime}=1}^{n}\cap_{m=1}^{n^{\prime}}C^{(m)}_{n^{\prime}}=\emptyset\\}=\\{\cap_{m=1}^{n}\cap_{n^{\prime}=m}^{n}C^{(m)}_{n^{\prime}}=\emptyset\\}=\\{\tau\leq
n\\}.$ (16)

Thus, our general strategy proposed in Definition 2.1 subsumes the SCD scheme
of Lorden (1971). We end our discussion with the following two remarks:

  * •

While we instantiated our scheme above for the case of a singleton null,
$\\{P_{0}\\}$, the exact same construction also applies to the case of a
composite null $\\{P_{\theta_{0}}:\theta_{0}\in\Theta_{0}\\}$, with
$\Theta_{0}\cap\Theta_{1}=\emptyset$. The only modification needed is to
update the stopping time $N(\alpha)$ to be equal to $\inf\\{n\geq
1:\Theta_{0}\cap C_{n}=\emptyset\\}$. By Theorem 2.2, the resulting SCD scheme
still controls the ARL at the required level $1/\alpha$.

  * •

Note that the e-detector framework, developed by Shin et al. (2022), also
generalizes Lorden’s scheme to work for composite, and nonparametric pre- and
post-change distribution classes ($\mathcal{P}_{0}$ and $\mathcal{P}_{1}$
respectively). However, the e-detectors were developed explicitly for 1.2
(partitioned); that is for a known class of pre-change distributions
$\mathcal{P}_{0}$ (although the general idea could be suitably adapted for the
non-separated formulation in some cases). This is unlike our scheme that is
easily applicable to both the partitioned and non-partitioned formulations of
the SCD problem.

##  4 Deferred proofs from Section 2

In this section, we present the proofs of the three technical results stated
in Section 2.

###  4.1 Proof of Theorem 2.2

The proof of this statement proceeds in three steps. In the first step, we
define an e-process (recalled in Definition 4.1) corresponding to every
confidence sequence $(C_{n}^{(m)})_{n\geq m}$ involved in our scheme. Then,
using these e-processes we introduce an e-detector $(M_{n})_{n\geq 1}$, that
is, a process adapted to the natural filtration $\mathcal{F}$ that satisfies
$\mathbb{E}_{\infty}[M_{\tau^{\prime}}]\leq\mathbb{E}_{\infty}[\tau^{\prime}]$
for all stopping times $\tau^{\prime}$. Finally, we show that our stopping
time $\tau$, introduced in Definition 2.1, is larger than
$\tau^{\prime}=\inf\\{n\geq 1:M_{n}\geq 1/\alpha\\}$, defined using the
e-detector. This allows us to leverage Shin et al. (2022, Proposition 2.4) to
conclude that $\mathbb{E}_{\infty}[\tau^{\prime}]\geq 1/\alpha$, which implies
required statement about the ARL of $\tau$. The details are in Section 4.1

Since we prove this result by attaching an e-process to every CS, we recall
their definition below.

######  Definition 4.1 (e-processes).

Given a class of probability measures $\mathcal{P}$, and a filtration
$\mathcal{F}\equiv(\mathcal{F}_{n})_{n\geq 1}$ defined on some measurable
space, a $\mathcal{P}$-e-process is a collection of nonnegative random
variables $(E_{n})_{n\geq 1}$ adapted to $\mathcal{F}$, satisfying
$\mathbb{E}_{P}[E_{\tau^{\prime}}]\leq 1$ for all $P\in\mathcal{P}$, and for
all stopping times $\tau^{\prime}$ (adapted to the same filtration).

#### Step 1. Construct an e-process for every CS.

For every CS starting after $m$ observations, denoted by $(C_{n}^{(m)})_{n\geq
m}$, we associate a process defined as

$\displaystyle E_{n}^{(m)}=\begin{cases}0,&\text{if }n<m,\\\ 0,&\text{if
}n\geq m,\text{ and }\theta_{0}\in C_{n}^{(m)},\\\ \frac{1}{\alpha},&\text{if
}n\geq m,\text{ and }\theta_{0}\not\in C_{n}^{(m)}.\end{cases}$ (17)

It is easy to verify that for every $m\geq 1$, the process
$\\{E_{n}^{(m)}:n\geq 1\\}$ is an e-process:

  * •

For every $n\geq 1$, the value of $E_{n}^{(m)}$ is
$\mathcal{F}_{n}=\sigma(X_{1},\ldots,X_{n})$ measurable.

  * •

For any stopping time $\tau^{\prime}$, adapted to the filtration
$\mathcal{F}$, we have

$\displaystyle\mathbb{E}_{\infty}[E_{\tau^{\prime}}^{(m)}]$
$\displaystyle=\mathbb{E}_{\infty}\left[0\times\boldsymbol{1}_{\tau^{\prime}<m}+\frac{1}{\alpha}\times\boldsymbol{1}_{\tau^{\prime}\geq
m}\boldsymbol{1}_{\theta_{0}\not\in
C_{\tau^{\prime}}^{(m)}}\right]=\frac{1}{\alpha}\times\mathbb{E}_{\infty}\left[\boldsymbol{1}_{\tau^{\prime}\geq
m}\boldsymbol{1}_{\theta_{0}\not\in C_{\tau^{\prime}}^{(m)}}\right]$ (18)
$\displaystyle\leq\frac{1}{\alpha}\times\mathbb{E}_{\infty}\left[\boldsymbol{1}_{\theta_{0}\not\in
C_{\tau^{\prime}}^{(m)}}\right]=\frac{1}{\alpha}\times\mathbb{P}_{\infty}\left(\theta_{0}\not\in
C_{\tau^{\prime}}^{(m)}\right)\leq\frac{1}{\alpha}\times\alpha=1.$ (19)

The last inequality uses the fact that $(C_{n}^{(m)})_{m\geq n}$ is a
level-$(1-\alpha)$ CS for $\theta_{0}$. Thus, for every $m\geq 1$, the process
$(E_{n}^{(m)})_{n\geq 1}$ is a valid e-process.

#### Step 2. Construct an e-detector.

For every $n\geq 1$, we define $M_{n}$ to be equal to
$\sum_{m=1}^{n}E_{n}^{(m)}$, and observe that the process $(M_{n})_{n\geq 1}$
is an _e-detector_ , as defined by Shin et al. (2022, Definition 2.2). More
formally, $(M_{n})_{n\geq 1}$ satisfies the following two properties:

  * •

$(M_{n})_{n\geq 1}$ is adapted to $(\mathcal{F}_{n})_{n\geq 1}$: since for any
$n\geq 1$, all the $M_{n}^{(m)}$ are $\mathcal{F}_{n}$ measurable by
construction.

  * •

For any stopping time $\tau^{\prime}$, we have
$\mathbb{E}_{\infty}[M_{\tau^{\prime}}]\leq\mathbb{E}_{\infty}[\tau^{\prime}]$,
as noted by Shin et al. (2022, Definition 2.6).

#### Step 3. Bound the ARL using the e-detector.

Finally, we translate the stopping criterion of our proposed scheme (stated as
the non-intersection of the confidence sequences) in terms of the e-detector
$(M_{n})_{n\geq 1}$. In particular, we have

$\displaystyle\\{\tau\leq
n\\}=\\{\cap_{m=1}^{n}C_{n}^{(m)}=\emptyset\\}\subset\\{\exists
m\in[n]:\theta_{0}\not\in C_{n}^{(m)}\\}=\\{\exists
m\in[n]:E_{n}^{(m)}=1/\alpha\\}=\\{M_{n}\geq 1/\alpha\\}.$ (20)

In words, when $T=\infty$, if the intersection of the CSs is empty prior to
some time $n$, it means that at least of of the CSs constructed prior to $n$
must miscover. This in turn implies that the value of at least one of the
e-processes at $n$ is equal to $1/\alpha$; or the value of the e-detector
$M_{n}$ is at least $1/\alpha$. Recall, that in the first equality above, we
have assumed that the sets in the confidence sequences are nested; that is,
$C_{n}^{(m)}\subset C_{n^{\prime}}^{(m)}$ for every $m\leq n^{\prime}<n$. This
allows us to look only at the intersection of the most recent sets to define
the stopping condition.

We now define a new stopping time

$\displaystyle\tau^{\prime}=\inf\\{n\geq 1:M_{n}\geq 1/\alpha\\},$ (21)

and observe that it is stochastically dominated by $\tau$; that is, (20)
implies that

$\displaystyle\\{\tau^{\prime}>n\\}=\\{M_{n}<1/\alpha\\}\subset\\{\tau>n\\},\quad\text{which
implies}\quad\mathbb{E}_{\infty}[\tau^{\prime}]\leq\mathbb{E}_{\infty}[\tau].$
(22)

Hence, by appealing to Shin et al. (2022, Proposition 2.4), we conclude that
$\mathbb{E}_{\infty}[\tau]\geq\mathbb{E}_{\infty}[\tau^{\prime}]\geq
1/\alpha$. ∎

###  4.2 Proof of Theorem 2.4

The proof of this result follows the general argument developed by Shekhar and
Ramdas (2023) for analyzing their BCS-Detector strategy, with some
modifications due to the use of forward CSs (instead of backward CSs used in
the BCS-Detector).

In particular, we consider blocks of the post-change observations, each of
length $u\equiv u(\theta_{0},\theta_{1},T)$, starting at time $T_{j}=T+ju$ for
$j\geq 0$. Note that all these blocks are independent of each other (since
$P_{1}$ is a product distribution), and also independent of the event
$\mathcal{E}=\\{\theta_{0}\in C^{(1)}_{T}\\}$. Now, observe that for $k\geq
1$, we have $\\{\tau>T_{k}\\}=\cap_{j=1}^{k}\\{\tau>T_{j}\\}$, which
furthermore implies

$\displaystyle\\{\tau>T_{k}\\}\cap\mathcal{E}\subset\cap_{j=1}^{k}\\{C_{T_{j}}^{(T_{j-1})}\cap
C_{T}^{(1)}\neq\emptyset\\}\cap\mathcal{E}\subset\cap_{j=1}^{k}\\{C^{(T_{j-1})}_{T_{j}}\text{
miscovers }\theta_{1}\\}\cap\mathcal{E}.$ (23)

The last inclusion follows from the definition of $u\equiv
u(\theta_{0},\theta_{1},T)$, and the event $\mathcal{E}$. Using this, we
obtain:

$\displaystyle\sum_{t=T_{k}+1}^{T_{k+1}}\mathbb{P}_{T}\left(\tau\geq
t|\mathcal{E}\right)$
$\displaystyle\leq\sum_{t=T_{k}+1}^{T_{k+1}}\mathbb{P}_{T}\left(\tau\geq
T_{k}|\mathcal{E}\right)=u\mathbb{P}_{T}\left(\tau>T_{k}|\mathcal{E}\right)\leq
u\times\mathbb{P}_{T}\left(\cap_{j=1}^{k}\\{C^{(T_{j-1})}_{T_{j}}\text{
miscovers }\theta_{1}\\}\cap\mathcal{E}|\mathcal{E}\right)$ (24)
$\displaystyle\stackrel{{\scriptstyle(i)}}{{\leq}}\frac{u}{\mathbb{P}_{T}(\mathcal{E})}\times\prod_{j=1}^{k}\mathbb{P}_{T}\left(\\{C^{(T_{j-1})}_{T_{j}}\text{
miscovers
}\theta_{1}\\}\cap\mathcal{E}\right)\leq\frac{u\alpha^{k}}{1-\alpha}.$ (25)

The inequality $(i)$ uses the fact that $\mathcal{E}$ only depends on the pre-
change observations, and hence is independent of the post-change CSs. We now
fix some $k_{0}>1$, and observe that

$\displaystyle\mathbb{E}_{T}[(\tau-T)+]\leq
k_{0}u+\sum_{k=k_{0}}^{\infty}\frac{u\alpha^{k}}{1-\alpha}=u\left(k_{0}+\frac{\alpha^{k_{0}}}{1-\alpha}\right).$
(26)

By setting $k_{0}=\lceil\log(1/1-\alpha)/\log(1/\alpha)\rceil$, we get the
required statement for 1.1.

To prove the second part of Theorem 2.4, we proceed as above, considering
blocks of post-change observations of length $u\equiv
u(\Theta_{0},\theta_{1})$ as defined in (6). We then define $T_{j}=T+ju$ for
$j\geq 0$, and note that

$\displaystyle\\{\tau>T_{k}\\}\subset\cap_{j=1}^{k}\\{C^{(T_{j-1})}_{T_{j}}\cap\Theta_{0}\neq\emptyset\\}\subset\cap_{j=1}^{k}\\{C^{(T_{j-1})}_{T_{j}}\text{
miscovers }\theta_{1}\\}.$ (27)

The rest of the argument, then proceeds as above.

###  4.3 Proof of Proposition 2.6

Before presenting the proof of Proposition 2.6, we first recall some of
details of the betting CS proposed by Waudby-Smith and Ramdas (2023).

#### Background on betting CS.

Given observations $X_{1},X_{2},\ldots$ drawn from an independent process with
mean $\theta$, the betting CS is defined as

$\displaystyle C_{n}=\\{s\in[0,1]:W_{n}(s)<1/\alpha\\},\quad\text{with}\quad
W_{n}(s)\coloneqq\prod_{i=1}^{n}(1+\lambda_{t}(s)(X_{t}-s)),\quad\text{for all
}s\in[0,1],$ (28)

where $\\{\lambda_{t}(s):t\geq 1,s\in[0,1]\\}$ are predictable bets, taking
values in $[-1/(1-s),1/s]$. For certain betting strategies, such as the
_mixture method_ (Hazan, 2016, § 4.3), the _regret_ is logarithmic for all
$s$. In particular, this implies that

$\displaystyle\sup_{\lambda\in\left[\frac{-1}{1-s},\frac{1}{s}\right]}\sum_{t=1}^{n}\log\left(1+\lambda(X_{t}-s)\right)-\sum_{t=1}^{n}\log\left(1+\lambda_{t}(s)(X_{t}-s)\right)\leq
2\log n,\quad\text{for all }n\geq 1.$ (29)

We now present the details of the proof of Proposition 2.6. First we show that
under the condition that $T\geq 64\log(64/\Delta^{2}\alpha)/\Delta^{2}$, the
analysis of the first setting (i.e., with unknown $\Theta_{0}$) can be reduced
to the second case (with known $\Theta_{0}$). Then, we present the details of
the proof of the second setting.

#### Proof of (7).

Using the fact that $\log(1+x)\geq x-x^{2}/2$, we can further lower bound
$\log W_{n}(s)$ with

$\displaystyle\log W_{n}(s)$
$\displaystyle\geq\sup_{\lambda\in\left[\frac{-1}{1-s},\frac{1}{s}\right]}\sum_{t=1}^{n}\lambda(X_{t}-s)-\frac{\lambda^{2}}{2}(X_{t}-s)^{2}-\log(n^{2}).$
(30)

By setting the value of $\lambda$ to $\frac{1}{n}\sum_{t=1}^{n}X_{t}-m$, and
on simplifying, we can show that the betting CS after $n$ observation
satisfies $|C^{(1)}_{n}|\leq 4\sqrt{\log(n/\alpha)/n}$. This implies that for
$T\geq 64\log(64/\Delta^{2}\alpha)/\Delta^{2}$, the width of the CS starting
at time $1$ must be smaller than $\Delta/2=|\theta_{1}-\theta_{0}|/2$. If the
event $\mathcal{E}=\\{\theta_{0}\in C^{(1)}_{T}\\}$ happens (recall that this
is a probability $1-\alpha$ event), then we know that
$\theta_{0}\in\widetilde{\Theta}_{0}\coloneqq\\{\theta:|\theta-\theta_{0}|\leq\Delta/2\\}$.
This set $\widetilde{\Theta}_{0}$ plays the role of the known pre-change
parameter class in the analysis. Hence the rest of the proof to obtain the
upper bound stated in (7) proceeds exactly as in the case when the pre-change
distribution class is known, and we present the details for the latter case
next.

#### Proof of (9).

Since the proof of this result is long, we break it down into four simpler
steps.

_Step 1: Bound $(\tau-T)$ with the maximum of a class of stopping times
$(N_{\theta})_{\theta\in\Theta_{0}}$._ Introduce the stopping times
$N_{m}=\inf\\{n\geq m:C^{(m)}_{n}\cap\Theta_{0}=\emptyset\\}$, and note that

$\displaystyle\tau\leq\min_{m\geq 1}N_{m},\quad\text{which implies
that}\quad\mathbb{E}_{T}[\tau]\leq\min_{m\geq T+1}\mathbb{E}_{T}[N_{m}].$ (31)

Since the post change observations are assumed to be drawn i.i.d. from a
distribution with mean $\theta_{1}$, all the $N_{m}$ for $m\geq T+1$, have the
same distribution, and thus, the same expected value. Hence, it suffices to
get an upper bound on $N_{T+1}$.

To simplify the ensuing argument, we will use $C_{n}$ to denote
$C_{n+T+1}^{(T+1)}$ for $n\geq 1$. Furthermore, we also assume that
$\theta_{1}<\theta$ for all $\theta\in\Theta_{0}$, and
$\inf_{\theta\in\Theta_{0}}\theta=\theta_{1}+\Delta$. By the definition of
betting CS, the stopping time $N_{T+1}$ can be written as the supremum of a
collection of stopping times:

$\displaystyle
N_{T+1}-T=\sup_{\theta\in\Theta_{0}}N_{\theta},\quad\text{where}\quad
N_{\theta}=\inf\\{n\geq 1:W_{n}(\theta)\geq 1/\alpha\\}.$ (32)

_Step 2: Bound $(N_{\theta})_{\theta\in\Theta_{0}}$ with a monotonic class of
stopping times._ Next, we will upper bound each $N_{\theta}$ with another
stopping time $\gamma_{\theta}$, which have the property that
$\gamma_{\theta^{\prime}}<\gamma_{\theta}$ for $\theta^{\prime}>\theta$. In
particular, using the regret guarantee of the betting strategy, observe the
following:

$\displaystyle\log(W_{n}(\theta))$
$\displaystyle\geq\sup_{\lambda\in\left[\frac{-1}{1-\theta},\frac{1}{\theta}\right]}\sum_{t=1}^{n}\log(1+\lambda(X_{t}-\theta))-2\log
n$ (33)
$\displaystyle\geq\sup_{\lambda\in\left[0,\frac{1}{1-\theta}\right]}\sum_{t=1}^{n}\log(1+\lambda(\theta-
X_{t})-2\log n\coloneqq Z_{n}(\theta)-2\log n.$ (34)

Define a new stopping time $\gamma_{\theta}=\inf\\{n\geq 1:Z_{n}(\theta)-2\log
n\geq\log(1/\alpha)\\}$, and note that the above display implies
$\gamma_{\theta}\geq N_{\theta}$, and thus we have
$N_{T+1}-T\leq\sup_{\theta\in\Theta_{0}}\gamma_{\theta}$. We now show the
monotonicity of $\gamma_{\theta}$.

For any $\theta^{\prime}>\theta$, we have
$\lambda(\theta^{\prime}-X_{t})\geq\lambda(\theta-X_{t})$ for any $\lambda>0$,
which implies that
$\sum_{t=1}^{n}\log(1+\lambda(\theta^{\prime}-X_{t}))\geq\sum_{t=1}^{n}\log(1+\lambda(\theta-
X_{t}))$. Thus, we have the following relation (for
$\theta^{\prime}\geq\theta$):

$\displaystyle Z_{n}(\theta^{\prime})$
$\displaystyle=\sup_{\lambda\in\left[0,\frac{1}{1-\theta^{\prime}}\right]}\sum_{t=1}^{n}\log(1+\lambda(\theta^{\prime}-X_{t}))\geq\sup_{\lambda\in\left[0,\frac{1}{1-\theta}\right]}\sum_{t=1}^{n}\log(1+\lambda(\theta^{\prime}-X_{t}))$
(35)
$\displaystyle\geq\sup_{\lambda\in\left[0,\frac{1}{1-\theta}\right]}\sum_{t=1}^{n}\log(1+\lambda(\theta-
X_{t}))=Z_{n}(\theta).$ (36)

Thus, $Z_{n}(\theta^{\prime})\geq Z_{n}(\theta)$, which implies that
$\gamma_{\theta^{\prime}}\leq\gamma_{\theta}$, and in particular,
$\gamma_{\theta}\leq\gamma_{\theta_{1}+\Delta}$ for all $\theta\in\Theta_{0}$.
This leads to the required conclusion

$\displaystyle
N_{T+1}-T\leq\sup_{\theta\in\Theta_{0}}N_{\theta}\leq\sup_{\theta\in\Theta_{0}}\gamma_{\theta}\leq\gamma_{\theta_{1}+\Delta}.$
(37)

This is a crucial step, as it reduces the task of analyzing the supremum of a
large collection of stopping times, into that of analyzing a single stopping
time $\gamma_{\theta_{1}+\Delta}$.

_Step 3: Bound $\gamma_{\theta_{1}+\Delta}$ with the ‘oracle’ stopping time
$\rho^{*}$._ Let $\lambda^{*}\equiv\lambda^{*}(\theta_{1}+\Delta)$ denote the
log-optimal betting fraction, defined as
$\operatorname*{argmax}_{\lambda\in[0,1/(1-\theta_{1}-\Delta)}\mathbb{E}[\log(1+\lambda(\theta_{1}+\Delta-X))]$,
where $X$ is drawn from the post-change distribution. By definition then, we
have

$\displaystyle Z_{n}(\theta_{1}+\Delta)\geq
Z_{n}^{*}(\theta_{1}+\Delta)\coloneqq\sum_{t=1}^{n}\log(1+\lambda^{*}(\theta_{1}+\Delta-
X_{t})),$ (38)

which immediately implies

$\displaystyle\gamma_{\theta_{1}+\Delta}\leq\rho^{*}\coloneqq\inf\\{n\geq
1:Z_{n}^{*}(\theta_{1}+\Delta)\geq\log(n^{2}/\alpha)\\}.$ (39)

The stopping time $\rho^{*}$ is much easier to analyze as it is the first
crossing of the boundary $\log(n^{2}/\alpha)$ by the random walk
$Z_{n}^{*}(\theta_{1}+\Delta)$ with i.i.d. increments.

_Step 4: Evaluate the expectation of $\rho^{*}$._ Observe that
$Z_{n}^{*}\equiv Z_{n}^{*}(\theta_{1}+\Delta)=\sum_{t=1}^{n}V_{t}$, with
$V_{t}=\log(1+\lambda^{*}(\theta_{1}+\Delta-X_{t}))$. Without loss of
generality, we can assume that $\lambda^{*}<1/(1-\theta_{1}-\Delta)$ (if not,
we simply repeat the argument with $\lambda^{*}-\epsilon$ for an arbitrarily
small $\epsilon>0$), and hence $(V_{t})_{t\geq 1}$ are i.i.d. and bounded
increments, which means that $\mathbb{E}[V_{t}]<\infty$. In fact, by the dual
definition of the information projections (Honda and Takemura, 2010), we have
$\mathbb{E}[V_{t}]=K_{2}\equiv K_{2}(P_{1},\Theta_{0})$. Next, with
$n_{0}\coloneqq\inf\\{n\geq 1:\log(n^{2}/\alpha)/n<K_{2}/2\\}$, we have for
$n\geq n_{0}$:

$\displaystyle\mathbb{P}\left(\rho^{*}>n\right)\leq\mathbb{P}\left(\frac{1}{n}\sum_{t=1}^{n}V_{t}-K_{2}\leq-\frac{K_{2}}{2}\right)\leq\exp\left(-c^{\prime\prime}n\right),$
(40)

for some $c^{\prime\prime}>0$. Hence, the expectation of $\rho^{*}$ satisfies

$\displaystyle\mathbb{E}[\rho^{*}]=\sum_{n\geq
0}\mathbb{P}\left(\rho^{*}>n\right)\leq n_{0}+\sum_{n\geq
n_{0}}\exp\left(-c^{\prime\prime}n\right)=n_{0}+\frac{e^{-c^{\prime\prime}n_{0}}}{1-e^{-c^{\prime\prime}}}<\infty.$
(41)

Thus, both $\rho^{*}$ and $(V_{t})_{t\geq 1}$ have bounded expectations, and
we can appeal to Wald’s lemma (Durrett, 2019, Theorem 2.6.2) to obtain
$\mathbb{E}[Z^{*}_{\rho^{*}}]=\mathbb{E}[\rho^{*}]K_{2}$. Furthermore, by the
definition of $\rho^{*}$, and the boundedness of $(V_{t})_{t\geq 1}$, we can
upper bound $\mathbb{E}[Z_{\rho^{*}}^{*}]$ with
$\log(1/\alpha)+2\log(\mathbb{E}_{T}[\rho^{*}])+c^{\prime}$, where
$c^{\prime}=\max\\{\log(1+\lambda^{*}_{\theta_{1}+\Delta}),\,\log(1-\lambda^{*}_{\theta_{1}+\Delta})\\}$.
In other words, we have

$\displaystyle\mathbb{E}[\rho^{*}]\leq\frac{\log(1/\alpha)+2\log(\mathbb{E}[\rho^{*}])+c^{\prime}}{K_{2}},\quad\text{which
implies}\quad\mathbb{E}[\rho^{*}]=\mathcal{O}\left(\frac{\log(1/\alpha
K_{2})}{K_{2}}\right).$ (42)

This completes the proof.

##  5 Conclusion

In this paper, we proposed a changepoint detection scheme that constructs a
new CS with every observation, and declares a detection as soon as the
intersection of the active CSs becomes empty. The design of our scheme was
motivated by the BCS-Detector of Shekhar and Ramdas (2023), which proceeds by
initializing new “backward CSs” with each new observation. We showed that our
new scheme matches the detection delay performance of BCS-Detector, while
improving the ARL lower bound by a factor of $2$. Furthermore, our scheme
achieves this improvement under weaker model assumptions (i.e., without
needing the ability to construct CSs in both forward and backward directions).
Interestingly, our proposed scheme can be seen as a nonparametric
generalization of Lorden’s reduction from SCD to repeated sequential testing,
due to the duality between sequential testing and CSs.

## References

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# X-ray Studies of Exoplanets:
A 2020 Decadal Survey White Paper

Scott J. Wolk1*, Jeremy J. Drake1, Graziella Branduardi-Raymont2, Katja
Poppenhaeger3,4, Vladimir Airapetian5, Kevin France 6, Salvatore Sciortino7,
Ignazio Pillitteri7, Rachel A. Osten8,9, Carey M. Lisse9, Vinay Kashyap1, Brad
Wargelin1, Brian Wood10, Willaim Dunn1, David Principe11, Moritz Günther11,
Damian J. Christian12, Julián David Alvarado-Gómez1, Chuanfei Dong13, Lidia
Oskinova3, Margarita Karovska1, Sofia P. Moschou1, Peter K. Williams1, Randall
Smith1, Bradford Snios1, Elena Gallo14, William Danchi15, John P. Pye16, Joel
Kastner17, Jose Dias Do Nascimento1,18, Jae-Sub Hong19

1Smithsonian Astrophysical Observatory, 2University College London, 3Queen’s
University Belfast, United Kingdom, 4University of Potsdam, Germany,
5NASA/GSFC & American University, 6JILA, University of Colorado,
7INAF–Osservatorio Astronomico di Palermo, 8Space Telescope Science Institute,
9Johns Hopkins University, 10Naval Research Laboratory, 11Massachusetts
Institute of Technology, 12California State University Northridge,
13Department of Astrophysical Sciences, Princeton University, 14University of
Michigan, 15NASA Goddard Space Flight Center, 16University of Leicester,
United Kingdom, 17Rochester Institute of Technology, 18Univ. Federal do Rio G.
do Norte, UFRN, Brazil, 19Harvard University *Corresponding author

(A White Paper submitted to the Astro2020 decadal survey committee, National
Academies of Sciences, Engineering and Medicine
March 5, 2019)

Over the last two decades, the discovery of exoplanets has fundamentally
changed our perception of the universe and humanity’s place within it. Recent
work indicates that a solar system’s X-ray and high energy particle
environment is of fundamental importance to the formation and development of
the atmospheres of close-in planets such as hot Jupiters, and Earth-like
planets around M stars. X-ray imaging and spectroscopy provide powerful and
unique windows into the high energy flux that an exoplanet experiences, and
X-ray photons also serve as proxies for potentially transfigurative coronal
mass ejections. Finally, if the host star is a bright enough X-ray source,
transit measurements akin to those in the optical and infrared are possible
and allow for direct characterization of the upper atmospheres of exoplanets.
In this brief white paper, we discuss contributions to the study of exoplanets
and their environs which can be made by X-ray data of increasingly high
quality that are achievable in the next 10–15 years.1[] Over the last two
decades, the discovery of exoplanets has fundamentally changed our perception
of the universe and humanity’s place within it. Recent work indicates that a
solar system’s X-ray and high energy particle environment is of fundamental
importance to the formation and development of the atmospheres of close-in
planets such as hot Jupiters, and Earth-like planets around M stars. X-ray
imaging and spectroscopy provide powerful and unique windows into the high
energy flux that an exoplanet experiences, and X-ray photons also serve as
proxies for potentially transfigurative coronal mass ejections. Finally, if
the host star is a bright enough X-ray source, transit measurements akin to
those in the optical and infrared are possible and allow for direct
characterization of the upper atmospheres of exoplanets. In this brief white
paper, we discuss contributions to the study of exoplanets and their environs
which can be made by X-ray data of increasingly high quality that are
achievable in the next 10–15 years.1

##  1 High Energy Photons in the Context of Exoplanets

The field of exoplanetary science is in transition from the discovery phase
which marked its first two and a half decades to the detailed characterization
of exoplanets hosted by stars across the Hertzprung-Russell (HR) diagram.
While some atomic species such as sodium have been detected in particular
exoplanets, measurements of oxygen, ozone, and carbon dioxide, currently in
their infancy, will transform the science in the coming decades. Part of that
characterization is necessary for the stellar ecosystem where the planet
lives. A star does not end at its photosphere; although more tenuous, its
substance extends to surrounding high temperature coronal plasma, high energy
flares, closed and open magnetic fields, stellar winds, and coronal mass
ejections (CMEs). In a very real sense, planets exist in the upper atmospheres
of stars.

Planetary characteristics, including habitability, are determined by much more
than the mass of the exoplanet and the bolometric flux of the host star.
Recent studies of atmospheric escape due to thermal and non-thermal processes
concluded that close-in planets would lose mass proportional to the incident
stellar X-ray and ultraviolet (XUV) flux (e.g. Lammer et al. 2003, Erkaev et
al. 2007, Sanz-Forcada et al. 2010). The effective diameter of planets is
larger at XUV wavelengths compared to optical wavelengths due to the
absorption of low density gas at XUV energies. Further, Roche lobe overflow
plays a critical role in mass loss and the related Hill (tidal) radius is
proportional to the -3 power of the orbital distance. Analogously, magnetic
interactions are proportional to the -3 power within the Alfvén radius,
varying as the inverse square relation further out. The result is that close-
in systems experience stronger influence by high energy particles and photons
than naïvely expected from scalings based simply on the stellar optical
luminosity.

Figure 1:  Left: distribution of planet radii and orbital periods. Right: same
as left but with insolation flux relative to Earth on the horizontal axis. The
dearth of exoplanets with periods less 3 days and radii between 2 and 20 Earth
masses (the “exoplanet evaporation desert”) is attributed to atmospheric
losses due to high energy radiation (Fulton & Petigura 2018).

High-energy photons likely drive the “exoplanet evaporation desert” (Figure
1). In one recent example, XMM-Newton observations indicate WASP-19b could
have already lost roughly half its current total mass in its $\sim$2.2 Gyr
life due to incident X-ray flux (Lalitha et al. 2018). Statistically this
effect has been shown by Kepler results (Fulton & Petigura 2018) and described
theoretically (Lopez & Fortney 2013, Owen & Wu 2017, Owen & Lai 2018).

Much of the current and near term search for habitable zone planets has
focused on M stars with relatively close-in planets. This includes tantalizing
cases such as TRAPPIST-1 (Gillon et al. 2017). Close-in planets lie in unique
environments and the impact of the X-rays from the host star changes the
evolution of the system. The effects can work several ways:
(1) the intense high energy flux heats the upper atmosphere of an exoplanet
via photoionization,
(2) the angular momentum and magnetic field of the planet can induce more
activity on the star, thereby leading to non-linear enhancements in the rates
of heating, ionization and various atmospheric chemical processes,
(3) chemical reactions are also driven by particles absorbed in the upper
atmosphere, and heating and dissociative reactions can lead to erosion and
evaporation of the planetary atmospheres, particularly for close-in planets
(Lammer et al. 2003, Yelle 2004, Tian et al. 2005, Murray-Clay et al. 2009,
Airapetian et al. 2016, Dong et al. 2018).

In fact, atmospheric escape has already been observed in HD 189733b, HD
209458b, WASP-12b and GJ436b (Vidal-Madjar et al.2003, Linsky et al. 2010,
Lecavelier Des Etangs et al. 2010, Fossati et al. 2010, Kulow et al. 2014,
Ehrenreich et al. 2015). The kinetic impact of stellar coronal mass ejections
(CMEs) and winds can be even more devastating, and penetrating particles can
directly damage organics at the cellular level. For a full understanding of
exoplanet atmospheres, their evolution, and planetary habitability, it is
therefore crucial that we characterize the host stars’ high energy emission.

In the coming decades, we may be able to utilize X-rays to characterize
exoplanets as we do planets in our own solar system. Planetary auroral X-ray
and radio emission can be used as a probe for difficult to detect planetary
magnetic fields, which are thought to be vital for protecting a planet’s
atmosphere from high energy particle stripping. Meanwhile, charge exchange
between stellar winds and planetary atmospheres can be used as a probe of
atmospheres and the interplanetary medium (Wargelin & Drake 2001, Cravens et
al. 2001, Bhardwaj et al. 2007). Advances in our understanding will come from
a combination of statistical studies and focused investigations of carefully
selected interesting systems. These will be chosen from the thousands of
extra-solar planets discovered and explored by the CoRoT, Kepler, TESS,
CHEOPS, ARIEL, and PLATO space missions, while Gaia will help to identify the
young, highly eccentric systems best suited to test models of UV and X-ray
irradiation effects as a function of star-planet separation and orbital phase.

##  2 Characterization of the High Energy Stellar Flux of Exoplanet Hosts

Stellar coronal activity and flaring are ubiquitous in solar-type and lower-
mass stars, and X-ray observations can be used to determine the temperature,
metallicity, and electron density of coronal plasma. Previous studies indicate
significant secular decrease of stellar XUV flux with age. Enhanced X-ray
emission and higher temperatures during early epochs means that X-rays at
early times will dissociate and ionize molecules in planetary thermospheres
and exospheres more readily (Güdel 2007; Penz, Micela & Lammer 2008). It is
not clear if this has a deleterious result or is a necessary mechanism of
clearing extremely dense proto-atmospheres.

Additional energy available due to star–planet interaction (SPI) is expected
to roughly scale with the strength of the individual magnetic fields ($B_{*}$
and $B_{P}$), the relative velocity of the star and planet ($v_{rel}$), and
their separation ($a$) according to $B_{*}B_{P}v_{rel}a^{-n}$ (Cuntz et al.
2000, Lanza 2009, 2012, Iro & Demming 2010). While the exponential $n$ is
$\sim$2 in the open-field region of the stellar wind, $n$ is $\sim$ 3 when
close to the star where the field is dipole-like. It is possible that such an
effect could be detected by statistical studies of large well chosen samples.
Indeed this was attempted using ROSAT All Sky Survey (RASS) data; a weak
dependence between X-ray flux and semi-major axis was noted (Kashyap et al.
2008). More recent work finds some evidence of measurable X-ray flux excesses
in close-in systems, but this is dominated by a few extreme systems where SPI
seems to play a significant role (Miller et al. 2015). Other phenomena related
to SPI include stellar spin-up effects (Poppenhaeger & Wolk 2014), dynamo
quenching (or obscuration) (Pillitteri et al. 2014, Fossati et al. 2018), and
coronal abundance alterations (Wood et al. 2018).

Current studies are severely constrained by instrumental limitations. Chandra
has limited effective area, and XMM-Newton has modest spatial resolution,
reducing its ability to study young stars in clusters. Both facilities use Si-
based X-ray detectors with imaging spectral resolution of $R=\frac{E}{\Delta
E}\approx 50$. In addition to surveys, extended monitoring of specific
interesting sources is needed to understand the range of conditions planets
may be subjected to; impulsive events are particularly easy to miss in
surveys.

While major observational advances will require a combination of more
sensitive detectors and larger collecting area, substantial progress in a few
areas can be made with focused missions using current technology. Surveys of
older, more isolated stars could benefit substantially from modest spatial
resolution X-ray instruments. The RASS revealed about 50,000 normal stars with
fluxes above $2\times 10^{-13}$erg cm-2s-1 in the soft X-ray band. The energy
resolution of modern detectors is $R\approx 100$, making them much less
sensitive to panchromatic background noise. Short observations of a few hours
could measure a luminosity and temperature for an exoplanet host star in the
RASS.

For the brightest X-ray sources, the effective area requirement is low and it
is possible to follow-up with specialized low cost small X-ray satellites
which can monitor exoplanet hosts for flares (and transits). Specifically,
Pillitteri et al. (2011, 2014) have suggested that some of the activity
observed on HD 189733A is tied to the planetary orbital period and not the
stellar rotation period. Similarly, Maggio et al. (2015) detected flaring in
HD 17156 at the periastron passage of its highly eccentric exoplanet.
Currently, reproducing such observations requires devoting scarce "Great
Observatory" scale resources to monitoring for a stochastic signal, which then
needs to be compared to an extended baseline to understand its significance.
Given a focused long term monitoring program and using a devoted low cost
mission it will be possible to confirm the relationship between the synoptic
flares and differentiate them from intrinsic stellar events, primarily on the
basis of timing.

Further advances in distinguishing activity induced by star–planet
interactions from intrinsic stellar activity, will require more effective area
coupled with higher spectral resolution (R$\sim$ 500–2000). This can be done
using either micro-calorimeters or dispersive optics. Such an experiment would
be able to collect more photons in less time and thus spectrally resolve
ephemeral events. With this moderate spectral resolution, coupled with
collecting area on the order of 1000-10,000 cm2, we will be able to spectrally
resolve flares on the time scale of minutes. Work on young stars and planets,
which are predicted to be the most affected by such events, will require
excellent spatial resolution as well; to perform background limited studies of
individual sources in the densest regions of even the closest massive young
star forming cluster (Orion), spatial resolution on the order of 1′′ would be
required.

Strong X-ray flares on the Sun are usually accompanied by the ejection of
cooler material (roughly 10,000 K) that had previously been confined by
magnetic fields near the solar surface. These coronal mass ejections (CMEs)
may also contain high energy protons accelerated in the flare and CME shock
front. CMEs differ from the quasi-steady solar wind in two respects: they are
orders of magnitude denser, and are spatially confined. CMEs have been
theorized to induce exotic chemical reactions in planetary atmospheres which
may be required for formation of complex molecular compounds, such as amino
acids (e.g., Airapetian et al. 2016, Lingam et al. 2018). Conversely, CMEs are
a threat to exoplanet habitability because their impacts can, over time, strip
off a significant fraction of a planet’s atmosphere, especially its ozone.
With similar collecting area but even higher resolution spectroscopy (R$\sim$
5000), X-ray astronomers could routinely and definitively observe the tell-
tale Doppler shifts of CMEs or their coronal compression waves and measure
their physical properties, including velocity, temperature, mass, and energy.

##  3 Transits

Remarkable progress has been made in studying the atmospheres of hot Jupiters
by observing transits in the optical and IR. Additional and unique insights
can be obtained via X-ray transits, although the much lower photon fluxes and
greater intrinsic variability mean that repeated observations will be
necessary. In the UV, higher energy measurements allow access to layers of the
atmosphere lying above the optical photosphere. In the case of UV this is
limited to about 10% above the optical cloud tops. X-rays are absorbed at far
lower densities and hence can resolve details up to twice the altitude of the
optical clouds (Vidal-Madjar et al. 2003, Linsky et al. 2010). Lecavelier des
Etangs et al. (2012) ascribe their detection of Ly${\alpha}$ transits to X-ray
flares, suggesting a role for X-ray observations in coordinated multiwavlength
observations with JWST and ALMA. More recently, Lavie et al. (2017) have
detected an enlarged asymmetric exosphere around the hot Neptune GJ 436b.

Figure 2: Left: Representation of the relative depth of light curves observed
at various energies due to different absorption cross-sections. Right:
Simulation of detection of the 0.5 keV oxygen absorption edge betraying
enhanced O abundance for multiple transits of a super-earth planet around an M
dwarf (by K. Poppenhaeger).

Using Chandra, Poppenhaeger et al. (2013) monitored multiple transits of HD
189733b. After excluding flares and co-adding the remaining lightcurves, they
were able to measure the transit width and its depth (7%), much deeper than
the $\approx\leavevmode\nobreak\ 2\%$ optical eclipse depth. This implied a
soft X-ray ($<$1 keV) planet radius of about 1.75 times the optical size. They
identified the source of the additional obscuration as due to C,N,O in a
spherically symmetric exosphere with a density of about $10^{11}$ cm-3 and a
temperature of about 20,000K. Future X-ray observations could potentially
measure asymmetries in transit lightcurves, pointing to stellar wind induced
atmospheric depletion, comet-like sublimation tails, or bow shocks around
planets. Using hot-Jupiter observations and modeling as a guide, one could
then constrain the loss of atmospheres in exoEarths around M-stars (Figure 2).

The more photons that arrive during the transit the greater the number of
samples along the light curve and the higher precision that is possible in
each individual sample or temporal bin. Statistically, we expect a few dozen
TESS-revealed exoplanet hosts to be detected in the RASS, which has a limiting
flux of $\approx$ 2 $\times 10^{-13}$ erg cm-2 sec-1. For these sources, about
10,000 cm2 is required to achieve a rate of 1 cnt sec-1. For a single time bin
to have a precision of 10% would require a minimum of 100 counts, or 104
counts for a precision of 1%.

The typical transit duration for a close-in system is 2–3 hours. In principle,
a single transit could reveal a small fluctuation, but in practice, repeated
observations are needed because of fluctuations of active regions on the
stellar surface (c.f. Llama & Shkolnik 2015). In addition, while the detection
of a small dip due to a transit is useful, the key science of atmospheric
modeling comes from the ability to measure the shape of the light curve.
Multiple transit observations allow the observer to correct for the underlying
variations and allow finer time bins. Given the 1000-10,000 cm2 described
above, 5-10 transit observations would allow for the creation of a high
resolution light curve with error bars approaching a few percent. If
dispersive gratings were used various ionic species could be sorted by
altitude (Figure 2). Alternatively, a dedicated small telescope with an
effective area on the order 100 cm2 could measure exospheric struture by co-
adding 50-100 transits.

Conversely, given greater collecting area ($\approx 20,000\leavevmode\nobreak\
$cm2) and better spatial resolution (to reduce noise, especially from cosmic
rays and secondary astrophysical sources in the field), the coming decades
will allow us to further investigate the vertical distribution of atmospheric
species. Extremely high resolution maps of individual planetary atmospheres
are possible. The transverse velocity of a close-in exoplanet is around 150
km/s. Poppenhaeger & Wolk (2014) found the X-ray diameter of HD 189733b at
around 800 eV is about 1.75 times the optical diameter or nearly 250,000 km.
The diameter should be even larger than this at lower energies near the carbon
edge ($\approx 300$ eV) where the opacity per particle is much higher. By
combining photons from multiple transits and concentrating on photons within
narrow energy bands corresponding to (e.g.) oxygen or carbon it will be
possible to map structures on scales perhaps 5–10% the planetary diameter.

##  4 Summary

The coming decades will open a new opportunity of exploring the habitability
of other worlds. X-ray observations of exoplanet hosts will be critical to
this understanding. Devoted X-ray instruments of modest size and resolution
will be able to contribute. They will be especially useful in monitoring and
transit studies of bright exoplanets and their hosts. However, the most vital
scientific questions will require large effective area and spatial and
spectral resolutions sufficient to distinguish stars, elemental species, and
dynamic structures at a variety of scales.

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