Second- and Third-Order Asymptotics of the Continuous-Time Poisson Channel

TL;DR

This paper establishes the optimal second-order coding rate for the continuous-time Poisson channel and provides bounds on the third-order rate, marking the first such result for a continuous-time channel.

Contribution

It introduces the first second-order asymptotics for a continuous-time channel and develops novel techniques for the converse and achievability proofs.

Findings

Optimal second-order coding rate derived

Bounds on third-order coding rate obtained

First second-order result for a continuous-time channel

Abstract

The paper derives the optimal second-order coding rate for the continuous-time Poisson channel. We also obtain bounds on the third-order coding rate. This is the first instance of a second-order result for a continuous-time channel. The converse proof hinges on a novel construction of an output distribution induced by Wyner's discretized channel and the construction of an appropriate $\epsilon$-net of the input probability simplex. While the achievability proof follows the general program to prove the third-order term for non-singular discrete memoryless channels put forth by Polyanskiy, several non-standard techniques -- such as new definitions and bounds on the probabilities of typical sets using logarithmic Sobolev inequalities -- are employed to handle the continuous nature of the channel.

Equations

$$\mathbb{P}\{\nu (0)=0\}$$

$$\mathbb{P}\{\nu (t+\tau )-\nu (t)=k\}$$

$$\displaystyle\Lambda=\Lambda(\lambda,\lambda_{0},t,\tau)\coloneqq\int_{t}^{t+\tau}\Big{(}\lambda(u)+\lambda_{0}\Big{)}\,\mathrm{d}u.$$

$$\frac{1}{T}{\int }_0^T\lambda (t)\mathrm{d}t\le \sigma A.$$

$$\frac{1}{M}\sum _{m=1}^M\mathbb{P}\{\psi ({\nu }_0^T)=m\mid \lambda =\varphi (m)\}\ge 1-\epsilon .$$

$$\log M_{\mathrm{avg}}^{\ast }({\lambda }_0,T,A,\sigma ,\epsilon )=T{C}^{\ast }+\mathrm{o}(T)(\mathrm{as}T\to \infty ),$$

$$C^{\ast }$$

$$\displaystyle\coloneqq A\,\bigg{(}(1-p^{\ast})\,s\log\frac{s}{p^{\ast}+s}+p^{\ast}\,(1+s)\log\frac{1+s}{p^{\ast}+s}\bigg{)},$$

$$s=s({\lambda }_0,A)$$

$$p^{\ast }=p^{\ast }({\lambda }_0,A,\sigma )$$

$$p_0=p_0({\lambda }_0,A)$$

$$\log M_{\mathrm{avg}}^{\ast }({\lambda }_0,T,A,\sigma ,\epsilon )=T{C}^{\ast }+\sqrt{T{V}^{\ast }}{\Phi }^{-1}(\epsilon )+{\rho }_T,$$

$$V^{\ast }$$

$$\displaystyle\coloneqq A\,\bigg{(}(1-p^{\ast})\,s\log^{2}\frac{s}{p^{\ast}+s}+p^{\ast}\,(1+s)\log^{2}\frac{1+s}{p^{\ast}+s}\bigg{)},$$

$$\Phi (u):=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^u{\mathrm{e}}^{-t^2/2}\mathrm{d}t.$$

$$\frac{1}{2}\log T-c_1\le {\rho }_T\le \log T+c_2$$

$$D_{\mathrm{s}}^{ϵ}(P\Vert Q)$$

$$D(P\Vert Q)$$

$$V(P\Vert Q)$$

$$\Xi (P\Vert Q)$$

$$D(W\Vert Q\mid P)$$

$$V(W\Vert Q\mid P)$$

$$\Xi (W\Vert Q\mid P)$$

$$(P\times W)(x,y):=P(x)W(y\mid x)$$

$$I(P,W)$$

$$V(P,W)$$

$$\tilde{V}(P,W)$$

$$\Xi (P,W)$$

$$\tilde{\Xi }(P,W)$$

$$PW(y):=\sum _{x^{\prime }\in \mathcal{X}}(P\times W)(x^{\prime },y)$$

$$W_{\Delta }(1\mid x):=\{\begin{array}{ll}sA\Delta {\mathrm{e}}^{-sA\Delta }& \mathrm{if}x=0,\\ (1+s)A\Delta {\mathrm{e}}^{-(1+s)A\Delta }& \mathrm{if}x=1,\end{array}$$

$$W_n:=W_{{\Delta }_n}$$

$${\Delta }_n:=\frac{T}{n}.$$

$$a_n$$

$$b_n$$

$$\frac{1}{n}\sum _{i=1}^n{x}_i=:P_{\mathbf{x}}(1)\le \sigma ,$$

$$\frac{1}{M}\sum _{m=1}^M{W}_n^n({\psi }^{-1}(m)\mid \varphi (m))\ge 1-\epsilon ,$$

$$W^n(\mathbf{y}\mid \mathbf{x}):=\prod _{i=1}^{n}W(y_i\mid x_i)$$

Full text

I Introduction

This study explores fundamental limits in Poisson communication theory [2] from the perspective of fixed-error probability asymptotics [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The continuous-time Poisson channel—simply referred to as the Poisson channel—is a canonical optical direct-detection communication model [13, 14]. The output of the Poisson channel is a Poisson counting process whose intensity is determined by the sum of a dark current noise and an input waveform subject to peak and average power constraints; for more details refer to Section II of this paper. Kabanov [15] derived the capacity, i.e., the optimal first-order coding rate, of the Poisson channel in the absence of an average power constraint. Davis [16] generalized Kabanov’s capacity formula with an average power constraint.

While Kabanov’s [15] and Davis’ [16] proofs involved martingale techniques, Wyner [17] provided an alternative proof based on a discretization technique. In particular, Wyner discretized the Poisson channel into a binary-input binary-output discrete memoryless channel. He then applied elementary information-theoretic techniques to analyze the capacity of this channel and related the capacity to that of the original Poisson channel. Using the same discretization argument, Wyner [17] derived the error exponent [18] of the Poisson channel for all rates below capacity.

Recently, to understand the finite blocklength performance of channel coding, refinements of asymptotic estimates on optimal coding rates with fixed error probability have gained increasing traction [3, 4, 5, 6, 7, 8, 10]. In this paper, we derive the optimal second-order coding rate for the Poisson channel. We also show an upper and a lower bounds on the optimal third-order coding rate. This is the first instance of a conclusive second-order result in continuous-time communications in information theory.

While martingale-based techniques for Poisson channels enjoy several advantages [19], our proof techniques are inspired by Wyner’s discretization argument [17]. In the converse part, particularly in the application of the meta-converse [5] to the discretized channel, we construct a somewhat artificial output distribution induced by Wyner’s discretized channel and the construction of an appropriate $\epsilon$-net of the input probability simplex. This construction differs from existing constructions in the literature [4, 7, 9] and appears to be essential in handling the continuous nature of the channel model. In the achievability part, we adapt Polyanskiy’s technique to prove the third-order asymptotics for non-singular111A channel $W:\mathcal{X}\to\mathcal{Y}$ is said to be singular if $W(y\mid x)\,W(y\mid z)>0$ implies that $W(y\mid x)=W(y\mid z)$ (cf. [8, Section 4.2.1]). A channel is said to be non-singular if it is not singular. discrete memoryless channels (DMCs) [6, Section 3.4.5] to our scenario. A straightforward generalization of the random coding union bound [5, Theorem 16] and a judicious choice of the input distribution inspired by the delta-convention [20, Convention 2.11] are proposed in view of the additive cost constraint imposed on the discretized channel. In addition, to bound a certain probability that results from the continuous nature of the Poisson channel and certain properties of its discretized version, a modified logarithmic Sobolev inequality (cf. [21, Chapter 6]) rather than Hoeffding’s inequality is employed.

The rest of this paper is organized as follows: Section II introduces the formal definition of the Poisson channel, its classical channel capacity [15, 16, 17] is stated in Theorem 1, and our second-order asymptotics for the Poisson channel—i.e., the main result of this study—is presented in Theorem 2 together with certain bounds on the third-order coding rate. Section III describes the proof of Theorem 2. This section is partitioned into various subsections. In Section III-A, we introduce the basic notations in the the study of second- and third-order asymptotics. In Section III-B, we revisit Wyner’s discretization argument [17]. The proofs of the converse and achievability parts of Theorem 2 are provided in Sections III-C and III-D, respectively. Section IV discusses technical novelties and contributions of our proofs; readers may benefit from reading through the discussion in Section IV before perusing the proofs.

II Continuous-Time Poisson Channel and Its Second- and Third-Order Asymptotics

We now introduce the mathematical model of the continuous-time Poisson channel. An input of the Poisson channel is an integrable function $\lambda:[0,\infty)\to[0,\infty)$ called a waveform. Given a waveform $\lambda$ and a constant $\lambda_{0}\geq 0$ called the dark current, the output of the Poisson channel is modeled by a Poisson counting process $\{\nu(t)\}_{t\geq 0}$ of intensity $\lambda(t)+\lambda_{0}$, i.e.,

[TABLE]

for every $0\leq t,\tau<\infty$ and every $k\in\mathbb{N}\cup\{0\}$, where

[TABLE]

Here, the Poisson counting process $\{\nu(t)\}_{t\geq 0}$ is defined as a random mapping $\nu:[0,\infty)\to\mathbb{N}\cup\{0\}$. For the sake of brevity, we denote the random mapping $\nu:[0,T]\to\mathbb{N}\cup\{0\}$ as $\nu_{0}^{T}$.

Let $T>0$, $A>0$, and $0<\sigma\leq 1$ be three constants. Denote by $\mathcal{S}(T)$ the collection of non-decreasing functions $g:[0,T]\to\mathbb{N}\cup\{0\}$ satisfying $g(0)=0$. In addition, denote by $\mathcal{W}(T,A,\sigma)$ the collection of waveforms $\lambda:[0,T]\to[0,A]$ satisfying the average power constraint

[TABLE]

Then, a channel code for the Poisson channel with dark current $\lambda_{0}$ can be defined as follows:

Definition 1**.**

Given an integer $M\geq 1$ and a real number $0\leq\varepsilon\leq 1$, a pair of encoder $\phi:\{1,\dots,M\}\to\mathcal{W}(T,A,\sigma)$ and decoder $\psi:\mathcal{S}(T)\to\{1,\dots,M\}$ is called a $(T,M,A,\sigma,\varepsilon)_{\mathrm{avg}}$-code if

[TABLE]

Here, $\lambda$ stands for the $\mathcal{W}(T,A,\sigma)$-valued random variable (r.v.) induced by the encoder $\phi$ and the uniformly distributed messages on $\{1,\dots,M\}$.

For $0<\varepsilon<1$, denote by $M_{\mathrm{avg}}^{\ast}(\lambda_{0},T,A,\sigma,\varepsilon)$ the maximum integer $M$ such that a $(T,M,A,\sigma,\varepsilon)_{\mathrm{avg}}$-code exists for the Poisson channel with dark current $\lambda_{0}$. Assume throughout this paper that all logarithms are the natural logarithm.

Theorem 1** ([15, 16, 17]).**

It holds that

[TABLE]

where the Poisson channel capacity $C^{\ast}$ is given by

[TABLE]

and three numbers $s$, $p^{\ast}$, and $p_{0}$ are given by

[TABLE]

Our goal is to refine the $+\mathrm{o}(T)$ term in (6). We do so via Wyner’s discretization argument [17, Section II in Part I] and finite blocklength analyses. The following theorem exactly characterizes the optimal second-order coding rate of the Poisson channel together with bounds on the third-order coding rate.

Theorem 2**.**

It holds that

[TABLE]

where $\rho_{T}=\mathrm{O}(\log T)$ as $T\to\infty$, the Poisson channel dispersion $V^{\ast}$ is given by

[TABLE]

and $\Phi^{-1}(\cdot)$ stands for the inverse function of the standard Gaussian cumulative distribution function

[TABLE]

More precisely, there exist positive constants $c_{1}$ and $c_{2}$ satisfying

[TABLE]

for sufficiently large $T$.

The converse and achievability parts of Theorem 2 are proved in Sections III-C and III-D, respectively. The technical contributions and novelties in the proofs of the converse and achievability parts will be discussed in detail in Sections IV-A and IV-B, respectively.

III Proof of Theorem 2

III-A Notations for Second-Order Asymptotic Analysis

Given two discrete distributions $P$ and $Q$ on the same space, define the following four divergences:

[TABLE]

where $Z$ is a r.v. satisfying222The notation $\mathbb{P}\circ Z^{-1}$ stands for the probability distribution induced by the r.v. $Z$. $\mathbb{P}\circ Z^{-1}=P$. Moreover, given countable alphabets $\mathcal{X}$ and $\mathcal{Y}$, a distribution $P$ on $\mathcal{X}$, a channel $W:\mathcal{X}\to\mathcal{Y}$, and a distribution $Q$ on $\mathcal{Y}$, define the following three conditional divergences:

[TABLE]

where $(X,Y)$ is a pair of r.v.’s satisfying $\mathbb{P}\circ(X,Y)^{-1}=P\times W$, and $P\times W$ stands for the joint distribution on $\mathcal{X}\times\mathcal{Y}$ defined as

[TABLE]

for each $(x,y)\in\mathcal{X}\times\mathcal{Y}$. In particular, we write

[TABLE]

for a distribution $P$ on $\mathcal{X}$ and a channel $W:\mathcal{X}\to\mathcal{Y}$, where $PW$ stands for the output distribution on $\mathcal{Y}$ defined as

[TABLE]

for each $y\in\mathcal{Y}$.

III-B Discretization of the Poisson Channel

Our analyses hinge on Wyner’s ad hoc assumption [17, Section II in Part I]. At a high level, this assumption says that the performance of the original channel is roughly equivalent to a discretized version of the Poisson channel. We now introduce the discretized version of the Poisson channel as follows: For $\Delta>0$, define the binary asymmetric channel $W_{\Delta}:\{0,1\}\to\{0,1\}$ by333If the dark current of the Poisson channel is zero, i.e., $\lambda_{0}=0$, then the channel $W_{\Delta}:\{0,1\}\to\{0,1\}$ is a Z-channel.

[TABLE]

where $s$ is defined in (8). In particular, we define

[TABLE]

for each $n\geq 1$, where the number $\Delta_{n}>0$ is given by

[TABLE]

It is clear that $W_{n}(\cdot\mid 0)$ and $W_{n}(\cdot\mid 1)$ are Bernoulli distributions. We write these Bernoulli parameters as

[TABLE]

Denote by $\mathcal{B}(n,\sigma)$ the set of $n$-length binary sequences $\boldsymbol{x}=(x_{1},\dots,x_{n})\in\{0,1\}^{n}$ satisfying the weight constraint

[TABLE]

where $P_{\boldsymbol{x}}$ stands for the type or empirical distribution of the binary sequence $\boldsymbol{x}\in\{0,1\}^{n}$ (see [20]). We define a channel code for the discretized channel $W_{n}^{n}$ under the weight constraint $\sigma$ as follows:

Definition 2**.**

Given an integer $M\geq 1$ and a real $0\leq\varepsilon\leq 1$, a pair of encoder $\phi:\{1,\dots,M\}\to\mathcal{B}(n,\sigma)$ and decoder $\psi:\{0,1\}^{n}\to\{1,\dots,M\}$ is called an $(n,M,\sigma,\varepsilon)_{\mathrm{avg}}$-code for the discretized channel $W_{n}^{n}$ if

[TABLE]

where $W^{n}:\mathcal{X}^{n}\to\mathcal{Y}^{n}$ stands for the $n$-fold product channel of a DMC $W:\mathcal{X}\to\mathcal{Y}$, i.e.,

[TABLE]

for each $\boldsymbol{x}=(x_{1},\dots,x_{n})\in\mathcal{X}^{n}$ and $\boldsymbol{y}=(y_{1},\dots,y_{n})\in\mathcal{Y}^{n}$.

Wyner’s ad hoc assumption [17, Section II in Part I] constrains a $(T,M,A,\sigma,\varepsilon)_{\mathrm{avg}}$-code for the Poisson channel in a certain way so that the resultant channel code is equivalent—in a sense to be made precise in Lemma 1—to an $(n,M,\sigma,\varepsilon)_{\mathrm{avg}}$-code for the discretized channel $W_{n}^{n}$. In fact, the Poisson channel and its discretized channel $W_{n}^{n}$ can be compared via a certain channel ordering, as shown in the following proposition.

Proposition 1**.**

The Poisson channel is better in the Shannon sense than its discretized channel $W_{n}^{n}$, where we say that a channel $V_{1}$ is better in the Shannon sense than another channel $V_{2}$ if for each $n\geq 1$ and each code for $V_{2}^{n}$, say $\mathcal{C}_{2}$, there exists a code for $V_{1}^{n}$, say $\mathcal{C}_{1}$, with the same message size and the average probability of error of $\mathcal{C}_{2}$ is no larger than that for $\mathcal{C}_{1}$ (cf. [20, Problem 6.17(a)], [22], [23]).

Proof:

See Appendix A. ∎

Therefore, it is clear that a $(T,M,A,\sigma,\varepsilon)_{\mathrm{avg}}$-code exists for the Poisson channel, provided that an $(n,M,\sigma,\varepsilon)_{\mathrm{avg}}$-code exists for the discretized channel $W_{n}^{n}$. This implies that

[TABLE]

where $M_{n}^{\ast}(\sigma,\varepsilon)$ stands for the maximum integer $M$ such that an $(n,M,\sigma,\varepsilon)_{\mathrm{avg}}$-code exists for the discretized channel $W_{n}^{n}$. Moreover, Wyner [17, Theorem 2.1 in Part II] showed that this discretization error is negligible for $n$ large enough. The following lemma is a direct consequence of [17, Theorem 2.1 in Part II].

Lemma 1**.**

There exist a sequence $\epsilon_{n}=\mathrm{o}(1)$ satisfying $0<\epsilon_{n}<1-\varepsilon$ and a subsequence $\{n_{k}\}_{k=1}^{\infty}$ such that

[TABLE]

for every $k\geq 1$.

III-C Proof of Converse Part of Theorem 2

It follows from Lemma 1 that

[TABLE]

therefore, it suffices to prove the following lemma to assert the converse part of Theorem 2.

Lemma 2**.**

For every $0<\varepsilon<1$, it holds that

[TABLE]

as $T\to\infty$.

Proof:

Let $P^{\ast}$ be the capacity-achieving input distribution (CAID) for the Poisson channel, i.e., it is the Bernoulli distribution with parameter $p^{\ast}$, where $p^{\ast}$ is defined in (9). By the definitions of $p^{\ast}$ and $p_{0}$ in (9) and (10), respectively, we see that

[TABLE]

For each $-p^{\ast}\leq u\leq 1-p^{\ast}$, we define the $u$-shifted distribution $P_{[u]}^{\ast}$ from the CAID $P^{\ast}$ as the Bernoulli distribution

[TABLE]

Now, construct the distribution $Q^{(n)}$ on $\{0,1\}^{n}$ as follows:444Throughout the proof of Lemma 2, we assume that $T$ is large enough so that there exists at least one $m$ satisfying the condition of the third sum in (43).

[TABLE]

for each $n\geq 1$ and $\boldsymbol{y}=(y_{1},\dots,y_{n})\in\{0,1\}^{n}$, where the constant $\kappa>0$ is given by

[TABLE]

the constant $\gamma>0$ will be specified later (specifically in (90) of Section III-C3), and $F$ is the normalization constant ensuring that

[TABLE]

Note that the third component of $Q^{(n)}$ is a convex combination of a set of exponentially-weighted distributions indexed by an appropriately constructed $\epsilon$-net in the input probability simplex. Also, we note that $F$ is positive, and it follows by the Gaussian integral that

[TABLE]

Let $\mathbb{N}\coloneqq\{1,2,\dots\}$ be the set of positive integers, and $\eta$ chosen so that555The maximum value of $\{\epsilon_{n}\}_{n=1}^{\infty}$ exists, because $\epsilon_{n}=\mathrm{o}(1)$ as $n\to\infty$.

[TABLE]

where $\{\epsilon_{n}\}_{n=1}^{\infty}$ is given in Lemma 1. For each $n\geq 1$, choose a sequence $\boldsymbol{x}^{(n)}=(x_{1}^{(n)},\dots,x_{n}^{(n)})\in\{0,1\}^{n}$ satisfying

[TABLE]

For short, we write

[TABLE]

Now consider the partition $\{\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3}\}$ of $\mathbb{N}$ given by

[TABLE]

Clearly, at least one of the subsets $\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3}\subset\mathbb{N}$ must be countably infinite. We shall divide the proof of Lemma 2 into three subsequences: $\{p_{n}\}_{n\in\mathcal{I}_{1}}$, $\{p_{n}\}_{n\in\mathcal{I}_{2}}$, and $\{p_{n}\}_{n\in\mathcal{I}_{3}}$, since the limit superior is the supremum of the set of subsequential limits.

III-C1 When $\mathcal{I}_{1}$ is countably infinite

Assuming that $\mathcal{I}_{1}$ is countably infinite, we now prove Lemma 2 for the subsequence $\{p_{n}\}_{n\in\mathcal{I}_{1}}$, where note from (48) that $\mathcal{I}_{1}$ is nonempty only if $p^{\ast}=p_{0}<\sigma$. For simplicity, we shall write $\mathcal{I}_{1}=\{n_{k}\}_{k=1}^{\infty}$ in this subsubsection. Firstly, it follows from the symbol-wise converse bound (cf. [7, Proposition 6] or [8, Proposition 4.4]) that

[TABLE]

for every $n\geq 1$. Secondly, it follows by the sifting property of the information spectrum divergence from a convex combination $Q^{(n)}$ (cf. [7, Lemma 3] or [8, Lemma 2.2]) that

[TABLE]

where $\kappa$ is given in (44). Thirdly, it follows by Chebyshev’s inequality (cf. [7, Equation (5)] or [8, Proposition 2.2]) that

[TABLE]

where $D_{n}$ and $V_{n}$ are given by

[TABLE]

As $P_{n}$ is the type of $\boldsymbol{x}^{(n)}=(x_{1}^{(n)},\dots,x_{n}^{(n)})$ (see (48) and (49)), note that

[TABLE]

As shown in Appendix B, it holds that

[TABLE]

as $n\to\infty$, where the mapping $\tilde{C}:[0,\sigma]\to[0,\infty)$ is defined by

[TABLE]

After some algebra, it follows from the definition of $p_{0}$ in (10) that

[TABLE]

where the last equality follows by the definition of $p_{0}$ in (10). Since $p^{\ast}=p_{0}<\sigma$, equality (64) implies that $\tilde{C}:[0,1]\to\mathbb{R}$ is a strictly decreasing function on $[0,1]$. Thus, since $p^{\ast}+\kappa\leq p_{n_{k}}\leq\sigma$ for every $k\geq 1$, we have

[TABLE]

where the second inequality is due to [17, Equation (2.11) in Part I]. On the other hand, as shown in Appendix C, it holds that

[TABLE]

Combining (55)–(57), (62), (65), and (66), we obtain

[TABLE]

as $T\to\infty$. Therefore, Lemma 2 holds for the subsequence $\{n_{k}\}_{k=1}^{\infty}=\mathcal{I}_{1}$.

III-C2 When $\mathcal{I}_{2}$ is countably infinite

The proof can be done in the same way as Section III-C1 by replacing the input distribution $P_{[\kappa]}^{\ast}$ by $P_{[-\kappa]}^{\ast}$ in (56). Then, replacing every occurrence of $p^{\ast}+\kappa$ by $p^{\ast}-\kappa$, we can verify that Lemma 2 holds for the subsequence $\{n_{k}\}_{k=1}^{\infty}=\mathcal{I}_{2}$, provided that $\mathcal{I}_{2}$ is countably infinite.

III-C3 When $\mathcal{I}_{3}$ is countably infinite

Assuming that $\mathcal{I}_{3}$ is countably infinite, we now prove Lemma 2 for the subsequence $\{p_{n}\}_{n\in\mathcal{I}_{3}}$. For brevity, we shall write $\mathcal{I}_{3}=\{n_{k}\}_{k=1}^{\infty}$ in this subsubsection. Define the integer $m_{k}$ as

[TABLE]

for each $k\geq 1$. Since $p_{n_{k}}<p^{\ast}+\kappa$ for every $k\geq 1$ (see (54)), it follows that

[TABLE]

Moreover, since $\tilde{m}_{T}/T\to\kappa$ as $T\to\infty$, there exists a $T_{0}>0$ satisfying

[TABLE]

for every $k\geq 1$ and $T\geq T_{0}$; henceforth, assume that $T\geq T_{0}$. Denote by

[TABLE]

for each $k\geq 1$. By the definition of $\tilde{p}_{k}$ in (72), it is clear that $\tilde{p}_{k}$ is bounded away from zero for all $k\geq 1$. It follows by the sifting property of the information spectrum divergence from a convex combination $Q^{(n)}$ (cf. [7, Lemma 3] or [8, Lemma 2.2]) that

[TABLE]

By the definition of $\tilde{m}_{T}$ in (69), it follows that

[TABLE]

that is, Inequality (74) can be relaxed as

[TABLE]

On the other hand, it follows by the Berry–Esseen theorem (cf. [7, Lemma 5] or [8, Proposition 2.2]) that

[TABLE]

where $D_{k}$, $V_{k}$, and $\Xi_{k}$ are given by

[TABLE]

As $P_{n}$ is the type of $\boldsymbol{x}^{(n)}=(x_{1}^{(n)},\dots,x_{n}^{(n)})$ (see (48) and (49)), it follows that

[TABLE]

As shown in Appendix D, it holds that

[TABLE]

as $k\to\infty$, where the constant $G_{1}>0$ (whose positivity is asserted and proved in Appendix D) is given by

[TABLE]

Moreover, as shown in Appendix E, by exploiting the Lipschitz properties of the information variances, there exist two constants $\beta_{1},\beta_{2}>0$ satisfying

[TABLE]

as $k\to\infty$. Furthermore, as shown in Appendix F, assuming that $\eta=1/\sqrt{T}$, there exist a constant $G_{2}>0$ and a positive sequence $\delta_{k}=\mathrm{o}(1)$ (as $k\to\infty$) such that

[TABLE]

for sufficiently large $k$ and $T$.

Finally, we obtain

[TABLE]

as $k\to\infty$ and for sufficiently large $T$, where

• •

(a) follows from (55),

• •

(b) follows from (76),

• •

(c) follows from (77),

• •

(d) follows from (87),

• •

(e) follows from (84),

• •

(f) follows from (86), and

• •

(g) follows by defining the function $f:\mathbb{R}\to\mathbb{R}$ as

[TABLE]

By choosing the constant $\gamma>0$ in (43) sufficiently small so that

[TABLE]

it follows by the maximization of the quadratic function (89), which has a negative leading coefficient, that

[TABLE]

for sufficiently large $k$ and $T$. Combining (88) and (91), we have

[TABLE]

as $T\to\infty$, proving Lemma 2 for the subsequence $\{n_{k}\}_{k=1}^{\infty}=\mathcal{I}_{3}$. This completes the proof of Lemma 2 ∎

III-D Proof of Achievability Part of Theorem 2

As shown in Section III-B, a channel coding problem for the Poisson channel can be reduced to that of a certain discrete memoryless channel under an additive cost constraint. Now, define a channel code under an average cost-constraint as follows:

Definition 3**.**

Given a cost function $\chi:\mathcal{X}\to\mathbb{R}$, a pair of encoder $f:\{1,\dots,M\}\to\mathcal{X}$ and decoder $g:\mathcal{Y}\to\{1,\dots,M\}$ is called an $(M,\varepsilon,\beta)_{\mathrm{avg}}$-code for a channel $W:\mathcal{X}\to\mathcal{Y}$ if

[TABLE]

and

[TABLE]

The following lemma is a straightforward generalization of the random coding union bound [5, Theorem 16].

Lemma 3**.**

Let $M$ be a positive integer and $\beta$ a real. For any distribution $P$ on $\mathcal{X}$ and any channel $W:\mathcal{X}\to\mathcal{Y}$, there exists an $(M,\varepsilon,\beta)_{\mathrm{avg}}$-code satisfying

[TABLE]

where the r.v.’s $X$, $Y$, and $\bar{X}$ are defined so that $\mathbb{P}\circ(X,Y)^{-1}=P\times W$, $\mathbb{P}\circ\bar{X}^{-1}=P$, and $(X,Y)\Perp\bar{X}$.

Proof:

See Appendix G. ∎

To ensure that $\Delta_{n}=\mathrm{o}(1)$ as $T\to\infty$, assume throughout the achievability proof that666As a matter of fact, it suffices to assume in the achievability proof that $n=\omega(T)$ as $T\to\infty$. We assume, however, here that (96) holds to give an explicit dependence between $n$ and $T$. This dependence affects some constants that appear later in the proof.

[TABLE]

where $\Delta_{n}$ is defined in (31), and $\lceil u\rceil\coloneqq\min\{z\in\mathbb{Z}\mid z\geq u\}$ stands for the ceiling function. Recall that the $u$-shifted distribution $P_{[u]}^{\ast}$ from the CAID $P^{\ast}$ is defined in (42). For each $n\geq 1$, define the Bernoulli distribution $P_{n}$ by777The choice of parameter $u_{n}=-p^{\ast}\,n^{-1/4}$ is inspired by the delta-convention [20, Convention 2.11].

[TABLE]

It is clear that $P_{n}$ converges to $P^{\ast}$ in the variational distance topology as $T\to\infty$. Denote by

[TABLE]

the Bernoulli parameters of $P_{n}$ and $P_{n}W_{n}$, respectively. Now, we have the following lemma.

Lemma 4**.**

For $n$ satisfying (96), the following asymptotic estimates hold:

[TABLE]

as $T\to\infty$, where $C^{\ast}$ and $V^{\ast}$ are defined in (7) and (12), respectively, and $\Xi^{\ast}$ is defined as

[TABLE]

Proof:

See Appendix H. ∎

For each $n\geq 1$, denote by

[TABLE]

i.i.d. pairs of r.v.’s with generic distribution $P_{n}\times W_{n}$. For short, we write the random vectors

[TABLE]

Lemma 5**.**

It holds that

[TABLE]

for sufficiently large $T$, where $N(a\mid\boldsymbol{z})$ denotes the number of occurrences of an element $a\in\mathcal{A}$ in the sequence $\boldsymbol{z}=(z_{1},\dots,z_{m})\in\mathcal{A}^{m}$, i.e.,

[TABLE]

Proof:

See Appendix I. ∎

Given a constant $0<\kappa<1$, define the event

[TABLE]

for each $T>0$, where note that

[TABLE]

because $Y_{n,1},\dots,Y_{n,n}$ are i.i.d. Bernoulli r.v.’s with the same parameter $r_{n}$. The following lemma asserts that (109) is a high probability set asymptotically in the sense that the probability of its complement decays exponentially fast in $T$.

Lemma 6**.**

There exists a positive constant $K_{0}=K_{0}(\kappa,\lambda_{0},A,\sigma)$ satisfying

[TABLE]

for sufficiently large $T$, where $\mathcal{S}^{\complement}$ stands for the complement of the set $\mathcal{S}$.

Proof:

See Appendix J. ∎

For each $n\geq 1$ and $\boldsymbol{x},\boldsymbol{y}\in\{0,1\}^{n}$, define the information density

[TABLE]

Then, the following two lemmas, which are analogous to [5, Lemma 47], hold.

Lemma 7**.**

There exists a positive constant $K_{1}=K_{1}(\kappa,\lambda_{0},A,\sigma)$ such that

[TABLE]

almost surely for every real $\gamma$ and for sufficiently large $T$, where $\boldsymbol{1}_{\mathcal{E}}$ stands for the indicator function of the event $\mathcal{E}$.

Proof:

See Appendix K. ∎

Lemma 8**.**

There exists a positive constant $K_{2}=K_{2}(\lambda_{0},A,\sigma)$ satisfying

[TABLE]

for every real $\gamma$ and for sufficiently large $T$.

Proof:

See Appendix L. ∎

The following lemma is a final tool to prove the achievability part of Theorem 2.

Lemma 9**.**

There exists a positive constant $K_{3}=K_{3}(\lambda_{0},A,\sigma)$ satisfying

[TABLE]

for every real $\gamma$ and for sufficiently large $T$.

Proof:

See Appendix M. ∎

Now, for each $n\geq 1$, we define the following numbers:

[TABLE]

where the constants $K_{0}$, $K_{1}$, $K_{2}$, and $K_{3}$ are given in Lemmas 6–9, respectively. Then, it follows from Lemma 4 that

[TABLE]

as $T\to\infty$, where in (a), we have also applied a Taylor series expansion of $\Phi^{-1}(\cdot)$ around $\varepsilon$.

Let $\bar{X}^{n}$ be a r.v. satisfying $\mathbb{P}\circ(\bar{X}^{n})^{-1}=P_{n}^{n}$ and $\bar{X}^{n}\Perp(X^{n},Y^{n})$. It follows from Lemma 3 that there exists an $(n,M_{n},\sigma,\varepsilon^{\prime})_{\mathrm{avg}}$-code for the discretized channel $W_{n}^{n}$ such that

[TABLE]

for sufficiently large $T$, where

• •

(a) follows from Lemma 5,

• •

(b) follows from Lemma 6,

• •

(c) follows from Lemma 7,

• •

(d) follows from Lemma 8,

• •

(e) follows from Lemma 9,

• •

(f) follows by the definition of $M_{n}$ in (116), and

• •

(g) follows by the definitions of $S_{n}$, $G_{n}$, and $\varepsilon_{n}$ in (117)–(119), respectively.

Therefore, it follows from (37) and (120) that the achievability bound of Theorem 2 is satisfied, completing the proof.

IV Concluding Remarks

We have derived the optimal second-order coding rate for the continuous-time Poisson channel. We have also obtained bounds on the third-order coding rate. This is the first instance of a second-order asymptotic result for continuous-time communication models in information theory. While the high-level proof ideas of Theorem 2 are based on Wyner’s discretization argument [17] and standard techniques in second-order asymptotics [3, 4, 5, 6, 7, 8], several novel finite blocklength techniques have to be introduced in both the converse and achievability arguments in view of the continuous-time nature of the Poisson channel. In the following two subsections, we summarize these technical contributions, partitioning our discussion into the converse and achievability parts of Theorem 2.

IV-A Technical Contributions in Proving Converse Part of Theorem 2

In the proof of Lemma 2, we constructed a somewhat artificial output distribution $Q^{(n)}$ in (43) to be substituted into the $\epsilon$-information spectrum divergence in (48). This construction of our choice of $Q^{(n)}$ is partly inspired by Tomamichel and Tan’s choice of the output distribution [7, Equation (6)], which was used to derive a tight third-order converse term for the fundamental limit of DMCs having positive channel dispersions [7, Proposition 8] (see also [8, Section 4.2.3]). Tomamichel and Tan’s choice is a hybrid between Hayashi’s choice of the output distribution used in [4, Section X-A] and another artificial output distribution based on the construction of an appropriate $\epsilon$-net in the output probability simplex. Tomamichel and Tan’s and Hayashi’s choices cannot, however, be used to prove Lemma 2 even up to the second-order converse term $\sqrt{T\,V^{\ast}}\,\Phi^{-1}(\varepsilon)$ in (40). This is because both choices require the application of the type counting lemma [20, Lemma 2.2] (see also [20, Problem 2.1]) in the approximation arguments. In contrast, due to the continuous nature of the Poisson channel and Wyner’s discretization argument, we have to take the limit superior in $n$ on the left-hand side of (40). To ameliorate this problem, we constructed $Q^{(n)}$ in (43), whose third part consists of a convex combination of exponentially-weighted output distributions indexed by elements of an $\epsilon$-net in the input probability simplex. This differs from Tomamichel and Tan’s work [7] in which they considered an $\epsilon$-net for the output probability simplex. Moreover, while Tomamichel and Tan’s construction of the $\epsilon$-net can yield a uniformly bounded normalization constant $F$, our construction in the third part of (43) cannot yield a uniformly bounded normalization constant $F$ with respect to $T>0$, but our construction yields an $F$ that scaled as $\mathrm{O}(\sqrt{T})$ as $T\to\infty$; see (46). This construction, and resulting unboundedness with respect to $T$, appears to be required to handle the continuous nature of Poisson channel. Because $F=\mathrm{O}(\sqrt{T})$ as $T\to\infty$, our upper bound on the third-order coding rate, as shown in (14), is $\leq\log T+\mathrm{O}(1)$ as $T\to\infty$, which differs slightly from the lower bound which reads $\geq(1/2)\log T+\mathrm{O}(1)$ as $T\to\infty$.

IV-B Technical Contributions in Proving Achievability Part of Theorem 2

The proof of the achievability part of Theorem 2 is inspired by Polyanskiy’s technique to prove the second- and third-order asymptotics for non-singular DMCs (cf. [6, Section 3.4.5]). We note, however, that Polyanskiy’s proof [6, Section 3.4.5] holds for codes without a cost constraint. Moreover, although the (symbol-wise) discretized channel $W_{n}$ is non-singular, it depends on $n$ (see (29)–(31)). Thus, Polyanskiy’s technique cannot be adapted in a straightforward manner to yield the achievability part of Theorem 2. To adapt Polyanskiy’s technique to prove the third-order term for the Poisson channel, we first slightly generalize the random coding union bound so that it is amenable to handling cost constraints; this is stated in Lemma 3. By constructing an input distribution that is slightly perturbed from the CAID in (97) and using the concentration bound in Lemma 5, we were able to bound the probability that the cost constraint is violated. The choice of (97) and Lemma 5 are inspired by the delta-convention888The delta-convention is a technical assumption frequently used in the method of types [20] to assert that the strongly typical set is a high-probability set. [20, Convention 2.11] and its consequent lemma [20, Lemma 2.12 or Problem 3.18(b)], respectively. Here, it is worth pointing out that if $p_{0}<\sigma$, i.e., if the weight constraint in (34) is not tight, then it suffices to employ the CAID $P^{\ast}$ (and not a perturbed version of it in (97)) to generate the random code. Next, we defined and analyzed a high-probability event $\mathcal{E}_{T}$ of (109); this event replaces Polyanskiy’s choice in [6, Equation (3.318)]. In Lemma 6, the probability of $\mathcal{E}_{T}^{\complement}$ was shown to be appropriately bounded by appealing to a modified logarithmic Sobolev inequality (see Remark 1 of Appendix J). We showed in Lemma 7 that conditioned on $\mathcal{E}_{T}$, each of the terms in the generalized random coding union bound can be appropriately bounded.

We note that there are other techniques to show that $+(1/2)\log n+\mathrm{O}(1)$ is third-order achievable for certain channels sources; see, for example [9, 10, 11, 12, 24]. For example, Tan and Tomamichel [10] showed that $+(1/2)\log n+\mathrm{O}(1)$ is third-order achievable for the additive white Gaussian noise channel. However, while the key idea in [10] is to use Laplace’s approximation to bound a certain probability, in the proof of Lemma 7 in Appendix K, we applied the Berry–Esseen theorem judiciously to bound the analogous probability.

Appendix A Proof of Proposition 1

Given two channels $W_{1}:\mathcal{X}\to\mathcal{Y}$ and $W_{2}:\mathcal{Y}\to\mathcal{Z}$, denote by $W_{1}W_{2}:\mathcal{X}\to\mathcal{Z}$ the concatenation999Clearly, this concatenation operator is associative, as in the matrix multiplication. of $W_{1}$ and $W_{2}$, i.e.,

[TABLE]

for each $(x,z)\in\mathcal{X}\times\mathcal{Z}$ with a certain r.v. $Y_{x}$ satisfying $\mathbb{P}\circ Y_{x}^{-1}=W_{1}(\cdot\mid x)$. Let $W_{\lambda_{0}}:\mathcal{W}(T,A,\sigma)\to\mathcal{S}(T)$ be the Poisson channel defined in Section II. By the same argument as Wyner’s ad hoc assumption [17, Section II in Part I], the discretized channel $W_{n}^{n}:\mathcal{B}(n,\sigma)\to\{0,1\}^{n}$ defined in Section III-B can be seen as a concatenation of four channels

[TABLE]

where the channel $U:\mathcal{B}(n,\sigma)\to\mathcal{W}(T,A,\sigma)$ is given by

[TABLE]

the channel $V_{1}:\mathcal{S}(T)\to(\mathbb{N}\cup\{0\})^{n}$ is given by

[TABLE]

and the channel $V_{2}:(\mathbb{N}\cup\{0\})^{n}\to\{0,1\}^{n}$ is given by

[TABLE]

Therefore, we can obtain Proposition 1 by considering appropriate stochastic encoder and decoder induced by $U$ and $V_{1}V_{2}$, respectively (cf. [20, Problem 6.17(b)], [22], [23]). This completes the proof of Proposition 1.

Appendix B Proof of (62)

Denote by

[TABLE]

the parameter of the Bernoulli distribution $P_{[\kappa]}^{\ast}W_{n}$, where $a_{n}$ and $b_{n}$ are defined in (32) and (33), respectively. Note that the following asymptotic equivalences hold:

[TABLE]

as $n\to\infty$, where $r_{n}$ is defined in (51). Then, we observe that

[TABLE]

as $n\to\infty$, where

• •

(a) follows by (60) and the definition of binary entropy function $h:u\mapsto-u\log u-(1-u)\log(1-u)$,

• •

(b) follows by the asymptotic equivalences as stated in (130)–(131) and $-\log(1-u)\sim u$ as $u\to 0$,

• •

(c) follows by the asymptotic equivalences as stated in (128)–(129) and $-(1-u)\log(1-u)\sim u$ as $u\to 0$,

• •

(d) follows by the definition of $r_{n}\coloneqq P_{n}W_{n}(1)=(1-p_{n})\,a_{n}+p_{n}\,b_{n}$ in (51), and

• •

(e) follows by the asymptotic equivalences as stated in (128)–(131).

Equation (132) indeed implies (62), as desired.

Appendix C Proof of (66)

Define the binary relative varentropy $v(p\,\|\,q)$ by

[TABLE]

for each $0\leq p,q\leq 1$, where $d(p\,\|\,q)$ stands for the binary relative entropy defined by

[TABLE]

After some algebra, we obtain

[TABLE]

Now, a direct calculation yields

[TABLE]

as $n\to\infty$, where

• •

(a) follows from (61) and the definition of $r_{n}^{\prime}$ in (127),

• •

(b) follows from (135), and

• •

(c) follows from the asymptotic equivalences as stated in (128)–(131).

We now employ the following inequality101010Lemma 10 is a generalization of the well-known information theoretic inequalities: $1-x^{-1}\leq\log x\leq x-1$..

Lemma 10** ([25, Lemma 1]).**

For every $u>0$ and $q>r$, it holds that $\ln_{q}u\leq\ln_{r}u$ with equality if and only if $u=1$, where $\ln_{q}:(0,\infty)\to\mathbb{R}$ stands for the $q$-logarithm function [26] defined by

[TABLE]

It follows from Lemma 10 with $q=1$ and $r=1/2$ that

[TABLE]

for every $u>0$. Now, we have

[TABLE]

where

• •

(a) follows from (136), and

• •

(b) follows from (138).

This completes the proof of (66).

Appendix D Proof of (84)

After some algebra, we observe that

[TABLE]

for each $k\geq 1$, and one has111111While Tomamichel and Tan [7, Property 4 of Lemma 7] gave an upper bound on the relative entropy $D(P\,\|\,Q)$ by the reverse Pinsker inequality $\leq|P-Q|^{2}/Q_{\min}$ to prove [7, Proposition 8], we cannot get a useful upper bound by the reverse Pinsker inequality to prove the second-order converse of the Poisson channel, because for the Poisson channel, it holds that $Q_{\min}=\Theta(n_{k}^{-1})$ as $k\to\infty$.

[TABLE]

as $k\to\infty$, where

• •

(a) follows from the definitions of $r_{n_{k}}$ and $\tilde{r}_{k}$ in (51) and (73), respectively,

• •

(b) follows from the facts that

[TABLE]

• •

(c) follows from the definitions of $\tilde{m}_{T}$ and $\tilde{p}_{k}$ in (69) and (72), respectively, implying that $p_{n_{k}}\leq\tilde{p}_{k}$ or

[TABLE]

• •

(d) follows from (196)–(143) and the fact that

[TABLE]

• •

(e) follows by the definition of $\tilde{p}_{k}$ in (72), and

• •

(f) follows by the definition of $m_{k}$ in (69).

On the other hand, one can see in the same way as Appendix B that

[TABLE]

as $k\to\infty$, where the function $g:[0,1]\to\mathbb{R}$ is defined by

[TABLE]

Direct calculations show

[TABLE]

where $p_{0}$ is defined in (10). Now, it follows from (9) that121212The first-order derivative in (152) is zero for the usual discrete memoryless channels (DMCs) without cost-constraint. However, the first-order derivative in (152) can be positive when there is a cost constraint on the codewords.

[TABLE]

and it follows from (10), (48), and (50) that $p_{n_{k}}<p^{\ast}$ if $p_{0}>\sigma$. Therefore, it follows by Taylor’s theorem applied to $g(\cdot)$ around $p^{\ast}$ that there exists a real number $\hat{p}$ between $p^{\ast}$ and $p_{n_{k}}$ such that131313The upper bound (153) is a counterpart of [7, Property 3 in Lemma 7].

[TABLE]

for each $k\geq 1$, where

• •

(a) follows from (152) and the fact that

[TABLE]

and

• •

(b) follows by the definition of $\mathcal{I}_{3}=\{n_{k}\}_{k=1}^{\infty}$ in (54).

Note from (152) that the second term

[TABLE]

in (153) is zero if $p_{0}\leq\sigma$, and is negative if $p_{0}>\sigma$. In any case, we have from (153) that $g(p_{n_{k}})\leq g(p^{\ast})-(\text{positive const.})\times(p_{n_{k}}-p^{\ast})^{2}$. By the definitions of $C^{\ast}$ and $g(\cdot)$ in (7) and (147), respectively, it is clear that

[TABLE]

Hence, we obtain

[TABLE]

as $k\to\infty$, where

• •

(a) follows from (140),

• •

(b) follows from (141), and

• •

(c) follows from (146), (153), and (156).

Here, note that the terms in parentheses in the second term in (157) is strictly positive, i.e.,

[TABLE]

because of (41), (44), and (152). This completes the proof of (84) with the identification of $G_{1}$ as the constant in (158).

Appendix E Proof of (86)

Similar to (136), we may observe that

[TABLE]

as $k\to\infty$, where the mapping $\tilde{v}:[0,1]\times[0,1]\to\mathbb{R}$ is defined by

[TABLE]

Since $\sqrt{\tilde{v}(t,u)}$ is a continuously differentiable function of $u\in(0,1]$ for each fixed $t\in[0,1]$, and since $\tilde{p}_{k}$ and $p_{n_{k}}$ are bounded away from zero for all $k\geq 1$, it follows by the Lipschitz continuity141414Note that every continuously differentiable function is Lipschitz continuous and the Lipschitz constant can be taken to be the supremum of the absolute value of the first derivative (over its domain). of $u\mapsto\sqrt{\tilde{v}(t,u)}$ that there exists a Lipschitz constant $\beta_{1}(t)>0$ satisfying

[TABLE]

for every $k\geq 1$ and $0\leq t\leq 1$, where

• •

(a) follows by the definition of $\tilde{p}_{k}$ in (72), and

• •

(b) follows by the definition of $m_{k}$ in (69).

Here, it can be verified by a direct calculation of the partial derivative of $u\mapsto\sqrt{\tilde{v}(t,u)}$ with fixed $t$ that the Lipschitz constant $\beta_{1}(t)$ is continuous in $t\in[0,1]$. Moreover, as $p_{n_{k}}$ belongs to the closed interval $[p^{\ast}-\kappa,p^{\ast}+\kappa]$ (see (54)), inequality (161) can be uniformly relaxed as

[TABLE]

for every $k\geq 1$, where the absolute constant $\beta_{1}>0$ is given by

[TABLE]

Analogously, since $\sqrt{\tilde{v}(t,t)}$ is a continuously differentiable function of $t\in(0,1]$, it follows by the Lipschitz continuity of $t\mapsto\sqrt{\tilde{v}(t,t)}$ that there exists an absolute constant $\beta_{2}>0$ satisfying

[TABLE]

for every $k\geq 1$. By the definitions of $V^{\ast}$ and $\tilde{v}(\cdot,\cdot)$ in (12) and (160), respectively, it is clear that

[TABLE]

Combining (159), (161), (164), and (165), it follows by the triangle inequality that

[TABLE]

as $k\to\infty$. This completes the proof of (86).

Appendix F Proof of (87)

Since $(t,u)\mapsto\sqrt{\tilde{v}(t,u)}$ is continuous and positive on $(0,1)\times(0,1)$, and since $p_{n_{k}}$ and $\tilde{p}_{k}$ are in the closed intervals $[p^{\ast}-\kappa,p^{\ast}+\kappa]$ and $[p^{\ast}-\kappa,p^{\ast}+2\kappa]$, respectively (see (54), (69), (70), and (72)), it follows by the extreme value theorem that there exists a constant $v_{\min}>0$ satisfying

[TABLE]

for every $k\geq 1$ yielding together with (159) that

[TABLE]

for some $\zeta=\mathrm{o}(1)$ as $k\to\infty$.

Now, define the binary absolute and central third-moment divergence by

[TABLE]

for each $0\leq p,q\leq 1$ and where $d(p\|\,q)$ is defined in (134). Similar to (135), we see that

[TABLE]

Then, we obtain

[TABLE]

as $k\to\infty$, where

• •

(a) follows from (83),

• •

(b) follows from (170), and

• •

(c) follows from the asymptotic equivalences as stated in (128)–(130) and (143).

Similar to (168), it follows by the extreme value theorem that there exists a constant $\xi_{\max}>0$ satisfying

[TABLE]

as $k\to\infty$. Combining (168) and (172), there exists a positive sequence $\delta_{k}=\mathrm{o}(1)$ (as $k\to\infty$) satisfying

[TABLE]

Supposing that $\eta=1/\sqrt{T}$, since

[TABLE]

for sufficiently large $k$ and $T$, it follows from a Taylor series expansion of $\Phi^{-1}(\cdot)$ around $\varepsilon+\epsilon_{n_{k}}+\delta_{k}$ that

[TABLE]

for sufficiently large $k$ and $T$, where the positive constant $\tilde{G}_{2}$ is given as

[TABLE]

depending only on the tolerated probability of error $0<\varepsilon<1$. This completes the proof of (87).

Appendix G Proof of Lemma 3

Let $X_{1},X_{2},\dots,X_{M}$ be i.i.d. r.v.’s with generic distribution $P$. Consider a maximum-likelihood decoder $g_{\mathrm{ML}}:\mathcal{Y}\to\mathcal{X}$ satisfying

[TABLE]

with probability $1$. Then, the error probability averaged over the ensemble of random codes $\{X_{m}\}_{m=1}^{M}$ is given by

[TABLE]

By the standard argument of the random coding union bound (see the proof of [5, Theorem 16]), the last term in the right-hand side on (178) can be bounded from above by

[TABLE]

which asserts Lemma 3 together with (178).

Appendix H Proof of Lemma 4

Since $n=\omega(T)$ as $T\to\infty$ (see (96)), it follows that $a_{n}=\mathrm{o}(1)$ and $b_{n}=\mathrm{o}(1)$ as $T\to\infty$. We readily see that

[TABLE]

as $T\to\infty$, where

• •

(a) follows from the fact that $-(1-u)\log(1-u)\sim u$ as $u\to 0$,

• •

(b) follows by the definition of $r_{n}=(1-p_{n})\,a_{n}+p_{n}\,b_{n}$ in (99),

• •

(c) follows from the facts that

[TABLE]

and

• •

(d) follows from the asymptotic equivalences as stated in (128)–(129).

Equation (180) implies (100) of Lemma 4.

We shall next verify (101) of Lemma 4. Noting the asymptotic equivalences stated in (128)–(129) and

[TABLE]

it follows from (180) that

[TABLE]

as $T\to\infty$. Therefore, we obtain

[TABLE]

as $T\to\infty$, implying (101) of Lemma 4. Finally, Equation (102) of Lemma 4 can be verified in the same way as (190). This completes the proof of Lemma 4.

Appendix I Proof of Lemma 5

We observe that

[TABLE]

where

• •

(a) follows from the fact that $\mathbb{E}[N(1\mid X^{n})]=n\,p^{\ast}\,(1-n^{-1/4})$,

• •

(b) follows by the definition of $p^{\ast}$ in (9), and

• •

(c) follows by Hoeffding’s inequality (cf. [21, Theorem 2.8]).

This completes the proof of Lemma 5 together with the hypothesis in (96) that $n=\lceil T^{2}\rceil$.

Appendix J Proof of Lemma 6

In this proof, we employ the following lemma, which is an application of the modified logarithmic Sobolev inequality (cf. [21, Chapter 6]).

Lemma 11** ([21, Theorem 6.12]).**

Let $Z_{1},\dots,Z_{n}$ be independent $\mathcal{A}$-valued r.v.’s. Consider a Borel-measurable mapping $f:\mathcal{A}^{n}\to\mathbb{R}$ satisfying the self-bounding property: for each $i=1,\dots,n$, there exists a Borel-measurable mapping $f_{i}:\mathcal{A}^{n-1}\to\mathbb{R}$ such that

[TABLE]

and

[TABLE]

for every $(z_{1},\dots,z_{n})\in\mathcal{A}^{n}$. Then, it holds that

[TABLE]

for every $0<t\leq\mathbb{E}[f(Z_{1},\dots,Z_{n})]$.

The following example shows a special case of Lemma 11.

Example 1**.**

If $Z_{1},\dots,Z_{n}$ are independent Bernoulli r.v.’s, then the mapping

[TABLE]

which follows a binomial distribution, satisfies the self-bounding property.

Recall that $n=\lceil T^{2}\rceil$ (see (96)). By the asymptotic equivalence

[TABLE]

it follows from the choice of constant $0<\kappa<1$ that there exists a $T_{0}=T_{0}(\kappa,\lambda_{0},A,\sigma)>0$ satisfying

[TABLE]

for every $T\geq T_{0}$. Now, we observe that

[TABLE]

where

• •

(a) holds for every $T\geq T_{0}$, because it follows from (110), the left-hand inequality of (197), and Lemma 11,

• •

(b) holds for every $T\geq T_{0}$, because of the right-hand inequality of (197), and

• •

(c) follows by choosing the constant

[TABLE]

This completes the proof of Lemma 6.

Remark 1**.**

While Lemma 5 is proved using Hoeffding’s inequality, the same inequality cannot be used to show Lemma 6. This is because the Bernoulli parameter $r_{n}$ given in (99) approaches to zero as $T$ goes to infinity, but Hoeffding’s inequality is independent of (the rate of decay of) $r_{n}$. We have avoided this issue via the modified logarithmic Sobolev inequality as stated in Lemma 11.

Appendix K Proof of Lemma 7

For each $n\geq 1$, let $B_{n,1},B_{n,2},\dots,B_{n,n}$ be i.i.d. Bernoulli r.v.’s with parameter

[TABLE]

For short, we write $B_{n}^{m}=(B_{n,1},\dots,B_{n,m})$. Lemma 6 tells us that there exists a $T_{0}>0$ such that $\mathbb{E}[N(1\mid Y^{n})]>\kappa\,(p^{\ast}+s)\,A\,T$ for every $T\geq T_{0}$. With probability one, for any reals $\alpha$ and $\beta$ and for all $T\geq T_{0}$, we have

[TABLE]

where

• •

(a) follows by defining the r.v. $\alpha(n,k_{0},Y^{n})$ so that

[TABLE]

• •

(b) follows from the fact that the r.v.

[TABLE]

follows the binomial distribution with parameters $N(1\mid Y^{n})$ and $p_{n}\,b_{n}/r_{n}$,

• •

(c) follows from the Berry–Esseen theorem with the identities:

[TABLE]

• •

(d) follows by the definition of $\mathcal{E}_{T}$ in (109), and

• •

(e) follows from (110).

Recall that $n=\omega(T)$ as $T\to\infty$. By the asymptotic equivalences

[TABLE]

as $T\to\infty$, we see that

[TABLE]

and

[TABLE]

Therefore, it follows from (201) that there exist $K_{1}=K_{1}(\kappa,\lambda_{0},A,\sigma)>0$ and $T_{1}=T_{1}(\kappa,\lambda_{0},A,\sigma)>0$ satisfying

[TABLE]

almost surely for every reals $\alpha$ and $\beta$ and every $T\geq T_{1}$. Now, as in [5, Equation (474)], it follows from (213) that

[TABLE]

almost surely for every $T\geq T_{1}$. This completes the proof of Lemma 7.

Appendix L Proof of Lemma 8

It follows from [5, Lemma 47] (see also [8, Theorem 1.7]) that

[TABLE]

where note that

[TABLE]

and

[TABLE]

Therefore, Lemma 4 yields Lemma 8.

Appendix M Proof of Lemma 9

It follows from the Berry–Esseen theorem (cf. [8, Theorem 1.6]) that

[TABLE]

The existence of the constant $K_{3}=K_{3}(\lambda_{0},A,\sigma)>0$ can be verified by Lemma 4. This completes the proof of Lemma 9.