# Broadcast Channels with Heterogeneous Arrival and Decoding Deadlines: Second-Order Achievability

Homa Nikbakht, Malcolm Egan, Jean-Marie Gorce, H. Vincent Poor

► To cite this version:

Homa Nikbakht, Malcolm Egan, Jean-Marie Gorce, H. Vincent Poor. Broadcast Channels with Heterogeneous Arrival and Decoding Deadlines: Second-Order Achievability. IEEE Transactions on Information Theory, 2025, 71 (3), pp.1-1. 10.1109/TIT.2025.3532649. hal-04360147

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# Broadcast Channels with Heterogeneous Arrival and Decoding Deadlines: Second-Order Achievability

Homa Nikbakht, Member, IEEE, Malcolm Egan, Jean-Marie Gorce, Senior Member, IEEE, and H. Vincent Poor, Life Fellow, IEEE

## Abstract

A standard assumption in the design of ultra-reliable low-latency communication systems is that the duration between message arrivals is larger than the number of channel uses before the decoding deadline. Nevertheless, this assumption fails when messages arrive rapidly and reliability constraints require that the number of channel uses exceed the time between arrivals. In this paper, we consider a broadcast setting in which a transmitter wishes to send two different messages to two receivers over Gaussian channels. Messages have different arrival times and decoding deadlines such that their transmission windows overlap. For this setting, we propose a coding scheme that exploits Marton's coding strategy. We derive rigorous bounds on the achievable rate regions. Those bounds can be easily employed in point-to-point settings with one or multiple parallel channels. In the point-to-point setting with one or multiple parallel channels, the proposed achievability scheme outperforms the Normal Approximation, especially when the number of channel uses is smaller than 200. In the broadcast setting, our scheme agrees with Marton's strategy for sufficiently large numbers of channel uses and shows significant performance improvements over standard approaches based on time sharing for transmission of short packets.

## I. INTRODUCTION

Mobile wireless networks in 5G and in 6G proposals are increasing intended for use in latency-critical and high-reliability systems, notably in industrial control applications, autonomous vehicles and remote surgery [1]–[6]. In such ultra-reliable low-latency communications (URLLC), packets are typically short. As a consequence, data transmission cannot be made reliable by increasing channel code blocklength arbitrarily.

A key challenge is, therefore, to design coding schemes that support high reliability requirements under finite blocklength. In recent years, a number of channel coding schemes have been proposed to address such requirements including short LDPC and polar codes [7], [8]. At the same time, new characterizations of fundamental tradeoffs among the size of the message set, the probability of error, and the length of the code have been obtained via achievability and converse bounds, thereby building on the work of [9]–[11].

A standard assumption in the design of coding schemes, even for URLLC, is that consecutive messages have distinct arrival times and decoding deadlines. As such, there is no choice but to encode the messages independently. However, this assumption is violated when a message arrives before the decoding deadline of a prior message. For example, a sensor in an unstable control system may send rapid measurements in order to stabilize the system [12]. In order to ensure reliability of the sensor observations, the channel uses allocated to each observation of the speed may partially overlap.

It is therefore desirable to consider joint encoding of multiple sensor observations, albeit with heterogeneous decoding deadlines. That is, if the channel uses for two separate observations overlap, it is not possible to wait until the entire transmission for both sensor observations is received before decoding. In this situation, code design must account for two issues:

(i) messages with close arrival times; and

(ii) messages with heterogeneous decoding deadlines.

In addition to heterogeneous arrival times and decoding deadlines, a transmitter may seek to send messages to distinct receivers. While this setting has clear relevance for URLLC applications, there are currently no known designs for appropriate coding schemes.

## A. Related Work

1) *Finite Blocklength Regime:* To capture both reliability and latency, investigation of coding bounds in the finite blocklength regime is a requirement that dates back to the work of Shannon, Gallager, and Berlekamp [13]. Most of the existing literature focuses on identifying the limits of communication between a single transmitter and a single receiver for a coding block of

H. Nikbakht and H. V. Poor are with the Department of Electrical and Computer Engineering, Princeton University, NJ, USA (email: homa_poor@princeton.edu). M. Egan and J-M. Gorce are with the INRIA and University of Lyon, CITI, INSA Lyon, 69100 Villeurbanne, France (e-mail: jean-marie.gorce@inria.fr, malcolm.egan@inria.fr). The work of H. Nikbakht and H. V. Poor has been supported by the U.S. National Science Foundation (NSF) under Grants CNS-2128448 and ECCS-2335876. The work of J-M. Gorce and M. Egan has been partly supported by the French National Agency for Research within the PEPR-5 program.
size $n$. Among the proposed bounds, the widely used bound is the Normal Approximation that for a given $n$ approximates the point-to-point transmission rate $R$ by [9], [14]:

$$R \approx C - \sqrt{\frac{V}{n} \log(e) Q^{-1}(\epsilon) + \frac{\log n}{2n}}, \quad (1)$$

where $C$ is the channel capacity, $V$ is the channel dispersion coefficient, $\epsilon$ is the average error probability and $Q^{-1}(\cdot)$ is the inverse of the Gaussian cumulative distribution function. However this approximation has been proved to be a valid $O(n^{-1})$ asymptotic approximation for converse and achievability bounds [15], but an $O(n^{-1})$ bound is not necessarily reliable for small values of $n$ corresponding to URLLC applications. Therefore, exact bounds are of considerable interest.

2) Heterogeneous Decoding Deadlines: The problem of designing a code to handle heterogeneous decoding deadlines was first considered in the context of static broadcasting [16], where a single message is decoded at multiple receivers under different relative decoding delay constraints. The work in [16] was recently generalized to multi-source and multi-terminal networks by Langberg and Effros in [17]. In particular, the notion of a time-rate region was introduced, which accounted for different decoding delay constraints for each message at each receiver.

The work in both [16] and [17] focused on the asymptotic regime. In the finite blocklength regime, a coding scheme for the Gaussian broadcast channel with heterogeneous blocklength constraints was introduced in [18], which decodes the messages at time-instances that depend on the realizations of the random channel fading. By employing an early decoding scheme, the authors showed that significant improvements over standard successive interference cancellation are possible. In [19] achievable rates and latency of the early-decoding scheme in [18] are improved by introducing concatenated shell codes.

3) Heterogeneous Arrival Times: The work in [16]–[19] focuses on the case where both messages are available at the time of encoding. In our previous works [20], [21], we introduced a coding scheme for the Gaussian point-to-point channel that encodes the first message before the second message arrives. The scheme proposed in [20] exploited power sharing for symbols between the arrival time of the second message and the decoding deadline of the first message. Under a Gaussian interference assumption, bounds on the error probabilities for each message were established based on the message set size and finite decoding deadline constraints. In [21], a coding scheme was proposed that exploits dirty-paper coding (DPC) [22]–[24]. We further derived rigorous bounds on the achievable error probabilities of the messages.

4) Broadcast Channels in the Finite Blocklength Regime: Finite blocklength analysis of broadcast channels (BCs) was studied in [25]–[28]. The second-order Gaussian BC setting was investigated in [26] where the authors studied the concatenate-and-code protocol [29] in which the transmitter concatenates the users’ message bits into a single data packet and each user decodes the entire packet to extract its own bits. This scheme was shown to outperform superposition coding and time division multiplexing (TDM) schemes. The work in [28] is an extension of [26] to K-user BCs. The work in [27] considered a two-user static BC and showed that under per-user reliability constraint, superposition coding combined with a rate splitting technique in which the message intended for the user with the lowest signal-to-noise ratio (SNR) (the cloud center message) is allocated to either users gives the largest second-order rate region.

## B. Main Contributions

While coding schemes for heterogeneous arrival and decoding deadlines have been developed for point-to-point channels, adapting these codes to broadcast channels remains an open problem. In this paper, we address this question by developing and analyzing a coding scheme tailored to broadcast channels with heterogeneous arrival times and decoding deadlines.

The main contributions of this work are:

* We introduce a coding scheme for two-user Gaussian BCs with heterogeneous arrival times and decoding deadlines in [20], which exploits Marton’s coding strategy [30]. Accounting for finite decoding deadline constraints (corresponding to fixed blocklengths), we first derive rigorous bounds on the achievable transmission rate for each of the messages. This is achieved by combining techniques to analyze the Gel’fand-Pinsker channel in the finite blocklength regime [23] and multiple parallel channels [15].

* With the developed bounds, we obtain further rigorous bounds for point-to-point settings with one or multiple parallel channels. In the point-to-point setting with one channel, we show that our achievability scheme outperforms the Normal Approximation in (1) proposed by Polyanskiy, Poor and Verdú in [9] especially when the number of channel uses (the size of coding block) is smaller than 200. In the point-to-point setting with multiple parallel channels our achievability bound outperforms the Normal Approximation bound proposed by Erseghe in [15].

* In the broadcast setup, we provide a second-order analysis for the achievable rate regions. We show that our scheme agrees with Marton’s bound in [30] for a sufficiently large number of channel uses.

* Finally, we show that our scheme outperforms the time-sharing scheme that transmits each message independently but over fewer number of channel uses.
<figure><img src="image_2.png" /><figcaption>Fig. 1: System model. (a) Arrival times and decoding deadlines of $m_1$ and $m_2$, (b) Problem setup with one transmitter and two receivers.</figcaption></figure>

### C. Organization

The rest of this paper is organized as follows. We end this section with some remarks on notation. Sections II and III describe the problem setup and the proposed coding scheme. Section IV and V present our main results and discussions on the related works. Section VI concludes the paper. Some technical proofs are referred to in appendices.

### D. Notation

The set of all integers is denoted by $\mathbb{Z}$, the set of positive integers by $\mathbb{Z}^+$ and the set of real numbers by $\mathbb{R}$. For other sets we use calligraphic letters, e.g., $\mathcal{X}$. Random variables are denoted by uppercase letters, e.g., $X$, and their realizations by lowercase letters, e.g., $x$. For vectors we use boldface notation, i.e., upper case boldface letters such as $\mathbf{X}$ for random vectors and lower case boldface letters such as $\mathbf{x}$ for deterministic vectors. Matrices are depicted with sans serif font, e.g., H. We also write $X^n$ for the tuple of random variables $(X_1, \dots, X_n)$ and $\mathbf{X}^n$ for the tuple of random vectors $(\mathbf{X}_1, \dots, \mathbf{X}_n)$.

## II. PROBLEM SETUP

Consider a transmitter $S$ that seeks to communicate with two receivers $Rx 1$ and $Rx 2$. It wishes to transmit message $m_1$ to receiver $Rx 1$ and message $m_2$ to receiver $Rx 2$. At time $t = a_1$, transmission commences for the first message $m_1$. At time $t = a_2$, transmission commences for the second message $m_2$. The two messages $m_1, m_2$ are assumed to be drawn independently and uniformly on $\{1, \dots, M_1\}$ and $\{1, \dots, M_2\}$, respectively.

Each message is subject to different decoding delay constraints.<fnref n="1" /> In particular, at time $d_1$, $Rx 1$ attempts to reconstruct the message $m_1$. Similarly, at time $d_2 > d_1$, $Rx 2$ attempts to reconstruct the message $m_2$. See Fig. 1a.

Under the assumption that $a_1 < a_2$ and $a_2 < d_1 < d_2$, the encoder outputs symbols at time $t \in \{a_1, \dots, d_2\}$ as

$$X_t = \begin{cases} h_t(m_1), & t \in \{a_1, \dots, a_2 - 1\} \\ \psi_t(m_1, m_2), & t \in \{a_2, \dots, d_1\} \\ \phi_t(m_2), & t \in \{d_1 + 1, \dots, d_2\}, \end{cases} \quad (2)$$

where $h, \psi, \phi$ are the encoding functions corresponding to the channel uses where only message $m_1$ has arrived but not $m_2$, where both $m_1, m_2$ are present, and $m_1$ has been decoded at $Rx 1$. We highlight that $m_2$ is not known before time $t = a_2$; i.e., encoding is causal. Define

$$n := d_2 - a_1 + 1, \quad n_{1,1} := a_2 - a_1, \quad n_{1,2} := d_1 - a_2 + 1 \quad \text{and} \quad n_{2,2} := d_2 - d_1. \quad (3)$$

<footnote n="1">¹In this work we do not consider the encoding/decoding time delays, but rather the delay due to transmission, i.e., in terms of number of channel uses.</footnote>
Choose $\beta_{1,1}$, $\beta_c$ and $\beta_{2,2}$ in $[0, 1]$ such that

$$n_{1,1}\beta_{1,1} + n_{1,2}\beta_c + n_{2,2}\beta_{2,2} = n. \quad (4)$$

We assume that the encoding functions satisfy an average block power constraint; namely,

$$\frac{1}{n} \sum_{i=a_1}^{d_2} X_i^2 \le P \quad (5)$$

and consequently

$$\frac{1}{n_{1,1}} \sum_{i=a_1}^{a_2-1} X_i^2 \le \beta_{1,1} P, \quad (6)$$

$$\frac{1}{n_{1,2}} \sum_{i=a_2}^{d_1} X_i^2 \le \beta_c P, \quad (7)$$

$$\frac{1}{n_{2,2}} \sum_{i=d_1+1}^{d_2} X_i^2 \le \beta_{2,2} P. \quad (8)$$

Denote the channel inputs by

$$\begin{aligned}
\mathbf{X}_{1,1} &= \{X_{a_1}, \dots, X_{a_2-1}\}, \\
\mathbf{X}_c &= \{X_{a_2}, \dots, X_{d_1}\}, \\
\mathbf{X}_{2,2} &= \{X_{d_1+1}, \dots, X_{d_2}\},
\end{aligned}
\quad (9)$$

and the corresponding channel outputs at Rx 1 by $\mathbf{Y}_{1,1}$ and $\mathbf{Y}_{1,2}$ and the channel outputs at Rx 2 by $\mathbf{Y}_{2,1}$ and $\mathbf{Y}_{2,2}$. The conditional distributions governing the four channels are then denoted by $f_{\mathbf{Y}_{1,1}|\mathbf{X}_{1,1}}$, $f_{\mathbf{Y}_{1,2}|\mathbf{X}_c}$, $f_{\mathbf{Y}_{2,1}|\mathbf{X}_c}$ and $f_{\mathbf{Y}_{2,2}|\mathbf{X}_{2,2}}$. We assume that each channel is additive, memoryless, stationary, and Gaussian; that is,

$$\mathbf{Y}_{1,1} = \mathbf{X}_{1,1} + \mathbf{Z}_{1,1}, \quad (10)$$

$$\mathbf{Y}_{1,2} = \mathbf{X}_c + \mathbf{Z}_{1,2}, \quad (11)$$

$$\mathbf{Y}_{2,1} = \mathbf{X}_c + \mathbf{Z}_{2,1}, \quad (12)$$

$$\mathbf{Y}_{2,2} = \mathbf{X}_{2,2} + \mathbf{Z}_{2,2}, \quad (13)$$

where $\mathbf{Z}_{i,j} \sim \mathcal{N}(0, \sigma_{i,j}^2 I)$ for $i=1,2$ and $j=1,2$.

This setup is illustrated in Fig. 1b.

Receiver Rx 1 attempts to reconstruct message $m_1$ based on the channel outputs $\mathbf{Y}_{1,1}$ and $\mathbf{Y}_{1,2}$ via the decoding function $g_1$; i.e.,

$$\hat{m}_1 = g_1(\mathbf{Y}_{1,1}, \mathbf{Y}_{1,2}). \quad (14)$$

Receiver Rx 2 attempts to reconstruct message $m_2$ based on the channel outputs $\mathbf{Y}_{2,1}$ and $\mathbf{Y}_{2,2}$ via the decoding function $g_2$; i.e.,

$$\hat{m}_2 = g_2(\mathbf{Y}_{2,1}, \mathbf{Y}_{2,2}). \quad (15)$$

Observe that both receivers are causal.

The average probability of error for each of the messages is then

$$\epsilon_1 = \mathbb{P}[\hat{m}_1 \neq m_1], \quad \epsilon_2 = \mathbb{P}[\hat{m}_2 \neq m_2]. \quad (16)$$

The focus of the remainder of this paper is on characterizing the tradeoff among the size of the message sets $M_1$ and $M_2$, the error probabilities $\epsilon_1, \epsilon_2$, and the decoding deadlines $d_1, d_2$. Formally, we study the achievable region defined as follows.

**Definition 1:** Given the power constraint $P$, a tuple $(a_1, a_2, d_1, d_2, M_1, M_2, \epsilon_1, \epsilon_2)$ is achievable if the messages $m_1$ and $m_2$ of cardinality $M_1$ and $M_2$, respectively, arriving at the $a_1$-th and $a_2$-th channel uses can be decoded by the $d_1$-th and $d_2$-th channel uses with an average probability of error not exceeding $\epsilon_1$ and $\epsilon_2$, respectively.

### III. RANDOM CODING SCHEME

In this section, we introduce our random coding scheme. Notice that instead of employing Gaussian codebooks, our analysis relies on the use of power-shell codebooks. A power-shell codebook of length $n$ consists of codewords that are uniformly distributed on the centered $(n-1)$-dimensional sphere with radius $\sqrt{n}P$ where $P$ is the average input power constraint.
## A. Encoding

The encoding process consists of three phases.

1) Transmitting only $m_1$: In the first channel, consisting of $n_{1,1}$ channel uses, only $m_1$ is known to the encoder. The channel input $\mathbf{X}_{1,1}$ corresponding to message $m_1$ is a codeword $\mathbf{X}_{1,1}(m_1) \in \mathbb{R}^{n_{1,1}}$, which is independently distributed on the sphere $\mathbb{S}^{n_{1,1}-1}$ with power $n_{1,1}\beta_{1,1}\text{P}$. That is, the probability density function of $\mathbf{X}_{1,1}$ is given by

$$f_{\mathbf{X}_{1,1}}(\mathbf{x}_{1,1}) = \frac{\delta(||\mathbf{x}_{1,1}||^2 - n_{1,1}\beta_{1,1}\text{P})}{S_{n_{1,1}}(\sqrt{n_{1,1}\beta_{1,1}\text{P}})}, \quad (17)$$

where $\delta(\cdot)$ is the Dirac delta function, and $S_n(r)$ is the surface area of a sphere of radius $r$ in $n$-dimensional space.

2) Transmitting both $m_1$ and $m_2$: Over the next $n_{1,2}$ channel uses, the encoder exploits Marton's coding strategy [30] to jointly encode $m_1$ and $m_2$. To this end, we choose $\beta_{2,1} \in [0, 1]$ and $\beta_{1,2} \in [0, 1]$ such that for a given $\rho \in [0, 1]$,

$$\beta_{1,2} + \beta_{2,1} + \rho\sqrt{\beta_{1,2}\beta_{2,1}} = \beta_c \quad (18)$$

The parameter $\rho$ is defined shortly.

The following two codebooks are then generated.

### Codebook Generation:

* Denote by $L_1$ the random coding parameter illustrating the number of auxiliary codewords for each message $m_1 \in \{1, \dots, M_1\}$. A random codebook $C_{\mathbf{X}_{1,2}}$ containing $M_1 L_1$ auxiliary codewords $\{\mathbf{X}_{1,2}(m_1, \ell_1)\}$ with $\ell_1 \in \{1, \dots, L_1\}$ and $m_1 \in \{1, \dots, M_1\}$ is generated where each codeword is independently distributed on the sphere $\mathbb{S}^{n_{1,2}-1}$ with power $n_{1,2}\beta_{1,2}\text{P}$.

* Denote by $L_2$ the random coding parameter illustrating the number of auxiliary codewords for each message $m_2 \in \{1, \dots, M_2\}$. A random codebook $C_{\mathbf{X}_{2,1}}$ containing $M_2 L_2$ auxiliary codewords $\{\mathbf{X}_{2,1}(m_2, \ell_2)\}$ with $\ell_2 \in \{1, \dots, L_2\}$ and $m_2 \in \{1, \dots, M_2\}$ is generated where each codeword is independently distributed on the sphere $\mathbb{S}^{n_{1,2}-1}$ with power $n_{1,2}\beta_{2,1}\text{P}$.

The codebooks are revealed to all terminals.

The transmitter chooses the pair $(\ell_1, \ell_2)$ such that $\mathbf{X}_{1,2}(m_1, \ell_1) \in C_{\mathbf{X}_{1,2}}$ and $\mathbf{X}_{2,1}(m_2, \ell_2) \in C_{\mathbf{X}_{2,1}}$ satisfy

$$\langle \mathbf{X}_{1,2}(m_1, \ell_1), \mathbf{X}_{2,1}(m_2, \ell_2) \rangle \in \mathcal{D} \quad (19)$$

where

$$\mathcal{D} \triangleq [n_{1,2}\sqrt{\beta_{1,2}\beta_{2,1}\text{P}}\rho : n_{1,2}\sqrt{\beta_{1,2}\beta_{2,1}\text{P}}] \quad (20)$$

and $\rho$ is the correlation parameter. If more that one such a pair exists, then one pair is selected arbitrarily.

The channel input over the $n_{1,2}$ symbols allocated for the joint transmission of $m_1$ and $m_2$ is given by

$$\mathbf{X}_c = \alpha (\mathbf{X}_{1,2}(m_1, \ell_1) + \mathbf{X}_{2,1}(m_2, \ell_2)), \quad (21)$$

where

$$\alpha := \sqrt{\frac{\beta_c}{\beta_c^*}}. \quad (22)$$

with

$$\beta_c^* := \beta_{1,2} + \beta_{2,1} + \rho^* \sqrt{\beta_{1,2}\beta_{2,1}}, \quad (23)$$

and $\rho^* \in [\rho, 1]$ is the correlation coefficient between the chosen codewords $\mathbf{X}_{1,2}(m_1, \ell_1)$ and $\mathbf{X}_{2,1}(m_2, \ell_2)$. Notice that $\alpha$ is a power normalization coefficient that ensures that the transmit signal satisfies the power constraint in (5).

We have the encoding error event:

$$\mathcal{E}_{1,2} \triangleq \{\text{no } (\ell_1, \ell_2) \text{ exists such that } (19) \text{ is satisfied}\}. \quad (24)$$

In our analysis, we set $\epsilon_{1,2} \in [0, 1]$ as the threshold on $\mathbb{P}[\mathcal{E}_{1,2}]$, i.e., given $n_{1,2}, \beta_{1,2}, \beta_{2,1}, \text{P}$ and $\rho$ we choose $L_1$ and $L_2$ such that $\mathbb{P}[\mathcal{E}_{1,2}] \le \epsilon_{1,2}$.

3) Transmitting only $m_2$: Over the last $n_{2,2}$ channel uses, the transmitter encodes only $m_2$ with a codeword $\mathbf{X}_{2,2}(m_2)$ which is independently distributed on the sphere $\mathbb{S}^{n_{2,2}-1}$ with power $n_{2,2}\beta_{2,2}\text{P}$.

## B. Decoding

Given the structure of the encoding functions, the output sequences at each receiver can be viewed as arising from two parallel channels: Rx 1 observes the two channel outputs $\mathbf{Y}_{1,1}$ and $\mathbf{Y}_{1,2}$ of $n_{1,1}$ and $n_{1,2}$ blocks, respectively; and Rx 2 observes the two channel outputs $\mathbf{Y}_{2,1}$ and $\mathbf{Y}_{2,2}$ of $n_{1,2}$ and $n_{2,2}$ blocks, respectively.
1) Decoding $m_1$: Given observations $\mathbf{Y}_{1,1}$ and $\mathbf{Y}_{1,2}$, Rx 1 estimates $m_1$ according to the pair $(\hat{m}_1, \hat{\ell}_1)$, such that the corresponding sequences $\mathbf{X}_{1,2}(\hat{m}_1, \hat{\ell}_1)$ and $\mathbf{X}_{1,1}(\hat{m}_1)$ maximize

$$i_1(\{\mathbf{x}_{1,j}\}_{j \in \{1,2\}}; \{\mathbf{y}_{1,j}\}_{j \in \{1,2\}}) := \log \prod_{j=1,2} \frac{f_{\mathbf{Y}_{1,j}|\mathbf{X}_{1,j}}(\mathbf{y}_{1,j}|\mathbf{x}_{1,j})}{f_{\mathbf{Y}_{1,j}}(\mathbf{y}_{1,j})} \quad (25)$$

over all pairs of $\mathbf{x}_{1,1}$ and $\mathbf{x}_{1,2} \in C_{\mathbf{X}_{1,2}}$. We have the following error event while decoding $m_1$:

$$\mathcal{E}_1 \triangleq \{\text{Rx 1 chooses a message } \hat{m}_1 \neq m_1\}. \quad (26)$$

Thus the average error of decoding $m_1$ is bounded by

$$\epsilon_1 \le \mathbb{P}[\mathcal{E}_{1,2}] + \mathbb{P}[\mathcal{E}_1 | \mathcal{E}_{1,2}^c]. \quad (27)$$

2) Decoding $m_2$: Given observations $\mathbf{Y}_{2,1}$ and $\mathbf{Y}_{2,2}$, Rx 2 estimates $m_2$ according to the pair $(\hat{m}_2, \hat{\ell}_2)$, such that the corresponding sequences $\mathbf{X}_{2,1}(\hat{m}_2, \hat{\ell}_2)$ and $\mathbf{X}_{2,2}(\hat{m}_2)$ maximize

$$i_2(\{\mathbf{x}_{2,j}\}_{j \in \{1,2\}}; \{\mathbf{y}_{2,j}\}_{j \in \{1,2\}}) := \log \prod_{j=1,2} \frac{f_{\mathbf{Y}_{2,j}|\mathbf{X}_{2,j}}(\mathbf{y}_{2,j}|\mathbf{x}_{2,j})}{f_{\mathbf{Y}_{2,j}}(\mathbf{y}_{2,j})} \quad (28)$$

over all pairs of $\mathbf{x}_{2,1} \in C_{\mathbf{X}_{2,1}}$ and $\mathbf{x}_{2,2}$. We have the following error event while decoding $m_2$:

$$\mathcal{E}_2 \triangleq \{\text{Rx 2 chooses a message } \hat{m}_2 \neq m_2\}. \quad (29)$$

Thus the average error of decoding $m_2$ is bounded by

$$\epsilon_2 \le \mathbb{P}[\mathcal{E}_{1,2}] + \mathbb{P}[\mathcal{E}_2 | \mathcal{E}_{1,2}^c]. \quad (30)$$

Remark 1: To improve the finite blocklength performance, all the codewords are uniformly distributed on the power shell. According to Shannon's observation, the optimal decay of the probability of error near capacity of the point-to-point Gaussian channel is achieved by codewords on the power-shell [31]. As a result of this code construction, the induced output distributions $f_{\mathbf{Y}_{i,j}}(\mathbf{y}_{i,j})$, with $i = 1, 2$, $j = 1, 2$, $f_{\mathbf{Y}_{1,2}|\mathbf{X}_{1,2}}(\mathbf{y}_{1,2}|\mathbf{x}_{1,2})$ and $f_{\mathbf{Y}_{2,1}|\mathbf{X}_{2,1}}(\mathbf{y}_{2,1}|\mathbf{x}_{2,1})$ are non-i.i.d.; thus we propose to bound the corresponding information density measure $i_i(\{\mathbf{x}_{i,j}\}_{j \in \{1,2\}}; \{\mathbf{y}_{i,j}\}_{j \in \{1,2\}})$, for each $i \in \{1, 2\}$, by

$$\tilde{i}_i(\{\mathbf{x}_{i,j}\}_{j \in \{1,2\}}; \{\mathbf{y}_{i,j}\}_{j \in \{1,2\}}) := \log \frac{f_{\mathbf{Y}_{i,i}|\mathbf{X}_{i,i}}(\mathbf{y}_{i,i}|\mathbf{x}_{i,i}) Q_{\mathbf{Y}_{i,j \neq i}|\mathbf{X}_{i,j \neq i}}(\mathbf{y}_{i,j \neq i}|\mathbf{x}_{i,j \neq i})}{\prod_{j=1}^2 Q_{\mathbf{Y}_{i,j}}(\mathbf{y}_{i,j})}, \quad (31)$$

where the $Qs$ are i.i.d Gaussian distributions. For each $i \in \{1, 2\}$, we thus introduce a parameter $J_i$, chosen such that

$$i_i(\{\mathbf{x}_{i,j}\}_{j \in \{1,2\}}; \{\mathbf{y}_{i,j}\}_{j \in \{1,2\}}) \ge \tilde{i}_i(\{\mathbf{x}_{i,j}\}_{j \in \{1,2\}}; \{\mathbf{y}_{i,j}\}_{j \in \{1,2\}}) + \log J_i. \quad (32)$$

## IV. MAIN RESULTS

Define $n_{2,1} = n_{1,2}$. For each $i \in \{1, 2\}$ let $j \in \{1, 2\} \setminus i$, and define

$$\Omega_{i,i} := \frac{\beta_{i,i} \mathbb{P}}{\sigma_{i,i}^2}, \quad (33a)$$

$$\Omega_{i,j} := \frac{(\beta_{i,j} + \rho^2 \beta_{j,i} + \rho \sqrt{\beta_{i,j} \beta_{j,i}}) \mathbb{P}}{\sigma_{i,j}^2 + (1 - \rho^2) \beta_{j,i} \mathbb{P}}, \quad (33b)$$

$$J_i := e^{\frac{n_{i,j}}{2} (\log 2 - \beta_{j,i} \mathbb{P})} \frac{4 \sqrt{\pi \beta_{i,j} \beta_{j,i} (1 + 2\Omega_{i,i})}}{243(1 + \Omega_{i,i})(\beta_{i,j} + \beta_{j,i})}. \quad (33c)$$

Our first result gives upper bounds on the achievable rates of the first and the second message.

Theorem 1: Given $n_{1,1}, n_{1,2}, n_{2,2}$ and $\mathbb{P}$, for each $i \in \{1, 2\}$, let $j \in \{1, 2\} \setminus i$, $\log L_i := n_{i,j} R_{L_i}$ and $\log M_i := R_i \sum_{j=1}^2 n_{i,j}$. For each $i \in \{1, 2\}$, we then have the following upper bound:

$$\log M_i + \log L_i \le \sum_{j=1}^2 n_{i,j} C(\Omega_{i,j}) - \sqrt{\sum_{j=1}^2 n_{i,j} V(\Omega_{i,j})} Q^{-1}(\epsilon_i - \Delta_i) + K_i \log \left( \sum_{j=1}^2 n_{i,j} \right) \quad (34)$$

subject to

$$\sum_{i=1}^{2} \log L_i \ge \log \frac{\log (\epsilon_{1,2})}{\log (1 - (1 - \rho^2)^{n_{1,2}-1})}, \quad (35)$$
where $C(x) := \frac{1}{2} \log(1+x)$ and $V(x) := \frac{x(2+x)}{2(1+x)^2} \log e^2$ with

$$ \Delta_i := \frac{6T_{\max,i}}{\sqrt{(\sum_{j=1}^2 n_{i,j} V(\Omega_{i,j}))^3}} + \frac{\prod_{j=1}^2 (1 + \Omega_{i,j})^{\frac{n_{i,j}}{2}}}{2^{E_{\min,i}} J_i (\sum_{j=1}^2 n_{i,j})^{-K_i}} + (1 - (1 - \rho^2)^{n_{i,j}-1})^{L_1 \cdot L_2}, \quad (36) $$

where $T_{\max,i}$ and $E_{\min,i}$ are defined in (37) and $\Phi(\cdot, \cdot, \cdot)$ is the Hurwitz Lerch transcendent, $\Gamma_{i,i} := \Gamma(0.5n_{i,i}+0.5)/\Gamma(0.5n_{i,i})$, $\Gamma_{i,j} := \Gamma(0.5n_{i,j}+0.5)/\Gamma(0.5n_{i,j})$, $\tilde{\beta}_c := \beta_{1,2} + \beta_{2,1} + \sqrt{\beta_{1,2}\beta_{2,1}}$, and for any $\kappa_i > 1$, $\zeta_i > 1$ and $\tilde{\zeta}_i > 1$.

*Proof: See Appendix A.*

*Proposition 1:* For sufficiently large $n_{1,1}, n_{1,2}, n_{2,2}$, and for $\rho > 0$, we have

$$ \log M_i + \log L_i \le \sum_{j=1}^{2} n_{i,j} C(\Omega_{i,j}) - \sqrt{\sum_{j=1}^{2} n_{i,j} V(\Omega_{i,j})} Q^{-1}(\epsilon_i) + O\left(\log\left(\sum_{j=1}^{2} n_{i,j}\right)\right) \quad (38) $$

subject to

$$ \sum_{i=1}^{2} \log L_i \ge \log(-\log \epsilon_{1,2}) - n_{1,2} \log(1 - \rho^2). \quad (39) $$

*Proof: See Appendix B.*

# V. DISCUSSION ON THE MAIN RESULTS AND RELATED WORKS

In this section, we review related settings that can be covered by Theorem 1.

$$ T_{\max,i} := 2^3 \kappa_i + 2^4 \max \left\{ \frac{1}{\Gamma(\frac{n_{i,i}}{2})} \zeta_i 2^{\frac{n_{i,i}-2}{4}} e^{-c_i} A(n_{i,i}, k_i, b_i, c_i), \frac{1}{\Gamma(\frac{n_{i,j}}{2})} \tilde{\zeta}_i 2^{\frac{n_{i,j}-2}{4}} e^{-\tilde{c}_i} A(n_{i,j}, \tilde{k}_i, \tilde{b}_i, \tilde{c}_i) \right\}, \quad (37a) $$

$$ E_{\min,i} := \sum_{j=1}^{2} \frac{n_{i,j}}{2} \log(1 + \Omega_{i,j}) + \frac{(1 - \sigma_{i,i}^2)n_{i,i}\Omega_{i,i}}{2(1 + \Omega_{i,i})} - \sqrt{2}\sigma_{i,i}^2 \Gamma_{i,i} k_i b_i - \frac{n_{i,j}\sigma_{i,j}^2\beta_c P}{2(\sigma_{i,j}^2 + \beta_c P)} - \frac{\beta_c P^2 n_{i,j} \beta_{j,i}}{2\sigma_{i,j}^2(\beta_c P + \sigma_{i,j}^2)} \\ + \frac{\beta_c P n_{i,j} \beta_{j,i}}{2\tilde{\beta}_c \sigma_{i,j}^2 (\beta_c P + \sigma_{i,j}^2)} \left( \sigma_{i,j}^2 + (\beta_c P - \sigma_{i,j}^2) \left( \sqrt{\tilde{\beta}_c \beta_c^{-1}} - 1 - \rho \sqrt{\beta_{j,i} \beta_{i,j}^{-1}} \right)^2 - \frac{\sqrt{2P} \tilde{\beta}_c \Gamma_{i,j}}{\sqrt{n_{i,j} \beta_{j,i}}} \right) \\ + \sqrt{n_{i,j} \beta_{i,j} P} \left( \sqrt{\beta_c \tilde{\beta}_c^{-1}} \left( \frac{1}{\sigma_{i,j}^2 + \beta_c P} + 1 + \rho \sqrt{\beta_{j,i} \beta_{i,j}^{-1}} \right) - \frac{1}{\sigma_{i,j}^2} \right) \left( \rho \sqrt{n_{i,j} \beta_{j,i} P} - \sigma_{i,j}^2 \sqrt{2\Gamma_{i,j}} \right), \quad (37b) $$

$$ A(n, k, b, c) := \left( (\tilde{\kappa} + 1)^3 - \frac{k^3 b^6}{8} - \frac{3}{2}(\tilde{\kappa} + 1)kb^2(\tilde{\kappa} + 1 - \frac{1}{2}kb^2) \right) \Phi(e^{-1}, -\frac{n}{2} + 1, c) \\ - 3k \left( 5k^2 b^4 + 4(\tilde{\kappa} + 1)((\tilde{\kappa} + 1) - 3kb^2) \right) \Phi(e^{-1}, -\frac{n}{2}, c) - 96\sqrt{2}k^3 b \Phi(e^{-1}, -\frac{n}{2} - \frac{3}{2}, c) \\ - 3kb \left( \frac{k^2 b^4}{\sqrt{2}} + 2\sqrt{2}(\tilde{\kappa} + 1)b((\tilde{\kappa} + 1) - b^2 k) \right) \Phi(e^{-1}, -\frac{n}{2} + \frac{1}{2}, c) - 64k^3 \Phi(e^{-1}, -\frac{n}{2} - 2, c) \\ + 8\sqrt{2}k^2 b(6(\tilde{\kappa} + 1) - 5kb^2) \Phi(e^{-1}, -\frac{n}{2} - \frac{1}{2}, c) + 24k(2(\tilde{\kappa} + 1)^2 - 5k^2 b^2) \Phi(e^{-1}, -\frac{n}{2} - 1, c), \quad (37c) $$

$$ \tilde{\kappa}_i := \frac{k_i b_i^2}{4} + \frac{\tilde{k}_i \tilde{b}_i^2}{4} - \frac{n_{i,i}}{2} \log(1 + \Omega_{i,i}) - \frac{n_{i,j}}{2} \log \frac{\beta_c P + \sigma_{i,j}^2}{\sigma_{i,j}^2} - \frac{n_{i,i} \Omega_{i,i}}{2(1 + \Omega_{i,i})} \\ - \tilde{k}_i \tilde{b}_i - \frac{n_{i,j} \beta_{i,j} P}{2(\beta_c P + \sigma_{i,j}^2)} + \frac{\sigma_{i,j}^2}{2} \left( \tilde{k}_i (\tilde{b}_i - \sqrt{n_{i,j} \beta_{j,i} P}) - \frac{\sqrt{n_{i,j} \beta_{i,j} P}}{\beta_c P + \sigma_{i,j}^2} \right)^2, \quad (37d) $$

$$ c_i := \frac{1}{2} \left( \sqrt{\frac{\tilde{\kappa}_i + \kappa_i}{2k_i}} - \frac{b_i}{2} \right)^2, \quad \tilde{c}_i := \frac{1}{2} \left( \sqrt{\frac{\tilde{\kappa}_i + \kappa_i}{2k_i}} - \frac{\tilde{b}_i}{2} \right)^2, \quad k_i := \frac{2 + \Omega_{i,i}}{2\sigma_{i,i}^2(1 + \Omega_{i,i})}, \quad \tilde{k}_i := \frac{2\sigma_{i,j}^2 + \beta_c P}{2\sigma_{i,j}^2(\beta_c P + \sigma_{i,j}^2)}, \quad (37e) $$

$$ b_i := \frac{\sqrt{n_{i,i}\Omega_{i,i}}}{k_i\sigma_{i,i}(1+\Omega_{i,i})}, \quad \tilde{b}_i := \sqrt{n_{i,j}\beta_{j,i}P} + \frac{\sqrt{n_{i,j}\beta_{i,j}P}}{\tilde{k}_i\sigma_{i,j}^2} \left( 1 - \sqrt{\frac{\beta_c}{\tilde{\beta}_c}} \left( 1 + \rho \sqrt{\frac{\beta_{j,i}}{\beta_{i,j}}} \right) \right) + \frac{\sqrt{n_{i,j}\beta_{i,j}P}}{\tilde{k}_i(\beta_c P + \sigma_{i,j}^2)}. \quad (37f) $$
<figure><img src="image_3.png" /><figcaption>Fig. 2: Converse and achievability bounds on $R_1$ versus $n_{1,1}$ in the point-to-point case with $\Omega_{1,1} = 0$ dB and $\epsilon_1 = 10^{-3}$. The value of $n_{1,1}$ varies from 70 to 2000 with a step size of 10.</figcaption></figure>

A. Point-to-Point Settings

1) Transmission of only $m_1$ over $n_{1,1}$ channel uses: In this setting, we have only the following channel outputs:

$$Y_{1,1} = X_{1,1} + Z_{1,1} \qquad (40)$$

with $||X_{1,1}||^2 = n_{1,1}P$ and $Z_{1,1} \sim N(0, \sigma_{1,1}^2 I_{n_{1,1}})$. Set

$$\Omega_{1,1} = \frac{P}{\sigma_{1,1}^2}. \qquad (41)$$

Theorem 2 (P2P, Only $m_1$): Given $n_{1,1}$ and $P$, $\log M_1 := n_{1,1}R_1$ is upper bounded as

$$\log M_1 \le n_{1,1}C(\Omega_{1,1}) - \sqrt{n_{1,1}V(\Omega_{1,1})}Q^{-1}(\epsilon_1 - \Delta_1) + K_1 \log(n_{1,1}), \qquad (42)$$

where

$$\Delta_1 := \frac{6T_{\max,1}}{\sqrt{(n_{1,1}V(\Omega_{1,1}))^3}} + \frac{n_{1,1}^{K_1}(1+\Omega_{1,1})^{\frac{n_{1,1}}{2}}}{J_1 \cdot 2^{E_{\min,1}}}, \qquad (43)$$

and by [32, Proposition 2]:

$$J_1 := \frac{\sqrt{8(\sigma_{1,1}^2 + 2P)}}{27\sqrt{\pi}(\sigma_{1,1}^2 + P)} \qquad (44)$$

with $K_1$ a constant. Notice that $T_{\max,1}$ and $E_{\min,1}$ can be easily calculated from (37a) and (37b) by setting $i = 1$ and $n_{i,j} = 0$.

Proof: The proof of this result follows the proof of Theorem 1 by setting

$$n_{1,2} = n_{2,1} = n_{2,2} = 0 \quad \text{and} \quad \beta_{1,1} = 1. \qquad (45)$$

Remark 2: By choosing $K_1$ such that $\Delta_1 = O(n_{1,1}^{-3/2})$, the bound in (42) matches the Normal Approximation bound in [9, Eq. 223] which is for Gaussian channels.

We now make a more specific comparison with the work of Polyanskiy, Poor and Verdú in [9]. To this end, note that, the authors in [9] show for both discrete memoryless channels and Gaussian channels, the Normal Approximation achieves
$$ \log M_1 = n_{1,1}C(\Omega_{1,1}) - \sqrt{n_{1,1}V(\Omega_{1,1})}Q^{-1}(\epsilon_1) + O(\log(n_{1,1})). \quad (46) $$

See [9, Eq.223]. The authors then show that for the Gaussian channels with equal-power and maximal-power constraints, the $O(\log(n_{1,1}))$ can be bounded as

$$ O(\log(n_{1,1})) \le \frac{1}{2} \log n_{1,1} + O(1), \quad (47) $$

and with the average power constraint as

$$ O(\log(n_{1,1})) \le \frac{3}{2} \log n_{1,1} + O(1). \quad (48) $$

See [9, Eq.294 and Eq.295]. To compare our achievability bound in (42) with the Normal Approximation, Fig. 2 illustrates the bound in (42) for $K_1 = 0.5$ and $K_1 = 1$ as well as the Normal Approximation with $O(\log(n_{1,1}))$ equal to 0 and $0.5 \log n_{1,1}$. The converse bound is the meta-converse bound proposed in [9, Theorem 41]. As can be seen from this figure, our achievability schemes with $K_1 = 1$ matches the converse bound for large values of $n_{1,1}$ and has an improvement over the Normal approximation especially for values of $n_{1,1}$ smaller than 200.

2) Transmission of only $m_1$ over two parallel channels of $n_{1,1}$ and $n_{1,2}$ channel uses: In this setting, we have the following two channel outputs:

$$ Y_{1,1} = X_{1,1} + Z_{1,1}, \quad Y_{1,2} = X_{1,2} + Z_{1,2}, \quad (49) $$

with $||X_{1,1}||^2 = n_{1,1}\beta_{1,1}P$, $||X_{1,2}||^2 = n_{1,2}\beta_{1,2}P$, and $Z_{1,1} \sim N(0, \sigma_{1,1}^2 I_{n_{1,1}})$ and $Z_{1,2} \sim N(0, \sigma_{1,2}^2 I_{n_{1,2}})$. The power sharing parameters $\beta_{1,1} \in [0, 1]$ and $\beta_{1,2} \in [0, 1]$ are chosen such that

$$ \beta_{1,1}n_{1,1} + \beta_{1,2}n_{1,2} = n. \quad (50) $$

Set

$$ \Omega_{1,1} = \frac{\beta_{1,1}P}{\sigma_{1,1}^2} \quad \text{and} \quad \Omega_{1,2} = \frac{\beta_{1,2}P}{\sigma_{1,2}^2}. \quad (51) $$

Theorem 3 (P2P, Parallel Channels, Only $m_1$): Given $n_{1,1}, n_{1,2}, \beta_{1,1}, \beta_{1,2}$ and $P$, $\log M_1 := R_1(n_{1,1} + n_{1,2})$ is upper bounded as

$$ \log M_1 \le \sum_{j=1}^{2} n_{1,j} C(\Omega_{1,j}) - \sqrt{\sum_{j=1}^{2} n_{1,j} V(\Omega_{1,j})} Q^{-1}(\epsilon_1 - \Delta_1) + K_1 \log \left( \sum_{j=1}^{2} n_{1,j} \right), \quad (52) $$

where

$$ \Delta_1 := \frac{6T_{\max,1}}{\sqrt{(\sum_{j=1}^2 n_{1,j} V(\Omega_{1,j}))^3}} + \frac{\prod_{j=1}^2 (1 + \Omega_{1,j})^{\frac{n_{1,j}}{2}}}{2^{E_{\min,1}} J_1 (\sum_{j=1}^2 n_{1,j})^{-K_1}} \quad (53) $$

with

$$ J_1 := \frac{\sqrt{8(\sigma_{1,1}^2 + 2\beta_{1,1}P)}}{27\sqrt{\pi}(\sigma_{1,1}^2 + \beta_{1,1}P)} \cdot \frac{\sqrt{8(\sigma_{1,2}^2 + 2\beta_{1,2}P)}}{27\sqrt{\pi}(\sigma_{1,2}^2 + \beta_{1,2}P)} \quad (54) $$

and $K_1$ a constant. Notice that $T_{\max,1}$ and $E_{\min,1}$ can be easily calculated from (37a) and (37b) by setting $i = 1, \beta_{2,1} = 0$ and $\beta_c = \beta_{1,2}$.

*Proof:* The proof follows the proof of Theorem 1 by setting

$$ n_{2,2} = 0, \quad \beta_{2,1} = 0, \quad \text{and} \quad \beta_{2,2} = 0. \quad (55) $$

*Remark 3:* By choosing $K_1$ such that $\Delta_1 = O(n_{1,1}^{-3/2})$, and $n_{1,1} = n_{1,2}$, the bound in (52) matches the Normal Approximation bound in [15, Eq. 54] that is for a two-parallel additive white Gaussian noise (AWGN) channels. This setting can be easily extended to a setting with more than two parallel channels.

We now make a more detailed comparison with the work of Erseghe in [15]. The leading idea of [15] is based on the fact that a probability of the form $P[\sum_{i=1}^n u_i \ge n\lambda]$ where $u_i$s are i.i.d continues random variables can be written and numerically evaluated using standard Laplace transforms. The author further employs this idea to propose new asymptotic approximations for the meta-converse and random union coding (RCU) achievability bounds for the parallel AWGN channels. In [15, Theorem 10] the author shows that in a $K$-parallel AWGN channels with each channel of $n_K$ channel uses, the meta-converse bound, proposed by Polyanskiy, Poor and Verdú in [9, Theorem 41], is consistent with the normal approximation (1) with

$$ V = \frac{1}{K} \sum_{k=1}^{K} \frac{\Omega_k(2 + \Omega_k)}{2(1 + \Omega_k)^2} \quad \text{and} \quad C = \frac{1}{K} \sum_{k=1}^{K} \frac{1}{2} \log(1 + \Omega_k), \quad (56) $$
<figure><img src="image_4.png" /><figcaption>Fig. 3: Converse and achievability bounds on $R_1$ versus $n_{1,1} + n_{1,2}$ in the two parallel Gaussian channel case with $\Omega_{1,1} = 20$ dB, $\Omega_{1,1} = \nu\Omega_{2,2}$ and $\epsilon_1 = 10^{-6}$. The value of $n = n_{1,1} + n_{1,2}$ varies from 20 to 2000 with a step size of 10.</figcaption></figure>

where $\Omega_k$ is the SNR of the k-th channel. We now numerically compare our bound in (52) with the normal approximation bound in [15] for the case of 2-parallel channels. Since in [15] all K channels are of equal blocklength thus we assume $n_{1,1} = n_{1,2}$ and $\Omega_{1,1} = \nu \cdot \Omega_{2,2}$ for some $\nu > 0$. Fig. 3 illustrates this comparison for $\Omega_{1,1} = 20$ dB, $\nu = 0.5$ and $\nu = 1$, $\epsilon_1 = 10^{-6}$ and $K_1 = 1.5$. In the case that channels are of equal SNRs, i.e., $\nu = 1$, the Normal Approximation for K-parallel channels each of blocklength $\frac{n}{K}$ is equivalent to the Normal Approximation for one channel of blocklength $n$. As can be seen from Fig. 3, for the case of $\nu = 1$, the Normal Approximation is not a good approximation of the meta-converse bound. Also, our achievability bound in (52) with $K_1 = 1.5$ outperforms the normal approximation for $\nu = 1$ and $\nu = 0.5$ specifically at small values of $n = n_{1,1} + n_{1,2}$.

## B. Broadcast Setting: Marton's Bounds

In this section, we compare our results with Marton's inner bound in [30] proposed for a general two-receiver broadcast channel. In [30], the Tx wishes to transmit message $m_1$ to Rx 1 and message $m_2$ to Rx 2 each over n channel uses. The Tx thus encodes $m_1$ using the codeword $U_1^n(m_1)$ and encodes $m_2$ using the codeword $U_2^n(m_2)$ where $U_1$ and $U_2$ are distributed according to $f_{U_1U_2}(u_1, u_2)$ and are correlated with a correlation parameter $\rho$ such that

$$ \langle U_1, U_2 \rangle = n\rho\sqrt{\beta_{1,2}\beta_{2,1}}\mathbf{P}. \quad (57) $$

With the codewords $U_1$ and $U_2$, the following codeword is formed:

$$ X = U_1 + U_2, \quad (58) $$

and is transmitted to Rx 1 over the channel $f_{Y_1|X}$ and to Rx 2 over the channel $f_{Y_2|X}$. Here, $Y_1$ and $Y_2$ denote the output sequences of the first and second channels, respectively.

Let $R_1 := \log M_1/n$ and $R_2 := \log M_2/n$, then the following inner bounds hold for this setting:

**Theorem 4 (Marton's bound):** The capacity region $C$ of this setting is a set of rate pairs $(R_1, R_2)$ satisfying

$$ R_1 \le I(U_1; Y_1), \quad (59a) $$

$$ R_2 \le I(U_2; Y_2), \quad (59b) $$

$$ R_1 + R_2 \le I(U_1; Y_1) + I(U_2; Y_2) - I(U_1; U_2). \quad (59c) $$

To compare this setup with ours, let $m_1$ and $m_2$ to be jointly sent over the entire n channel uses. Rx 1 observes the channel outputs $Y_{1,1}$ and Rx 2 observes the channel outputs $Y_{2,2}$ given by

$$ Y_{1,1} = \alpha(X_{1,2} + X_{2,1}) + Z_{1,1}, \quad Y_{2,2} = \alpha(X_{1,2} + X_{2,1}) + Z_{2,2} \quad (60) $$
<figure><img src="image_5.png" /><figcaption>Fig. 4: Effect of the correlation parameter $\rho$ on the bounds in Theorem 1 and Proposition 2.</figcaption></figure>

where $X_{2,1} \sim \mathcal{N}(0, \beta_{1,2}PI_n)$ and $X_{1,2} \sim \mathcal{N}(0, \beta_{2,1}I_n)$ where $\beta_{1,2}$ and $\beta_{2,1}$ are chosen such that

$$ \beta_{1,2} + \beta_{2,1} + \rho\sqrt{\beta_{1,2}\beta_{2,1}} = 1. \qquad (61) $$

For this Gaussian case, Theorem 4 can be written as the following proposition.

**Proposition 2:** Let $R_1 := \frac{\log M_1}{n}$, $R_2 := \frac{\log M_2}{n}$, $R_{L_1} := \frac{\log L_1}{n}$, $R_{L_2} := \frac{\log L_2}{n}$, $X_{1,2} \sim \mathcal{N}(0, \beta_{1,2}PI_n)$ and $X_{2,1} \sim \mathcal{N}(0, \beta_{2,1}PI_n)$, we have the following inner bounds:

$$ R_1 + R_{L_1} \le \frac{1}{2} \log \left( \frac{\sigma_{1,1}^2 + P}{\sigma_{1,1}^2 + (1-\rho^2)\beta_{2,1}P} \right) \qquad (62a) $$

(62b)

and

$$ R_2 + R_{L_2} \le \frac{1}{2} \log \left( \frac{\sigma_{2,2}^2 + P}{\sigma_{2,2}^2 + (1-\rho^2)\beta_{1,2}P} \right) \qquad (62c) $$

subject to

$$ R_{L_1} + R_{L_2} \ge -\frac{1}{2} \log(1 - \rho^2). \qquad (63) $$

**Remark 4:** By setting $n_{1,1} = n_{2,2} = 0$, $n_{1,2} = n$ and

$$ \epsilon_{1,2} = \frac{1}{2(1-\rho^2)^{\frac{n_{1,2}}{2}}}, \qquad (64) $$

**Proposition 1** matches **Proposition 2** in the asymptotic regime (i.e., when $n_{1,2} \to \infty$) and when all the codewords are i.i.d Gaussian.

**Remark 5:** If all the codewords are i.i.d Gaussian, by setting either $L_1$ or $L_2$ to 0, then in the asymptotic regime **Proposition 1** matches the DPC results of Costa [22] and in the finite blocklength regime matches the results of Scarlett [23, Theorem 2].

The correlation parameter $\rho$ is of a vital importance in both our scheme and Marton's scheme. This is due to the fact that increasing $\rho$ increases the right-hand side of (34) and (62) as well as the right-hand side of (35) and (63) of Theorem 1 and Proposition 2, respectively. More specifically, increasing $\rho$ increases the upper bound on $\log M_i + \log L_i$ as well as the lower bound on $\sum_{i=1}^2 \log L_i$. Hence, increasing $\rho$ will not always increase the upper bound on $R_i$. Fig. 4 illustrates the effects of increasing $\rho$ on upper bounds on $R_1$ and $R_2$ in (34) and Marton's bounds in Proposition 2. To make a comparison with Marton's bounds, in this figure, we set $n = n_{1,2}$, $n_{1,1} = n_{2,2} = 0$, $\epsilon_{1,2} = 10^{-3}$, $K_1 = K_2 = 1.5$, $\beta_{1,2} = \beta_{2,1}$, $L_1 = L_2$, $R_1 = R_2$, and $\sigma_{1,2}^2 = \sigma_{2,1}^2$. Fig. 4a is for the case where $n = 5000$ and $P/\sigma_{1,2}^2$ is equal to 10 dB and 0 dB. As can be seen from Fig. 4a, when $P/\sigma_{1,2}^2$ is equal to 10 dB, the transmission rate under our scheme and Marton's scheme maximizes around $\rho = 0.9$ and decays for the values of $\rho$ larger than 0.9. Whereas, for $P/\sigma_{1,2}^2$ equal to 0 dB, the transmission rate under our
<figure><img src="image_6.png" /><figcaption>Fig. 5: Example of comparing our scheme with the time-sharing scheme for $n = 300$ and $n_{1,2}$ taking values from 200 to 0 with a step size of 50.</figcaption></figure>

scheme and Marton's scheme maximizes at $\rho = 0$. A similar trend can be seen for very small values of $n$. See Fig. 4b. Notice that the upper bound is the case where $R_{L_1}$ (or $R_{L_2}$) is set at 0 thus $R_1$ (or $R_2$) can be maximized.

### C. Time-sharing: Transmission of both $m_1$ and $m_2$

In this setting, we divide the $n_{1,2}$ channel uses into two parts $\eta n_{1,2}$ and $(1 - \eta)n_{1,2}$ for $\eta \in [0, 1]$. Therefore, $m_1$ is transmitted over $n_{1,1} + \eta n_{1,2}$ channel uses and $m_2$ is transmitted over $(1 - \eta)n_{1,2} + n_{2,2}$ channel uses. Transmissions of $m_1$ and $m_2$ are thus independent. In this setting, we have the following two channel outputs:

$$ Y_{1,1} = X_{1,1} + Z_{1,1}, \quad Y_{2,2} = X_{2,2} + Z_{2,2}, \qquad (65) $$

with $||X_{1,1}||^2 = (n_{1,1} + \eta n_{1,2})\beta_{1,1}P$, $||X_{2,2}||^2 = ((1 - \eta)n_{1,2} + n_{2,2})\beta_{2,2}P$, and $Z_{1,1} \sim N(0, \sigma_{1,1}^2 I_{n_{1,1}+\eta n_{1,2}})$ and $Z_{2,2} \sim N(0, \sigma_{2,2}^2 I_{(1-\eta)n_{1,2}+n_{2,2}})$. The power sharing parameters $\beta_{1,1} \in [0, 1]$ and $\beta_{2,2} \in [0, 1]$ are chosen such that

$$ (n_{1,1} + \eta n_{1,2})\beta_{1,1} + ((1 - \eta)n_{1,2} + n_{2,2})\beta_{2,2} = n. \qquad (66) $$

Set

$$ \Omega_{1,1} = \frac{\beta_{1,1}P}{\sigma_{1,1}^2} \quad \text{and} \quad \Omega_{2,2} = \frac{\beta_{2,2}P}{\sigma_{2,2}^2}. \qquad (67) $$

**Theorem 5 (Time-Sharing, Both $m_1$ and $m_2$):** Given $n_{1,1}$, $n_{2,2}$, $\beta_{1,1}$, $\beta_{2,2}$ and $P$, $\log M_1 := (n_{1,1} + \eta n_{1,2})R_1$ and $\log M_2 := (n_{2,2} + (1 - \eta)n_{1,2})R_2$ are upper bounded as

$$ \log M_1 \le (n_{1,1} + \eta n_{1,2})C(\Omega_{1,1}) - \sqrt{(n_{1,1} + \eta n_{1,2})V(\Omega_{1,1})}Q^{-1}(\epsilon_1 - \Delta_1) + \tilde{K}_1 \log((n_{1,1} + \eta n_{1,2})), \qquad (68) $$

and

$$ \log M_2 \le (n_{2,2} + (1 - \eta)n_{1,2})C(\Omega_{2,2}) - \sqrt{(n_{2,2} + (1 - \eta)n_{1,2})V(\Omega_{2,2})}Q^{-1}(\epsilon_2 - \Delta_2) + \tilde{K}_2 \log((n_{2,2} + (1 - \eta)n_{1,2})), \qquad (69) $$

where

$$ \Delta_1 := \frac{6T_{\max,1}}{\sqrt{(n_{1,1} + \eta n_{1,2})V(\Omega_{1,1})}} + \frac{(n_{1,1} + \eta n_{1,2})^{\tilde{K}_1} (1 + \Omega_{1,1})^{\frac{(n_{1,1}+\eta n_{1,2})}{2}}}{J_1 \cdot 2^{E_{\min,1}}} \qquad (70) $$
<figure><img src="image_7.png" /><figcaption>Fig. 6: Comparison between the transmission rate pairs ($R_1, R_2$) obtained in our scheme and the time-sharing scheme under the example shown in Fig. 5. (a)In our scheme: $R_1 = \log M_1/(n_{1,1} + n_{1,2})$ and $R_2 = \log M_2/(n_{1,1} + n_{2,2})$, in the time-sharing scheme: $R_1 = \log M_1/(n_{1,1} + \eta n_{1,2})$ and $R_2 = \log M_2/(n_{1,1} + (1 - \eta)n_{1,2})$ with $\eta = 0.5$, (b) In both schemes: $R_1 = \log M_1/n$ and $R_2 = \log M_2/n$.</figcaption></figure>

$$ \Delta_2 := \frac{6T_{\max,2}}{\sqrt{(n_{2,2} + (1-\eta)n_{1,2})V(\Omega_{2,2})}^3} + \frac{(n_{2,2} + (1-\eta)n_{1,2})^{\tilde{K}_2}(1+\Omega_{2,2})^{\frac{(n_{2,2}+1-\eta)n_{1,2}}{2}}}{J_2 \cdot 2^{E_{\min,2}}} \quad (71) $$

(72)

with

$$ J_1 := \frac{\sqrt{8(\sigma_{1,1}^2 + 2\beta_{1,1}P)}}{27\sqrt{\pi}(\sigma_{1,1}^2 + \beta_{1,1}P)}, \quad J_2 := \frac{\sqrt{8(\sigma_{2,2}^2 + 2\beta_{2,2}P)}}{27\sqrt{\pi}(\sigma_{2,2}^2 + \beta_{2,2}P)}, \quad (73) $$

with $\tilde{K}_1$ and $\tilde{K}_2$ being constants. Notice that $T_{\max,1}$ and $E_{\min,1}$ can be easily calculated from (37a) and (37b) by setting $i=1, n_{1,2}=0$ and replacing $n_{1,1}$ by $n_{1,1} + \eta n_{1,2}$. Similarly, $T_{\max,2}$ and $E_{\min,2}$ can be calculated from (37a) and (37b) by setting $i=2, n_{2,1}=0$ and replacing $n_{2,2}$ by $n_{2,2} + (1 - \eta)n_{1,2}$.

*Proof:* Follows the proof of Theorem 1 by setting

$$ n_{1,2} = n_{2,1} = 0, \quad \beta_{1,2} = 0, \quad \beta_{2,1} = 0, \quad (74) $$

and replacing $n_{1,1}$ by $n_{1,1} + \eta n_{1,2}$ and $n_{2,2}$ by $n_{2,2} + (1 - \eta)n_{1,2}$.

To compare our scheme with the time-sharing scheme we consider a scenario where the total number of available channel uses is equal to $n = 300$. The first message $m_1$ arrives at the beginning of the first channel use (i.e., $a_1 = 1$) and is sent over 200 channel uses (i.e., $d_1 = 200$). The second message $m_2$ arrives at the time $a_2 \in [1:200]$ and has to be decoded at the end of 300 channel uses (i.e., $d_2 = 300$). Fig. 5 illustrates a schematic representation of this example under our scheme (Fig. 5.a) and the time sharing scheme with $\eta = 0.5$ (Fig. 5.b).

In Fig. 6, we plot the transmission rates $R_1$ and $R_2$ of our scheme and the time-sharing scheme for this example. Each point corresponds to a different value of $n_{1,2}$ shown in the example starting from $n_{1,2} = 200$ to $n_{1,2} = 0$ with a step size of 50. At $n_{1,2} = 0$, our scheme coincides with the time-sharing scheme. In this figure, $P = 5$, $\rho = 0.9$, $\sigma_{1,1}^2 = \sigma_{1,2}^2 = \sigma_{2,1}^2 = \sigma_{2,2}^2 = 1$, $K-1 = K_2 = \tilde{K}_1 = \tilde{K}_2 = 0.5$ and the values of the parameters $\beta_{1,1}, \beta_c, \beta_{2,2}, \beta_{1,2}, \beta_{2,1}$ are optimized to obtain the maximum sum transmission rates. In Fig. 6a, in our scheme $R_1 = \log M_1/(n_{1,1} + n_{1,2})$ and $R_2 = \log M_2/(n_{1,1} + n_{2,2})$ and in the time sharing scheme $R_1 = \log M_1/(n_{1,1} + \eta n_{1,2})$ and $R_2 = \log M_2/(n_{1,1} + (1 - \eta)n_{1,2})$. In Fig. 6b, in both schemes $R_1 = \log M_1/n$ and $R_2 = \log M_2/n$. As can be seen from this figure, our scheme significantly outperforms the time-sharing scheme.

An important difference between our scheme and the time-sharing scheme appears when the channels to one receiver are stronger than the channels to another receiver. Under our scheme, it is possible to adjust the design parameters $L_1$ and $L_2$ such that the rate to the weaker receiver increases while keeping the sum-rate approximately constant. In Theorem 1, it is required that parameters $L_1$ and $L_2$ to be chosen such the condition in (35) is satisfied. It is clear that, large values of $L_1$ and $L_2$ reduce the transmission rates $R_1$ and $R_2$ and vice versa. In the analysis provided in the previous Section V-B, $L_1$ and $L_2$
<figure><img src="image_8.png" /><figcaption>Fig. 7: The figure illustrates the effect of changing $\lambda = \frac{1}{n_{1,2}} \log(\frac{L_2}{L_1})$ on the achievable rates $R_1$ and $R_2$ in a scenario where $\sigma_{1,1}^2 = \sigma_{1,2}^2 = 1$, and $\sigma_{2,1}^2 = \sigma_{2,2}^2 = 0.1$ and (a) $n_{1,1} = n_{1,2} = n_{2,2} = 60$, (b) $n_{1,1} = n_{1,2} = n_{2,2} = 180$. In this figure, $P = 1$, $\epsilon_1 = \epsilon_2 = 10^{-5}$, $\epsilon_{12} = 10^{-7}$ and $\rho = 0.8$.</figcaption></figure>

are set to be equal. In this section, we focus on the effect of changing the values of $L_1$ and $L_2$ on the achievable rates $R_1$ and $R_2$. To this end, we introduce a parameter $\lambda$ as

$$ \lambda := \frac{1}{n_{1,2}} \log \left( \frac{L_2}{L_1} \right). \qquad (75) $$

By (35),

$$ \log L_1 \ge \frac{1}{2} \log \frac{\log(\epsilon_{1,2})}{\log(1 - (1 - \rho^2)^{n_{1,2}-1})} - \frac{n_{1,2}\lambda}{2}, \qquad (76) $$

and

$$ \log L_2 \ge \frac{1}{2} \log \frac{\log(\epsilon_{1,2})}{\log(1 - (1 - \rho^2)^{n_{1,2}-1})} + \frac{n_{1,2}\lambda}{2}. \qquad (77) $$

Therefore, increasing $\lambda$, increases $L_2$ (decreases $L_1$) which results in decreasing $R_2$ (increasing $R_1$).

To numerically evaluate the effect of $\lambda$, we consider a case where the channels of the first receiver are weaker than the channels of the second receiver. In Fig. 7, we assume that $\sigma_{1,1}^2 = \sigma_{1,2}^2 = 1$ and $\sigma_{2,1}^2 = \sigma_{2,2}^2 = 0.1$. In Fig. 7a, we set $n_{1,1} = n_{1,2} = n_{2,2} = 60$ and in Fig. 7b we set $n_{1,1} = n_{1,2} = n_{2,2} = 180$. In this figure, $P = 1$, $\epsilon_1 = \epsilon_2 = 10^{-5}$, $\epsilon_{12} = 10^{-7}$, $K_1 = K_2 = 0.5$, $\rho = 0.8$ and the values of power coefficients $\beta_{1,1}, \beta_{c}, \beta_{2,2}, \beta_{1,2}, \beta_{2,1}$ are set such that the sum-rate $R_1$ and $R_2$ is maximized. We then increase $\lambda$ from 0 (i.e., $L_1 = L_2$) to 1 with step sizes of 0.2. As can be seen from this figure, by increasing $L_2$ and consequently decreasing $L_1$, it is possible to increase the rate of $R_1$ while keeping the sum-rate $R_1 + R_2$ constant. For example, under our scheme, it is possible to achieve $R_1 = R_2 = 0.31$ by setting $\lambda$ at approximately 0.95.

## VI. CONCLUSIONS

We have considered a broadcast setting in which a transmitter sends two different messages to two receivers. Messages are considered to have different arrival times and decoding deadlines such that their transmission windows overlap. For this setting, we have proposed a coding scheme that exploits Marton's coding strategy. We have derived rigorous bounds on the achievable rate regions for broadcast setting and point-to-point settings with one or multiple parallel channels. In the point-to-point setup with one channel and more parallel channels, the proposed achievability scheme outperforms the Normal Approximation especially when the number of channel uses is smaller than 200. In the broadcast setup, our scheme agreed with Marton's strategy for sufficiently large numbers of channel uses. Our numerical analysis have shown significant performance improvements over standard approaches based on time sharing for transmission of short packets.

# APPENDIX A

## PROOF OF THEOREM 1

In the following subsections we analyze the probability that events $\mathcal{E}_{1,2}$ and $\mathcal{E}_1|\mathcal{E}_{1,2}^c$ occur. The analysis related to $\mathbb{P}[\mathcal{E}_2|\mathcal{E}_{1,2}^c]$ is similar to that of $\mathbb{P}[\mathcal{E}_1|\mathcal{E}_{1,2}^c]$.
A. Analyzing $\mathbb{P}[\mathcal{E}_{1,2}]$

Recall the definition of the error event $\mathcal{E}_{1,2}$ from (24). Let

$$ \cos(\theta) = \frac{\langle \mathbf{X}_{1,2}(m_1, \ell_1), \mathbf{X}_{2,1}(m_2, \ell_2) \rangle}{n_{1,2} \sqrt{\beta_{1,2}\beta_{2,1}} \mathbf{P}}. \qquad (78) $$

Then,

$$ \mathbb{P}[\mathcal{E}_{1,2}] = \mathbb{P}[\langle \mathbf{X}_{1,2}, \mathbf{X}_{2,1} \rangle \notin \mathcal{D}] \qquad (79) $$

$$ = \mathbb{P}[\langle \mathbf{X}_{1,2}, \mathbf{X}_{2,1} \rangle \notin [n_{1,2} \sqrt{\beta_{1,2}\beta_{2,1}} \mathbf{P} \rho : n_{1,2} \sqrt{\beta_{1,2}\beta_{2,1}} \mathbf{P}]] \qquad (80) $$

$$ = \prod_{\ell_1=1}^{L_1} \prod_{\ell_2=1}^{L_2} \left(1 - \mathbb{P}\left[\langle \mathbf{X}_{1,2}(m_1, \ell_1), \mathbf{X}_{2,1}(m_2, \ell_2) \rangle \in [n_{1,2} \sqrt{\beta_{1,2}\beta_{2,1}} \mathbf{P} \rho : n_{1,2} \sqrt{\beta_{1,2}\beta_{2,1}} \mathbf{P}]\right]\right) \qquad (81) $$

$$ = \left(1 - \mathbb{P}\left[\langle \mathbf{X}_{1,2}(m_1, 1), \mathbf{X}_{2,1}(m_2, 1) \rangle \in \left[n_{1,2} \sqrt{\beta_{1,2}\beta_{2,1}} \mathbf{P} \rho : n_{1,2} \sqrt{\beta_{1,2}\beta_{2,1}} \mathbf{P}\right]\right]\right)^{L_1 L_2} \qquad (82) $$

$$ = \left(1 - \mathbb{P}\left[\rho \le \cos(\theta) \le 1\right]\right)^{L_1 L_2} \qquad (83) $$

$$ = (1 + \mathbb{P}[\cos(\theta) \le \rho] - \mathbb{P}[\cos(\theta) \le 1])^{L_1 L_2} \qquad (84) $$

$$ = (1 + F_{\cos(\theta)}(\rho) - F_{\cos(\theta)}(1))^{L_1 L_2} \qquad (85) $$

$$ = (F_{\cos^2(\theta)}((\rho)^2))^{L_1 L_2} \qquad (86) $$

$$ = (I_{(\rho)^2}(1, n_{1,2} - 1))^{L_1 L_2} \qquad (87) $$

$$ = \left(1 - (1 - \rho^2)^{n_{1,2}-1}\right)^{L_1 L_2}. \qquad (88) $$

Note that in (85), $\cos^2(\theta) \sim \text{Beta}(1, n_{1,2} - 1)$. As such, it follows that

$$ F_{\cos^2(\theta)}(x) = I_x(1, n_{1,2} - 1) = 1 - (1 - x)^{n_{1,2}-1}, \qquad (89) $$

where $I_x(\cdot, \cdot)$ is regularized incomplete beta function. It then follows that

$$ \mathbb{P}[\mathcal{E}_{1,2}] = \left(1 - (1 - \rho^2)^{n_{1,2}-1}\right)^{L_1 L_2}. \qquad (90) $$

In order to upper bound this error probability by a given threshold $\epsilon_{1,2}$, $L_1$ and $L_2$ should be chosen such that

$$ L_1 \cdot L_2 \ge \frac{\log(\epsilon_{1,2})}{\log(1 - (1 - \rho^2)^{n_{1,2}-1})}. \qquad (91) $$

This proves the bound in (35).

B. Analyzing $\mathbb{P}[\mathcal{E}_1|\mathcal{E}_{1,2}^c]$

To analyze $\mathbb{P}[\mathcal{E}_1|\mathcal{E}_{1,2}^c]$, we use the threshold-based metric bound [9]. Let $\gamma_1 \in \mathbb{R}$, since the first decoder selects among $M_1 L_1$ codewords, thus

$$ \mathbb{P}[\mathcal{E}_1|\mathcal{E}_{1,2}^c] \le \mathbb{P}[i_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \le \gamma_1] + M_1 L_1 \mathbb{P}[i_1(\bar{\mathbf{X}}_{1,1}, \bar{\mathbf{X}}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \ge \gamma_1] \qquad (92) $$

where $\bar{\mathbf{X}}_{1,1} \sim f_{\mathbf{X}_{1,1}}(\mathbf{x}_{1,1})$ and $\bar{\mathbf{X}}_{1,2} \sim f_{\mathbf{X}_{1,2}}(\mathbf{x}_{1,2})$ and are independent of $(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})$. Throughout our analysis, we interpret $\mathbb{P}[i_1(\bar{\mathbf{X}}_{1,1}, \bar{\mathbf{X}}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \le \gamma_1]$ as the *missed-detection* probability and $\mathbb{P}[i_1(\bar{\mathbf{X}}_{1,1}, \bar{\mathbf{X}}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \ge \gamma_1]$ as the *false alarm* probability.

1) Analyzing the *missed-detection probability*: To calculate the *missed-detection probability*, i.e., $\mathbb{P}[i_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \le \gamma_1]$, first we define the following Gaussian distributions:

$$ Q_{\mathbf{Y}_{1,j}}(\mathbf{y}_{1,j}) \sim \mathcal{N}(\mathbf{y}_{1,j} : \mathbf{0}, \sigma_{\mathbf{y}_{1,j}}^2 I_{n_{1,1}}), \quad \text{for } j = 1, 2 \qquad (93) $$

$$ Q_{\mathbf{Y}_{2,1}|\mathbf{X}_{1,2}}(\mathbf{y}_{1,2}|\mathbf{x}_{1,2}) \sim \mathcal{N}(\mathbf{Y}_{2,1} : \mu_{1,2}, \sigma_{\mathbf{y}_{1,2}|\mathbf{x}_{1,2}}^2 I_{n_{1,2}}), \qquad (94) $$

where

$$ \sigma_{\mathbf{y}_{1,1}}^2 = \beta_{1,1} \mathbf{P} + \sigma_{1,1}^2, \qquad (95a) $$

$$ \sigma_{\mathbf{y}_{1,2}}^2 = \beta_{c} \mathbf{P} + \sigma_{1,2}^2, \qquad (95b) $$

$$ \sigma_{1,2}^2 \le \sigma_{\mathbf{y}_{1,2}|\mathbf{x}_{1,2}}^2 \le (1 - \rho^2)\beta_{2,1} \mathbf{P} + \sigma_{1,2}^2, \qquad (95c) $$
$$ \sqrt{\frac{\beta_c}{\tilde{\beta}_c}} \left( 1 + \rho \sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}} \right) \mathbf{X}_{1,2} \le \mu_{1,2} \le \left( 1 + \sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}} \right) \mathbf{X}_{1,2}. \qquad (95d) $$

with $\tilde{\beta}_c := \beta_{1,2} + \beta_{2,1} + \sqrt{\beta_{1,2}\beta_{2,1}}$. We then introduce

$$ \tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \triangleq \log \frac{f_{\mathbf{Y}_{1,1}|\mathbf{X}_{1,1}}(\mathbf{y}_{1,1}|\mathbf{x}_{1,1}) Q_{\mathbf{Y}_{2,1}|\mathbf{X}_{1,2}}(\mathbf{y}_{1,2}|\mathbf{x}_{1,2})}{\prod_{j=1}^2 Q_{\mathbf{Y}_{1,j}}(\mathbf{y}_{1,j})}. \qquad (96) $$

**Lemma 1:** The following inequality holds:

$$ i_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \ge \tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) + \log J_1, \qquad (97) $$

where $J_1$ is defined in (33c).

*Proof:* See Appendix D.

Thus

$$ \mathbb{P}[i_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \le \gamma_1] \qquad (98) $$

$$ \le \mathbb{P}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \le \gamma_1 - \log J_1]. \qquad (99) $$

Denote by $E_1$, $V_1$ and $T_1$ the first, second and third central moments of $\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})$. By the Berry-Esseen theorem, we have

$$ \begin{aligned} & \mathbb{P}[i_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \le \gamma_1] \\ & \le \mathbb{P}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \le \gamma_1 - \log J_1] \\ & \le \mathbb{Q}(-K) + \frac{6T_1}{\sqrt{V_1^3}}, \end{aligned} \qquad (100) $$

where $K > 0$ is chosen such that

$$ \gamma_1 = K\sqrt{V_1} + E_1 + \log J_1. \qquad (101) $$

We now seek lower bounds on the first and second central moments $E_1$ and $V_1$ and an upper bound on the third central moment $T_1$ since

$$ K = \frac{1}{\sqrt{V_1}} (\gamma_1 - \log J_1 - E_1), \qquad (102) $$

and

$$ \mathbb{P}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) - E_1 \le K\sqrt{V_1}] - \mathbb{Q}(-K) \le \frac{6T_1}{\sqrt{V_1^3}}. \qquad (103) $$

**Lemma 2:** The following inequalities hold:

$$ E_1 \ge E_{\min,1}, \qquad (104a) $$

$$ V_1 \ge n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2}), \qquad (104b) $$

$$ T_1 \le T_{\max,1}, \qquad (104c) $$

where $V(x) := \frac{x(2+x)}{2(1+x)^2} \log e^2$ and $T_{\max,1}$, $E_{\min,1}$ are defined in (37a) and (37b), respectively.

*Proof:* See Appendix C.

Set

$$ \gamma_1 := \log M_1 + \log L_1 - K_1 \log(n_{1,1} + n_{1,2}) - n_{1,1}C(\Omega_{1,1}) - n_{1,2}C(\Omega_{1,2}) + E_1 + \log J_1 \qquad (105) $$

for some $K_1$ and where $C(x) := \frac{1}{2} \log(1+x)$. Thus

$$ K = \frac{\log M_1 + \log L_1 - K_1 \log(n_{1,1} + n_{1,2}) - n_{1,1}C(\Omega_{1,1}) - n_{1,2}C(\Omega_{1,2})}{\sqrt{V_1}}. \qquad (106) $$

We then have

$$ \begin{aligned} & \mathbb{P}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \le \gamma_1 - \log J_1] \\ & = \mathbb{P}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) - E_1 \le \log M_1 + \log L_1 - K_1 \log(n_{1,1} + n_{1,2}) - n_{1,1}C(\Omega_{1,1}) - n_{1,2}C(\Omega_{1,2})] \\ & \le \frac{6T_1}{\sqrt{V_1^3}} + \mathbb{Q}\left(\frac{-\log M_1 - \log L_1 + K_1 \log(n_{1,1} + n_{1,2}) + n_{1,1}C(\Omega_{1,1}) + n_{1,2}C(\Omega_{1,2})}{\sqrt{V_1}}\right) \end{aligned} \qquad (107) $$
$$ \le \frac{6T_{\max,1}}{\sqrt{(n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2}))^3}} \\ + \mathbb{Q}\left( \frac{-\log M_1 - \log L_1 + K_1 \log(n_{1,1} + n_{1,2}) + n_{1,1}C(\Omega_{1,1}) + n_{1,2}C(\Omega_{1,2})}{\sqrt{n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2})}} \right) \quad (108) $$

where step (a) follows by Lemma 2.

2) Analyzing the false alarm probability: To bound the false alarm probability, i.e., $\mathbb{P}[i_1(\bar{X}_{1,1}, \bar{X}_{1,2}; Y_{1,1}, Y_{2,1}) \ge \gamma_1]$, we first use Bayes' formula to write

$$ \begin{aligned} f_{\mathbf{X}_{1,1}}(\bar{\mathbf{x}}_{1,1}) f_{\mathbf{X}_{1,2}}(\bar{\mathbf{x}}_{1,2}) &= \frac{f_{\mathbf{Y}_{1,1}}(\mathbf{y}_{1,1}) f_{\mathbf{Y}_{1,2}}(\mathbf{y}_{1,2}) f_{\mathbf{X}_{1,1}|\mathbf{Y}_{1,1}}(\bar{\mathbf{x}}_{1,1}|\mathbf{y}_{1,1}) f_{\mathbf{X}_{1,2}|\mathbf{Y}_{1,2}}(\bar{\mathbf{x}}_{1,2}|\mathbf{y}_{1,2})}{f_{\mathbf{Y}_{1,1}|\mathbf{X}_{1,1}}(\mathbf{y}_{1,1}|\bar{\mathbf{x}}_{1,1}) f_{\mathbf{Y}_{1,2}|\mathbf{X}_{1,2}}(\mathbf{y}_{1,2}|\bar{\mathbf{x}}_{1,2})} \\ &= f_{\mathbf{X}_{1,1}|\mathbf{Y}_{1,1}}(\bar{\mathbf{x}}_{1,1}|\mathbf{y}_{1,1}) f_{\mathbf{X}_{1,2}|\mathbf{Y}_{1,2}}(\bar{\mathbf{x}}_{1,2}|\mathbf{y}_{1,2}) 2^{-i_1(\bar{\mathbf{x}}_{1,1}, \bar{\mathbf{x}}_{1,2}; \mathbf{y}_{1,1}, \mathbf{y}_{1,2})}. \end{aligned} \quad (109) $$

Fix the sequences $\mathbf{y}_{1,1} \in \mathbb{R}^n$ and $\mathbf{y}_{1,2} \in \mathbb{R}^n$. We then have

$$ \begin{aligned} & \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \mathbf{1}\{i_1(\bar{\mathbf{x}}_{1,1}, \bar{\mathbf{x}}_{1,2}; \mathbf{y}_{1,1}, \mathbf{y}_{1,2}) > \gamma_1\} f_{\mathbf{X}_{1,1}}(\bar{\mathbf{x}}_{1,1}) f_{\mathbf{X}_{1,2}}(\bar{\mathbf{x}}_{1,2}) d\bar{\mathbf{x}}_{1,1} d\bar{\mathbf{x}}_{1,2} \\ &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} 2^{-i_1(\bar{\mathbf{x}}_{1,1}, \bar{\mathbf{x}}_{1,2}; \mathbf{y}_{1,1}, \mathbf{y}_{1,2})} \mathbf{1} \left\{ \prod_{j=1}^2 \frac{f_{\mathbf{Y}_{1,j}|\mathbf{X}_{1,j}}(\mathbf{y}_{1,j}|\bar{\mathbf{x}}_{1,j})}{f_{\mathbf{Y}_{1,j}}(\mathbf{y}_{1,j})} > 2^{\gamma_1} \right\} \\ & \quad \cdot f_{\mathbf{X}_{1,1}|\mathbf{Y}_{1,1}}(\bar{\mathbf{x}}_{1,1}|\mathbf{y}_{1,1}) f_{\mathbf{X}_{1,2}|\mathbf{Y}_{1,2}}(\bar{\mathbf{x}}_{1,2}|\mathbf{y}_{1,2}) d\bar{\mathbf{x}}_{1,1} d\bar{\mathbf{x}}_{1,2} \\ & \le \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} 2^{-i_1(\bar{\mathbf{x}}_{1,1}, \bar{\mathbf{x}}_{1,2}; \mathbf{y}_{1,1}, \mathbf{y}_{1,2})} \prod_{j=1}^2 \frac{f_{\mathbf{Y}_{1,j}|\mathbf{X}_{1,j}}(\mathbf{y}_{1,j}|\bar{\mathbf{x}}_{1,j})}{f_{\mathbf{Y}_{1,j}}(\mathbf{y}_{1,j})} 2^{-\gamma_1} \\ & \quad \cdot f_{\mathbf{X}_{1,1}|\mathbf{Y}_{1,1}}(\bar{\mathbf{x}}_{1,1}|\mathbf{y}_{1,1}) f_{\mathbf{X}_{1,2}|\mathbf{Y}_{1,2}}(\bar{\mathbf{x}}_{1,2}|\mathbf{y}_{1,2}) d\bar{\mathbf{x}}_{1,1} d\bar{\mathbf{x}}_{1,2} \\ &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} f_{\mathbf{X}_{1,1}|\mathbf{Y}_{1,1}}(\bar{\mathbf{x}}_{1,1}|\mathbf{y}_{1,1}) f_{\mathbf{X}_{1,2}|\mathbf{Y}_{1,2}}(\bar{\mathbf{x}}_{1,2}|\mathbf{y}_{1,2}) 2^{-\gamma_1} d\bar{\mathbf{x}}_{1,1} d\bar{\mathbf{x}}_{1,2} \\ &= 2^{-\gamma_1}. \end{aligned} \quad (110) $$

As a consequence,

$$ \mathbb{P}[i_1(\bar{\mathbf{X}}_{1,1}, \bar{\mathbf{X}}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \ge \gamma_1] \le 2^{-\gamma_1}. \quad (111) $$

By combining (108) and (111), we can bound the error probability in (92) by

$$ \begin{aligned} \mathbb{P}[\mathcal{E}_1 | \mathcal{E}_{1,2}^c] \le & \frac{6T_{\max,1}}{\sqrt{(n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2}))^3}} \\ & + \mathbb{Q}\left( \frac{-\log M_1 - \log L_1 + K_1 \log(n_{1,1} + n_{1,2}) + n_{1,1}C(\Omega_{1,1}) + n_{1,2}C(\Omega_{1,2})}{\sqrt{n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2})}} \right) \\ & + \frac{(n_{1,1} + n_{1,2})^{K_1} (1 + \Omega_{1,1})^{\frac{n_{1,1}}{2}} (1 + \Omega_{1,2})^{\frac{n_{1,2}}{2}}}{J_1 2^{E_{\min,1}}} \end{aligned} \quad (112) $$

As a result

$$ \begin{aligned} \epsilon_1 \le & \frac{6T_{\max,1}}{\sqrt{(n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2}))^3}} \\ & + \mathbb{Q}\left( \frac{-\log M_1 - \log L_1 + K_1 \log(n_{1,1} + n_{1,2}) + n_{1,1}C(\Omega_{1,1}) + n_{1,2}C(\Omega_{1,2})}{\sqrt{n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2})}} \right) \\ & + \frac{(n_{1,1} + n_{1,2})^{K_1} (1 + \Omega_{1,1})^{\frac{n_{1,1}}{2}} (1 + \Omega_{1,2})^{\frac{n_{1,2}}{2}}}{J_1 2^{E_{\min,1}}} + (1 - (1 - \rho^2)^{n_{1,2}-1})^{L_1 \cdot L_2}. \end{aligned} \quad (113) $$

Equivalently,

$$ \log M_1 + \log L_1 \le n_{1,1}C(\Omega_{1,1}) + n_{1,2}C(\Omega_{1,2}) + K_1 \log(n_{1,1} + n_{1,2}) \\ - \sqrt{n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2})} \mathbb{Q}^{-1}(\epsilon_1 - \Delta_1) \quad (114) $$

where $\Delta_1$ is defined in (36). This concludes the proof of Theorem 1.
## APPENDIX B
### PROOF OF PROPOSITION 1

For large values of $n_{1,1}$ and $n_{1,2}$, choose $K_1$ such that

$$ \lim_{n_{1,1}, n_{1,2} \to \infty} \frac{(n_{1,1} + n_{1,2})^{K_1} (1 + \Omega_{1,1})^{-\frac{n_{1,1}}{2}} (1 + \Omega_{1,2})^{-\frac{n_{1,2}}{2}}}{J_1 2^{E_{\min,1}}} \to 0. \quad (115) $$

Hence,

$$ \Delta_1 = O \left( \frac{1}{\sqrt{(n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2}))^3}} \right) \quad (116) $$

as $n_{1,1}, n_{1,2} \to \infty$. This proves the bound in (38). To prove the bound in (39), define

$$ A := 1 - (1 - \rho^2)^{n_{1,2}-1}. \quad (117) $$

We then use the approximation $\log(1-x) \approx -x$ which holds when $x$ is very small. Hence, for large values of $n_{1,2}$

$$ \log L_1 + \log L_2 \ge \log(-\log \epsilon_{1,2}) - n_{1,2} \log(1 - \rho^2) \quad (118) $$

which concludes the proof.

## APPENDIX C
### PROOF OF LEMMA 2

#### A. Bounding the first moment

We start by lower bounding the first moment. To this end, we have

$$ \mathbb{E}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})] \quad (119) $$

$$ \begin{align} \tag{ii} \stackrel{(i)}{=} \mathbb{E} & \left[ \frac{n_{1,1}}{2} \log \frac{\sigma_{y_{1,1}}^2}{\sigma_{1,1}^2} + \frac{n_{1,2}}{2} \log \frac{\sigma_{y_{1,2}}^2}{\sigma_{y_{1,2}|x_{1,2}}^2} - \frac{\|\mathbf{Z}_{1,1}\|^2}{2\sigma_{1,1}^2} + \frac{\|\mathbf{X}_{1,1} + \mathbf{Z}_{1,1}\|^2}{2\sigma_{y_{1,1}}^2} \right. \nonumber \\ & \left. \qquad - \frac{\|\alpha(\mathbf{X}_{1,2} + \mathbf{X}_{2,1}) + \mathbf{Z}_{1,2} - \mu_{1,2}\|^2}{2\sigma_{y_{1,2}|x_{1,2}}^2} + \frac{\|\alpha(\mathbf{X}_{1,2} + \mathbf{X}_{2,1}) + \mathbf{Z}_{1,2}\|^2}{2\sigma_{y_{1,2}}^2} \right] \tag{120} \end{align} $$

$$ \begin{align} \stackrel{(ii)}{\ge} \frac{n_{1,1}}{2} \log \frac{\sigma_{y_{1,1}}^2}{\sigma_{1,1}^2} + \frac{n_{1,2}}{2} \log \frac{\sigma_{y_{1,2}}^2}{((1-\rho^2)\beta_{2,1}\mathbf{P} + \sigma_{1,2}^2)} - \frac{\mathbb{E}[\|\mathbf{Z}_{1,1}\|^2]}{2\sigma_{1,1}^2} &+ \frac{\|\mathbf{X}_{1,1}\|^2 + \mathbb{E}[\|\mathbf{Z}_{1,1}\|^2] + 2\mathbb{E}\langle\mathbf{Z}_{1,1}, \mathbf{X}_{1,1}\rangle}{2\sigma_{y_{1,1}}^2} \nonumber \\ & - \frac{\|\alpha(\mathbf{X}_{1,2} + \mathbf{X}_{2,1}) + \mathbf{Z}_{1,2} - \sqrt{\frac{\beta_c}{\beta_c}}(1 + \rho\sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})\mathbf{X}_{1,2}\|^2}{2\sigma_{1,2}^2} + \frac{\|\alpha(\mathbf{X}_{1,2} + \mathbf{X}_{2,1}) + \mathbf{Z}_{1,2}\|^2}{2\sigma_{y_{1,2}}^2} \tag{121} \end{align} $$

$$ \begin{align} = \frac{n_{1,1}}{2} \log \frac{\sigma_{y_{1,1}}^2}{\sigma_{1,1}^2} + \frac{n_{1,2}}{2} \log \frac{\sigma_{y_{1,2}}^2}{((1-\rho^2)\beta_{2,1}\mathbf{P} + \sigma_{1,2}^2)} &+ \frac{1}{2} \left( \frac{1}{\sigma_{y_{1,1}}^2} - \frac{1}{\sigma_{1,1}^2} \right) \mathbb{E}[\|\mathbf{Z}_{1,1}\|^2] + \frac{\|\mathbf{X}_{1,1}\|^2}{2\sigma_{y_{1,1}}^2} \nonumber \\ & + \frac{\mathbb{E}[\langle\mathbf{Z}_{1,1}, \mathbf{X}_{1,1}\rangle]}{\sigma_{y_{1,1}}^2} + \frac{1}{2} \left( \frac{\alpha^2}{\sigma_{y_{1,1}}^2} - \frac{\left(\alpha\sqrt{\tilde{\beta}_c} - \sqrt{\beta_c}(1 + \rho\sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})\right)^2}{\tilde{\beta}_c\sigma_{1,2}^2} \right) \|\mathbf{X}_{1,2}\|^2 \nonumber \\ & + \left( \frac{\alpha}{\sigma_{y_{1,1}}^2} - \frac{\alpha\sqrt{\tilde{\beta}_c} - \sqrt{\beta_c}(1 + \rho\sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})}{\sqrt{\tilde{\beta}_c}\sigma_{1,2}^2} \right) (\mathbb{E}[\langle\mathbf{X}_{1,2}, \mathbf{X}_{2,1}\rangle] + \mathbb{E}[\langle\mathbf{X}_{1,2}, \mathbf{Z}_{1,2}\rangle]) \nonumber \\ & + \frac{1}{2} \left( \frac{1}{\sigma_{y_{1,2}}^2} - \frac{1}{\sigma_{1,2}^2} \right) (\alpha^2 \|\mathbf{X}_{2,1}\|^2 + \mathbb{E}[\|\mathbf{Z}_{1,2}\|^2] + 2\mathbb{E}[\langle\mathbf{X}_{2,1}, \mathbf{Z}_{1,2}\rangle]) \tag{122} \end{align} $$

$$ \begin{align} \stackrel{(iii)}{\ge} \frac{n_{1,1}}{2} \log \frac{\sigma_{y_{1,1}}^2}{\sigma_{1,1}^2} + \frac{n_{1,2}}{2} \log \frac{\sigma_{y_{1,2}}^2}{((1-\rho^2)\beta_{2,1}\mathbf{P} + \sigma_{1,2}^2)} &+ \frac{\sigma_{1,1}^4 n_{1,1}}{2} \left( \frac{1}{\sigma_{y_{1,1}}^2} - \frac{1}{\sigma_{1,1}^2} \right) + \frac{n_{1,1}\beta_{1,1}\mathbf{P}}{2\sigma_{y_{1,1}}^2} \nonumber \\ & - \frac{\sigma_{1,1}^2 \sqrt{2n_{1,1}\beta_{1,1}\mathbf{P}}}{\sigma_{y_{1,1}}^2} \frac{\Gamma\left(\frac{n_{1,1}+1}{2}\right)}{\Gamma\left(\frac{n_{1,1}}{2}\right)} + \frac{1}{2} \left( \frac{\beta_c}{\tilde{\beta}_c\sigma_{y_{1,1}}^2} - \frac{\left(\sqrt{\tilde{\beta}_c} - \sqrt{\beta_c}(1 + \rho\sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})\right)^2}{\tilde{\beta}_c\sigma_{1,2}^2} \right) n_{1,2}\beta_{1,2}\mathbf{P} \end{align} $$
$$
\begin{align}
&+ \left( \frac{\sqrt{\beta_c}}{\sqrt{\tilde{\beta}_c} \sigma_{y_{1,1}}^2} - \frac{\sqrt{\tilde{\beta}_c} - \sqrt{\beta_c}(1 + \rho \sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})}{\sqrt{\tilde{\beta}_c} \sigma_{1,2}^2} \right) \left( n_{1,2} \sqrt{\beta_{1,2} \beta_{2,1}} \rho \mathbf{P} - \sigma_{1,2}^2 \sqrt{2n_{1,2} \beta_{1,2} \mathbf{P}} \frac{\Gamma\left(\frac{n_{1,2}+1}{2}\right)}{\Gamma\left(\frac{n_{1,2}}{2}\right)} \right) \nonumber \\
&+ \frac{1}{2} \left( \frac{1}{\sigma_{y_{1,2}}^2} - \frac{1}{\sigma_{1,2}^2} \right) \left( n_{1,2} \beta_{2,1} \mathbf{P} + n_{1,2} \sigma_{1,2}^4 - 2\sqrt{2}\sigma_{1,2}^2 \sqrt{n_{1,2} \beta_{2,1} \mathbf{P}} \frac{\Gamma\left(\frac{n_{1,2}+1}{2}\right)}{\Gamma\left(\frac{n_{1,2}}{2}\right)} \right) \nonumber \\
&= \frac{n_{1,1}}{2} \log(1 + \Omega_{1,1}) + \frac{n_{1,2}}{2} \log(1 + \Omega_{1,2}) + \frac{(1 - \sigma_{1,1}^2) n_{1,1} \Omega_{1,1}}{2(1 + \Omega_{1,1})} - \Gamma_{1,1} \cdot \frac{\sqrt{2n_{1,1} \sigma_{1,1}^2 \Omega_{1,1}}}{1 + \Omega_{1,1}} - \frac{n_{1,2} \sigma_{1,2}^2 \beta_c \mathbf{P}}{2(\sigma_{1,2}^2 + \beta_c \mathbf{P})} \nonumber \\
&+ \sqrt{n_{1,1} \beta_{1,2} \mathbf{P}} \left( \sqrt{\frac{\beta_c}{\tilde{\beta}_c}} \left( \frac{1}{\sigma_{1,2}^2 + \beta_c \mathbf{P}} + 1 + \rho \sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}} \right) - \frac{1}{\sigma_{1,2}^2} \right) \left( \rho \sqrt{n_{1,2} \beta_{2,1} \mathbf{P}} - \sigma_{1,2}^2 \sqrt{2\Gamma_{1,2}} \right) \nonumber \\
&+ \frac{\beta_c n_{1,2} \beta_{1,2} \mathbf{P} \left( \sigma_{1,2}^2 + (\beta_c \mathbf{P} - \sigma_{1,2}^2) \left( \sqrt{\frac{\beta_c}{\tilde{\beta}_c}} - 1 - \rho \sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}} \right)^2 \right)}{2 \tilde{\beta}_c \sigma_{1,2}^2 (\beta_c \mathbf{P} + \sigma_{1,2}^2)} - \frac{\beta_c \mathbf{P} \sqrt{n_{1,2} \beta_{2,1} \mathbf{P}}}{2 \sigma_{1,2}^2 (\beta_c \mathbf{P} + \sigma_{1,2}^2)} \left( \sqrt{n_{1,2} \beta_{2,1} \mathbf{P}} - 2\sqrt{2}\sigma_{1,2}^2 \Gamma_{1,2} \right) \tag{124}
\end{align}
$$

where (i) is by (96), (ii) is by (95), (iii) is by (20) and the fact that $||Z_{1,1}||^2/\sigma_{1,1}^4$ and $||Z_{1,2}||^2/\sigma_{1,2}^4$ each follow a central chi-squared distribution with degree of $n_{1,1}$ and $n_{1,2}$, respectively, and $||Z_{1,1}||/\sigma_{1,1}^2$ and $||Z_{1,2}||/\sigma_{1,2}^2$ each follow a central chi-distribution with degree of $n_{1,1}$ and $n_{1,2}$, respectively. The last equality is by (33) and by defining $\Gamma_{1,1} := \Gamma(\frac{n_{1,1}+1}{2})/\Gamma(\frac{n_{1,1}}{2})$ and $\Gamma_{1,2} := \Gamma(\frac{n_{1,2}+1}{2})/\Gamma(\frac{n_{1,2}}{2})$.

## B. Bounding the second moment

We continue by lower bounding the variance of $\tilde{i}_1(X_{1,1}, X_{1,2}; Y_{1,1}, Y_{2,1})$. One can write

$$
\begin{align}
& \mathrm{Var}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})] \tag{125} \\
&= \mathrm{Var} \left[ \frac{n_{1,1}}{2} \log \frac{\sigma_{y_{1,1}}^2}{\sigma_{1,1}^2} + \frac{n_{1,2}}{2} \log \frac{\sigma_{y_{1,2}}^2}{\sigma_{1,2}^2} - \frac{\|\mathbf{Z}_{1,1}\|^2}{2\sigma_{1,1}^2} + \frac{\|\mathbf{X}_{1,1} + \mathbf{Z}_{1,1}\|^2}{2\sigma_{y_{1,1}}^2} \right. \nonumber \\
& \qquad \left. - \frac{\|\alpha(\mathbf{X}_{1,2} + \mathbf{X}_{2,1}) + \mathbf{Z}_{1,2} - \mu_{1,2}\|^2}{2\sigma_{y_{1,2}}^2} + \frac{\|\alpha(\mathbf{X}_{1,2} + \mathbf{X}_{2,1}) + \mathbf{Z}_{1,2}\|^2}{2\sigma_{y_{1,2}}^2} \right] \tag{126}
\end{align}
$$

$$
\begin{align}
& \stackrel{(i)}{\ge} \mathrm{Var} \left[ \frac{1}{2} \left( \frac{1}{\sigma_{y_{1,1}}^2} - \frac{1}{\sigma_{1,1}^2} \right) \|\mathbf{Z}_{1,1}\|^2 + \frac{1}{2} \left( \frac{1}{\sigma_{y_{1,2}}^2} - \frac{1}{\sigma_{1,2}^2} \right) \|\mathbf{Z}_{1,2}\|^2 + \frac{\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle}{\sigma_{y_{1,1}}^2} \right. \nonumber \\
& \qquad \left. + \left( \frac{\sqrt{\beta_c}}{\sqrt{\tilde{\beta}_c} \sigma_{y_{1,2}}^2} - \frac{1}{\sigma_{1,2}^2} \right) \langle \mathbf{X}_{2,1}, \mathbf{Z}_{1,2} \rangle + \left( \frac{\sqrt{\beta_c}}{\sqrt{\tilde{\beta}_c} \sigma_{y_{1,2}}^2} - \frac{1 - \sqrt{\frac{\beta_c}{\tilde{\beta}_c}(1 + \rho \sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})}}{\sigma_{1,2}^2} \right) \langle \mathbf{X}_{1,2}, \mathbf{Z}_{1,2} \rangle \right] \tag{127}
\end{align}
$$

$$
\begin{align}
& \stackrel{(ii)}{\ge} \frac{\sigma_{1,1}^4 n_{1,1}}{2} \left( \frac{1}{\sigma_{y_{1,1}}^2} - \frac{1}{\sigma_{1,1}^2} \right)^2 + \frac{\sigma_{1,2}^4 n_{1,2}}{2} \left( \frac{1}{\sigma_{y_{1,2}}^2} - \frac{1}{\sigma_{1,2}^2} \right)^2 + \frac{\mathrm{Var}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]}{(\sigma_{y_{1,1}}^2)^2} \nonumber \\
& \qquad + \left( \frac{\sqrt{\beta_c}}{\sqrt{\tilde{\beta}_c} \sigma_{y_{1,2}}^2} - \frac{1}{\sigma_{1,2}^2} \right)^2 \mathrm{Var}[\langle \mathbf{X}_{2,1}, \mathbf{Z}_{1,2} \rangle] + \left( \frac{\sqrt{\beta_c}}{\sqrt{\tilde{\beta}_c} \sigma_{y_{1,2}}^2} - \frac{1 - \frac{\beta_c}{\tilde{\beta}_c}(1 + \rho \sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})}{\sigma_{1,2}^2} \right)^2 \mathrm{Var}[\langle \mathbf{X}_{1,2}, \mathbf{Z}_{1,2} \rangle]. \tag{128}
\end{align}
$$

where (i) is by (95), and (ii) is by the fact that $\mathrm{Var}[||Z_{1,1}||^2] = 2n_{1,1}\sigma_{1,1}^4$ and $\mathrm{Var}[||Z_{1,2}||^2] = 2n_{1,2}\sigma_{1,2}^4$. To bound the remaining variance terms in (128), we use the following Chebyshev inequality.

**Theorem 6 (Chebyshev's inequality):** Let $X$ be a random variable with expectation $\mu$ and variance $\sigma^2$. Then

$$ \mathbb{P}[|X - \mu| \ge c] \le \frac{\sigma^2}{c^2}, \qquad (129) $$

for all $c > 0$.

We start by bounding $\mathrm{Var}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]$. For $c_1 > 0$

$$
\mathrm{Var}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle] \ge c_1^2 \mathbb{P}[|\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle - \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]| \ge c_1]. \quad (130)
$$

The right-hand side of (141) can be bounded as follows:

$$ \mathbb{P}[|\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle - \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]| \ge c_1] \quad (131) $$
$$= \mathbb{P}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle \ge c_1 + \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]] + \mathbb{P}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle \le -c_1 + \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]] \quad (132)$$

$$= \mathbb{P}\left[\|\mathbf{Z}_{1,1}\| \cos(\theta) \ge \frac{c_1 + \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]}{\|\mathbf{X}_{1,1}\|}\right] + \mathbb{P}\left[\|\mathbf{Z}_{1,1}\| \cos(\theta) \le \frac{-c_1 + \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]}{\|\mathbf{X}_{1,1}\|}\right] \quad (133)$$

$$\ge \mathbb{P}\left[-\|\mathbf{Z}_{1,1}\| \ge \frac{c_1 + \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]}{\|\mathbf{X}_{1,1}\|}\right] + \mathbb{P}\left[\|\mathbf{Z}_{1,1}\| \le \frac{-c_1 + \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]}{\|\mathbf{X}_{1,1}\|}\right] \quad (134)$$

$$= \mathbb{P}\left[\|\mathbf{Z}_{1,1}\| \le -\frac{c_1 + \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]}{\|\mathbf{X}_{1,1}\|}\right] + \mathbb{P}\left[\|\mathbf{Z}_{1,1}\| \le \frac{-c_1 + \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]}{\|\mathbf{X}_{1,1}\|}\right] \quad (135)$$

$$\stackrel{(a)}{\ge} \mathbb{P}\left[\|\mathbf{Z}_{1,1}\| \le -\frac{c_1 - \|\mathbf{X}_{1,1}\| \cdot \mathbb{E}[\|\mathbf{Z}_{1,1}\|]}{\|\mathbf{X}_{1,1}\|}\right] + \mathbb{P}\left[\|\mathbf{Z}_{1,1}\| \le \frac{-c_1 + \|\mathbf{X}_{1,1}\| \cdot \mathbb{E}[\|\mathbf{Z}_{1,1}\|]}{\|\mathbf{X}_{1,1}\|}\right] \quad (136)$$

$$\stackrel{(b)}{=} 2F_{\|\mathbf{Z}_{1,1}/\sigma_{1,1}^2\|} \left( -\frac{c_1}{\sigma_{1,1}^2 \sqrt{n_{1,1} \beta_{1,1} \mathbf{P}}} + \frac{\sqrt{2} \Gamma(\frac{n_{1,1}+1}{2})}{\sigma_{1,1}^2 \Gamma(\frac{n_{1,1}}{2})} \right) \quad (137)$$

$$= 2G \left( \frac{n_{1,1}}{2}, \frac{1}{2} \left( -\frac{c_1}{\sigma_{1,1}^2 \sqrt{n_{1,1} \beta_{1,1} \mathbf{P}}} + \frac{\sqrt{2}}{\sigma_{1,1}^2} \frac{\Gamma(\frac{n_{1,1}+1}{2})}{\Gamma(\frac{n_{1,1}}{2})} \right)^2 \right) \quad (138)$$

In step (a) we used the following bounds on $\mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle]$:

$$-\|\mathbf{X}_{1,1}\| \cdot \mathbb{E}[\|\mathbf{Z}_{1,1}\|] \le \mathbb{E}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle] \le \|\mathbf{X}_{1,1}\| \cdot \mathbb{E}[\|\mathbf{Z}_{1,1}\|] \quad (139)$$

In step (b) we used the fact that $Z_{1,1}/\sigma_{1,1}^2$ follows a chi distribution of degree $n_{1,1}$ and $F_{\|Z_{1,1}/\sigma_{1,1}^2\|}(\cdot)$ is the CDF of the chi distribution of degree $n_{1,1}$ and is equal to

$$F(x; n) = G\left(\frac{n}{2}, \frac{x^2}{2}\right) \quad (140)$$

where $G(\cdot, \cdot)$ is the regularized gamma function.

Thus

$$\mathrm{Var}[\langle \mathbf{X}_{1,1}, \mathbf{Z}_{1,1} \rangle] \ge 2c_1^2 G \left( \frac{n_{1,1}}{2}, \frac{1}{2} \left( -\frac{c_1}{\sigma_{1,1}^2 \sqrt{n_{1,1} \beta_{1,1} \mathbf{P}}} + \frac{\sqrt{2}}{\sigma_{1,1}^2} \frac{\Gamma(\frac{n_{1,1}+1}{2})}{\Gamma(\frac{n_{1,1}}{2})} \right)^2 \right) \quad (141)$$

for all $c_1 > 0$.

Following the same procedure, we prove that

$$\mathrm{Var}[\langle \mathbf{X}_{2,1}, \mathbf{Z}_{1,2} \rangle] \ge 2c_2^2 G \left( \frac{n_{1,2}}{2}, \frac{1}{2} \left( -\frac{c_2}{\sigma_{1,2}^2 \sqrt{n_{1,2} \beta_{2,1} \mathbf{P}}} + \frac{\sqrt{2}}{\sigma_{1,2}^2} \frac{\Gamma(\frac{n_{1,2}+1}{2})}{\Gamma(\frac{n_{1,2}}{2})} \right)^2 \right) \quad (142)$$

$$\mathrm{Var}[\langle \mathbf{X}_{1,2}, \mathbf{Z}_{1,2} \rangle] \ge 2c_3^2 G \left( \frac{n_{1,2}}{2}, \frac{1}{2} \left( -\frac{c_3}{\sigma_{1,2}^2 \sqrt{n_{1,2} \beta_{1,2} \mathbf{P}}} + \frac{\sqrt{2}}{\sigma_{1,2}^2} \frac{\Gamma(\frac{n_{1,2}+1}{2})}{\Gamma(\frac{n_{1,2}}{2})} \right)^2 \right) \quad (143)$$

for all $c_2 > 0$ and $c_3 > 0$. Thus

$$\mathrm{Var}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})] \quad (144)$$

$$
\begin{aligned}
&\ge \frac{\sigma_{1,1}^4 n_{1,1}}{2} \left( \frac{1}{\sigma_{y_{1,1}}^2} - \frac{1}{\sigma_{1,1}^2} \right)^2 + \frac{\sigma_{1,2}^4 n_{1,2}}{2} \left( \frac{1}{\sigma_{y_{1,2}}^2} - \frac{1}{\sigma_{1,2}^2} \right)^2 \\
&\quad + \frac{2c_1^2}{(\sigma_{y_{1,1}}^2)^2} G \left( \frac{n_{1,1}}{2}, \frac{1}{2} \left( -\frac{c_1}{\sigma_{1,1}^2 \sqrt{n_{1,1} \beta_{1,1} \mathbf{P}}} + \frac{\sqrt{2}}{\sigma_{1,1}^2} \frac{\Gamma(\frac{n_{1,1}+1}{2})}{\Gamma(\frac{n_{1,1}}{2})} \right)^2 \right) \\
&\quad + 2c_2^2 \left( \frac{\sqrt{\beta_c}}{\sqrt{\tilde{\beta}_c} \sigma_{y_{1,2}}^2} - \frac{1}{\sigma_{1,2}^2} \right)^2 G \left( \frac{n_{1,2}}{2}, \frac{1}{2} \left( -\frac{c_2}{\sigma_{1,2}^2 \sqrt{n_{1,2} \beta_{2,1} \mathbf{P}}} + \frac{\sqrt{2}}{\sigma_{1,2}^2} \frac{\Gamma(\frac{n_{1,2}+1}{2})}{\Gamma(\frac{n_{1,2}}{2})} \right)^2 \right) \\
&\quad + 2c_3^2 \left( \frac{\sqrt{\beta_c}}{\sqrt{\tilde{\beta}_c} \sigma_{y_{1,2}}^2} - \frac{1 - \frac{\beta_c}{\beta_c}(1 + \rho\sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})}{\sigma_{1,2}^2} \right)^2 G \left( \frac{n_{1,2}}{2}, \frac{1}{2} \left( -\frac{c_3}{\sigma_{1,2}^2 \sqrt{n_{1,2} \beta_{1,2} \mathbf{P}}} + \frac{\sqrt{2}}{\sigma_{1,2}^2} \frac{\Gamma(\frac{n_{1,2}+1}{2})}{\Gamma(\frac{n_{1,2}}{2})} \right)^2 \right)
\end{aligned}
\quad (145)
$$
$$= n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2}) - \frac{n_{1,1}\Omega_{1,1}}{(1+\Omega_{1,1})^2} - \frac{n_{1,2}(2\sigma_{1,2}^2\beta_c\mathbf{P} - (1-\rho^2)\beta_{2,1}\mathbf{P}(2\sigma_{1,2}^2 + (1-\rho^2)\beta_{2,1}\mathbf{P}))}{2(\sigma_{1,2}^2 + \beta_c\mathbf{P})^2}$$
$$
\begin{align}
&+\frac{2c_1^2}{(\sigma_{y1,1}^2)^2} G \left( \frac{n_{1,1}}{2}, \frac{1}{2} \left( -\frac{c_1}{\sigma_{1,1}^2 \sqrt{n_{1,1} \beta_{1,1} \mathbf{P}}} + \frac{\sqrt{2} \Gamma(\frac{n_{1,1}+1}{2})}{\sigma_{1,1}^2 \Gamma(\frac{n_{1,1}}{2})} \right)^2 \right) \nonumber \\
&+2c_2^2 \left( \frac{\sqrt{\beta_c}}{\sqrt{\tilde{\beta}_c} \sigma_{y1,2}^2} - \frac{1}{\sigma_{1,2}^2} \right)^2 G \left( \frac{n_{1,2}}{2}, \frac{1}{2} \left( -\frac{c_2}{\sigma_{1,2}^2 \sqrt{n_{1,2} \beta_{2,1} \mathbf{P}}} + \frac{\sqrt{2} \Gamma(\frac{n_{1,2}+1}{2})}{\sigma_{1,2}^2 \Gamma(\frac{n_{1,2}}{2})} \right)^2 \right) \nonumber \\
&+2c_3^2 \left( \frac{\sqrt{\beta_c}}{\sqrt{\tilde{\beta}_c} \sigma_{y1,2}^2} - \frac{1-\frac{\beta_c}{\beta_c}(1+\rho\sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})}{\sigma_{1,2}^2} \right)^2 G \left( \frac{n_{1,2}}{2}, \frac{1}{2} \left( -\frac{c_3}{\sigma_{1,2}^2 \sqrt{n_{1,2} \beta_{1,2} \mathbf{P}}} + \frac{\sigma_{1,2}^2 \Gamma(\frac{n_{1,2}+1}{2})}{\sqrt{2} \Gamma(\frac{n_{1,2}}{2})} \right)^2 \right) \tag{146}
\end{align}
$$

for all $c_1 > 0$, $c_2 > 0$ and $c_3 > 0$ with $V(x) := \frac{x(2+x)}{2(1+x)^2} \log e^2$. Given $n_{1,1}, n_{1,2}, \mathbf{P}, \beta_{1,1}, \beta_{1,2}, \beta_{2,1}, \rho$, since $c_1, c_2, c_3 > 0$ are arbitrary, we may choose them such that

$$
\begin{align}
& \frac{n_{1,1} \Omega_{1,1}}{(1 + \Omega_{1,1})^2} + \frac{n_{1,2} (2 \sigma_{1,2}^2 \beta_c \mathbf{P} - (1 - \rho^2) \beta_{2,1} \mathbf{P} (2 \sigma_{1,2}^2 + (1 - \rho^2) \beta_{2,1} \mathbf{P}))}{2 (\sigma_{1,2}^2 + \beta_c \mathbf{P})^2} \nonumber \\
&= \frac{2c_1^2}{(\sigma_{y1,1}^2)^2} G \left( \frac{n_{1,1}}{2}, \frac{1}{2} \left( -\frac{c_1}{\sigma_{1,1}^2 \sqrt{n_{1,1} \beta_{1,1} \mathbf{P}}} + \frac{\sqrt{2} \Gamma(\frac{n_{1,1}+1}{2})}{\sigma_{1,1}^2 \Gamma(\frac{n_{1,1}}{2})} \right)^2 \right) \nonumber \\
&\quad + 2c_2^2 \left( \frac{1}{\sigma_{y1,2}^2} - \frac{1}{\sigma_{1,2}^2} \right)^2 G \left( \frac{n_{1,2}}{2}, \frac{1}{2} \left( -\frac{c_2}{\sigma_{1,2}^2 \sqrt{n_{1,2} \beta_{2,1} \mathbf{P}}} + \frac{\sqrt{2} \Gamma(\frac{n_{1,2}+1}{2})}{\sigma_{1,2}^2 \Gamma(\frac{n_{1,2}}{2})} \right)^2 \right) \nonumber \\
&\quad + 2c_3^2 \left( \frac{1}{\sigma_{y1,2}^2} + \frac{\frac{\beta_c}{\beta_c}(1 + \rho \sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})}{(1 - \rho^2) \beta_{2,1} \mathbf{P} + \sigma_{1,2}^2} \right)^2 G \left( \frac{n_{1,2}}{2}, \frac{1}{2} \left( -\frac{c_3}{\sigma_{1,2}^2 \sqrt{n_{1,2} \beta_{1,2} \mathbf{P}}} + \frac{\sigma_{1,2}^2 \Gamma(\frac{n_{1,2}+1}{2})}{\sqrt{2} \Gamma(\frac{n_{1,2}}{2})} \right)^2 \right). \tag{147}
\end{align}
$$

Note that such $c_1, c_2, c_3 > 0$ always exist since the corresponding factors in (147) are positive and $\Omega_{1,1}, \Omega_{1,2} > 0$. Therefore,

$$
\mathrm{Var}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})] \geq n_{1,1}V(\Omega_{1,1}) + n_{1,2}V(\Omega_{1,2}) \quad (148)
$$

with $V(x) := \frac{x(2+x)}{2(1+x)^2} \log e^2$. This proves (104b).

### C. Bounding the third moment

In this section, we upper bound the third moment. To this end, we employ the following inequality:

$$
\mathbb{E}[|\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) - \mathbb{E}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})]|^3] \le 2^3 \mathbb{E}[|\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})|^3], \quad (149)
$$

which follows from the Minkowski inequality [34].

Let $f_{\tilde{i}}(\cdot)$ be the probability density function of $\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})$ with $F_i$ as its cumulative density function. For simplicity, define $Z := \tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})$. For any $\kappa > 1$ we have

$$
\begin{align}
& \mathbb{E}[|Z|^3] \nonumber \\
&= \int_{-\infty}^{\infty} |Z|^3 f_{\tilde{i}}(z) dz \nonumber \\
&\le \kappa + \int_{\kappa}^{\infty} Z^3 f_{\tilde{i}}(z) dz + \int_{-\infty}^{-\kappa} |Z|^3 f_{\tilde{i}}(z) dz \nonumber \\
&= \kappa + \int_{\kappa}^{\infty} Z^3 (f_{\tilde{i}}(z) - f_{\tilde{i}}(-z)) dz \nonumber \\
&= \kappa + \sum_{j=0}^{\infty} \int_{\kappa+j}^{\kappa+j+1} Z^3 (f_{\tilde{i}}(z) - f_{\tilde{i}}(-z)) dz \nonumber \\
&\le \kappa + \sum_{j=0}^{\infty} (\kappa + j + 1)^3 \int_{\kappa+j}^{\kappa+j+1} (f_{\tilde{i}}(z) - f_{\tilde{i}}(-z)) dz \nonumber \\
&= \kappa + \sum_{j=0}^{\infty} (\kappa + j + 1)^3 (F_{\tilde{i}}(\kappa + j + 1) - F_{\tilde{i}}(\kappa + j) + F_{\tilde{i}}(-\kappa - j) - F_{\tilde{i}}(-\kappa - j - 1)) \nonumber \\
&\le \kappa + \sum_{j=0}^{\infty} (\kappa + j + 1)^3 (1 - F_{\tilde{i}}(\kappa + j) + F_{\tilde{i}}(-\kappa - j)). \tag{150}
\end{align}
$$
Notice that

$$1 - F_{\tilde{i}}(\kappa + j) = \mathbb{P}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) > \kappa + j] \quad (151)$$

and

$$F_{\tilde{i}}(-\kappa - j) = \mathbb{P}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) \leq -\kappa - j]. \quad (152)$$

Hence,

$$\mathbb{E}[|\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})|^3] \leq \kappa + 2 \sum_{j=0}^{\infty} (\kappa + j + 1)^3 \mathbb{P}[|\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})| > \kappa + j]. \quad (153)$$

**Lemma 3:** The following inequality holds:

$$\begin{aligned} & \mathbb{P}[|\tilde{i}(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})| > \kappa + j] \\ & \leq \max \left\{ \frac{2^{1-\frac{n_{1,1}}{2}} \zeta}{\Gamma(\frac{n_{1,1}}{2})} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2k}} - \frac{b}{2} \right)^{n_{1,1}-2} e^{-\frac{1}{2} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2k}} - \frac{b}{2} \right)^2}, \right. \\ & \qquad \left. \frac{2^{1-\frac{n_{1,2}}{2}} \tilde{\zeta}}{\Gamma(\frac{n_{1,2}}{2})} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2\tilde{k}}} - \frac{\tilde{b}}{2} \right)^{n_{1,2}-2} e^{-\frac{1}{2} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2\tilde{k}}} - \frac{\tilde{b}}{2} \right)^2} \right\} \end{aligned} \quad (154)$$

for $\zeta > 1$ and $\tilde{\zeta} > 1$ satisfying

$$\frac{\zeta}{(\zeta - 1)(\frac{n_{1,1}}{2} - 1)} < \frac{1}{2} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2k}} - \frac{b}{2} \right)^2, \quad (155)$$

$$\frac{\tilde{\zeta}}{(\tilde{\zeta} - 1)(\frac{n_{1,2}}{2} - 1)} < \frac{1}{2} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2\tilde{k}}} - \frac{\tilde{b}}{2} \right)^2, \quad (156)$$

where $k, \tilde{k}, b, \tilde{b}$ and $\tilde{\kappa}$ are defined in (37).

*Proof:* See Appendix E.

<figure><img src="image_9.png" /></figure>

**Lemma 4:** It holds that

$$\begin{aligned} & \sum_{j=0}^{\infty} (\kappa + j + 1)^3 \mathbb{P}[|\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})| > \kappa + j] \quad (157) \\ & \leq \max \left\{ \frac{\zeta \sqrt{2}^{\frac{n_{1,1}}{2}-1}}{\Gamma(\frac{n_{1,1}}{2})} e^{-c} A(n_{1,1}, k, b, c), \frac{\tilde{\zeta} \sqrt{2}^{\frac{n_{1,2}}{2}-1}}{\Gamma(\frac{n_{1,2}}{2})} e^{-\tilde{c}} A(n_{1,2}, \tilde{k}, \tilde{b}, \tilde{c}) \right\} \end{aligned}$$

where $c, \tilde{c}$ and $A(n, k, b, c)$ are defined in (37).

*Proof:* The proof is based on the following equality [35]:

$$\sum_{j=c}^{\infty} j^n e^{-j} = e^{-c} \Phi(e^{-1}, -n, c). \quad (158)$$

<figure><img src="image_10.png" /></figure>

Hence,

$$\mathbb{E}[|\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})|^3] \quad (159)$$

$$\leq \kappa + 2 \max \left\{ \frac{\zeta \sqrt{2}^{\frac{n_{1,1}}{2}-1}}{\Gamma(\frac{n_{1,1}}{2})} e^{-c} A(n_{1,1}, k, b, c), \frac{\tilde{\zeta} \sqrt{2}^{\frac{n_{1,2}}{2}-1}}{\Gamma(\frac{n_{1,2}}{2})} e^{-\tilde{c}} A(n_{1,2}, \tilde{k}, \tilde{b}, \tilde{c}) \right\}. \quad (160)$$

As a result

$$\begin{aligned} & \mathbb{E}[|\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})| - \mathbb{E}[\tilde{i}_1(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})]|^3] \\ & \leq 2^3 \kappa + 2^4 \max \left\{ \frac{\zeta \sqrt{2}^{\frac{n_{1,1}}{2}-1}}{\Gamma(\frac{n_{1,1}}{2})} e^{-c} A(n_{1,1}, k, b, c), \frac{\tilde{\zeta} \sqrt{2}^{\frac{n_{1,2}}{2}-1}}{\Gamma(\frac{n_{1,2}}{2})} e^{-\tilde{c}} A(n_{1,2}, \tilde{k}, \tilde{b}, \tilde{c}) \right\} \end{aligned} \quad (161)$$

for any $\kappa > 1$. This concludes the proof.
## APPENDIX D
### PROOF OF LEMMA 1

By [32, Proposition 2], we have the following bounds:

$$ \frac{f_{\mathbf{Y}_{1,1}}(\mathbf{Y}_{1,1})}{Q_{\mathbf{Y}_{1,1}}(\mathbf{y}_{1,1})} \le 27 \sqrt{\frac{\pi}{8} \frac{\sigma_{1,1}^2 + \beta_{1,1} \mathbf{P}}{\sigma_{1,1}^2 + 2\beta_{1,1} \mathbf{P}}} \quad (162) $$

$$ \frac{f_{\mathbf{Y}_{2,1}}(\mathbf{Y}_{2,1})}{Q_{\mathbf{Y}_{2,1}}(\mathbf{y}_{1,2})} \le \frac{9(\beta_{1,2} + \beta_{2,1})}{2\pi\sqrt{2\beta_{1,2}\beta_{2,1}}}. \quad (163) $$

By [33, Lemma 5], we have the following bound:

$$ \frac{f_{\mathbf{Y}_{2,1}|\mathbf{X}_{1,2}}(\mathbf{y}_{1,2}|\mathbf{x}_{1,2})}{Q_{\mathbf{Y}_{2,1}|\mathbf{X}_{1,2}}(\mathbf{y}_{1,2}|\mathbf{x}_{1,2})} \ge 2^{\frac{n_{1,2}-2}{2}} e^{-\frac{n_{1,2}\beta_{2,1}P}{2}}. \quad (164) $$

Thus,

$$ \log \frac{f_{\mathbf{Y}_{1,1}|\mathbf{X}_{1,1}}(\mathbf{y}_{1,1}|\mathbf{x}_{1,1}) f_{\mathbf{Y}_{2,1}|\mathbf{X}_{1,2}}(\mathbf{y}_{1,2}|\mathbf{x}_{1,2})}{f_{\mathbf{Y}_{1,1}}(\mathbf{y}_{1,1}) f_{\mathbf{Y}_{2,1}}(\mathbf{y}_{1,2})} \quad (165) $$

$$ \ge \log \frac{f_{\mathbf{Y}_{1,1}|\mathbf{X}_{1,1}}(\mathbf{y}_{1,1}|\mathbf{x}_{1,1}) Q_{\mathbf{Y}_{2,1}|\mathbf{X}_{1,2}}(\mathbf{y}_{1,2}|\mathbf{x}_{1,2})}{Q_{\mathbf{Y}_{1,1}}(\mathbf{y}_{1,1}) Q_{\mathbf{Y}_{2,1}}(\mathbf{y}_{1,2})} + \log \frac{2^{\frac{n_{1,2}+4}{2}} \sqrt{\pi \beta_{1,2} \beta_{2,1} (\sigma_{1,1}^2 + 2\beta_{1,1} \mathbf{P})}}{e^{\frac{n_{1,2}\beta_{2,1}P}{2}} 243 (\sigma_{1,1}^2 + \beta_{1,1} \mathbf{P}) (\beta_{1,2} + \beta_{2,1})} \quad (166) $$

which concludes the proof.

## APPENDIX E
### PROOF OF LEMMA 3

Notice that

$$ \begin{aligned} \tilde{i}(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1}) = & \frac{n_{1,1}}{2} \log \frac{\sigma_{y_{1,1}}^2}{\sigma_{1,1}^2} + \frac{n_{1,2}}{2} \log \frac{\sigma_{y_{1,2}}^2}{\sigma_{y_{1,2}|x_{1,2}}^2} - \frac{\|\mathbf{Z}_{1,1}\|^2}{2\sigma_{1,1}^2} + \frac{\|\mathbf{X}_{1,1} + \mathbf{Z}_{1,1}\|^2}{2\sigma_{y_{1,1}}^2} \\ & - \frac{\|\alpha(\mathbf{X}_{1,2} + \mathbf{X}_{2,1}) + \mathbf{Z}_{1,2} - \mu_{1,2}\|^2}{2\sigma_{y_{1,2}|x_{1,2}}^2} + \frac{\|\alpha(\mathbf{X}_{1,2} + \mathbf{X}_{2,1}) + \mathbf{Z}_{1,2}\|^2}{2\sigma_{y_{1,2}}^2}, \end{aligned} \quad (167) $$

and by (95),

$$ \begin{aligned} |\tilde{i}(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})| \le & \frac{n_{1,1}}{2} \log \frac{\sigma_{y_{1,1}}^2}{\sigma_{1,1}^2} + \frac{n_{1,2}}{2} \log \frac{\sigma_{y_{1,2}}^2}{\sigma_{1,2}^2} + \frac{\|\mathbf{Z}_{1,1}\|^2}{2\sigma_{1,1}^2} + \frac{\|\mathbf{X}_{1,1} + \mathbf{Z}_{1,1}\|^2}{2\sigma_{y_{1,1}}^2} \\ & + \frac{\|\mathbf{X}_{2,1} + \mathbf{Z}_{1,2} + \left(1 - \sqrt{\frac{\beta_c}{\beta_c}}(1 + \rho\sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})\right) \mathbf{X}_{1,2}\|^2}{2\sigma_{1,2}^2} + \frac{\|\mathbf{X}_{1,2} + \mathbf{X}_{2,1} + \mathbf{Z}_{1,2}\|^2}{2\sigma_{y_{1,2}}^2}. \end{aligned} \quad (168) $$

Recall the definitions of $k, \tilde{k}, b, \tilde{b}$ and $\tilde{\kappa}$ from (37). Thus,

$$ \begin{aligned} & \mathbb{P}[|\tilde{i}(\mathbf{X}_{1,1}, \mathbf{X}_{1,2}; \mathbf{Y}_{1,1}, \mathbf{Y}_{2,1})| > \kappa + j] \\ \le & \mathbb{P} \left[ \frac{n_{1,1}}{2} \log \frac{\sigma_{y_{1,1}}^2}{\sigma_{1,1}^2} + \frac{n_{1,2}}{2} \log \frac{\sigma_{y_{1,2}}^2}{\sigma_{1,2}^2} + \frac{\|\mathbf{Z}_{1,1}\|^2}{2\sigma_{1,1}^2} + \frac{\|\mathbf{X}_{1,1} + \mathbf{Z}_{1,1}\|^2}{2\sigma_{y_{1,1}}^2} \right. \\ & \left. + \frac{\|\mathbf{X}_{2,1} + \mathbf{Z}_{1,2} + \left(1 - \sqrt{\frac{\beta_c}{\beta_c}}(1 + \rho\sqrt{\frac{\beta_{2,1}}{\beta_{1,2}}})\right) \mathbf{X}_{1,2}\|^2}{2\sigma_{1,2}^2} + \frac{\|\mathbf{X}_{1,2} + \mathbf{X}_{2,1} + \mathbf{Z}_{1,2}\|^2}{2\sigma_{y_{1,2}}^2} > \kappa + j \right] \end{aligned} \quad (169) $$

$$ \le \mathbb{P} \left[ k\left(\|\mathbf{Z}_{1,1}\| + \frac{b}{2}\right)^2 + \tilde{k}\left(\|\mathbf{Z}_{1,2}\| + \frac{\tilde{b}}{2}\right)^2 > \tilde{\kappa} + \kappa + j \right] \quad (170) $$

$$ \le \max \left\{ \mathbb{P} \left[ k\left(\|\mathbf{Z}_{1,1}\| + \frac{b}{2}\right)^2 > \frac{\tilde{\kappa} + \kappa + j}{2} \right], \mathbb{P} \left[ \tilde{k}\left(\|\mathbf{Z}_{1,2}\| + \frac{\tilde{b}}{2}\right)^2 > \frac{\tilde{\kappa} + \kappa + j}{2} \right] \right\} \quad (171) $$

$$ \le \max \left\{ \mathbb{P} \left[ k\left(\|\mathbf{Z}_{1,1}\| + \frac{b}{2}\right)^2 > \frac{\tilde{\kappa} + \kappa + j}{2} \right], \mathbb{P} \left[ \tilde{k}\left(\|\mathbf{Z}_{1,2}\| + \frac{\tilde{b}}{2}\right)^2 > \frac{\tilde{\kappa} + \kappa + j}{2} \right] \right\} \quad (172) $$
$$ = 1 - \min \left\{ \mathbb{P} \left[ \left\| Z_{1,1} \right\| \le \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2k}} - \frac{b}{2} \right], \mathbb{P} \left[ \left\| Z_{1,2} \right\| \le \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2\tilde{k}}} - \frac{\tilde{b}}{2} \right] \right\} \quad (173) $$

$$ \stackrel{(i)}{=} 1 - \min \left\{ 1 - \frac{\Gamma\left(\frac{n_{1,1}}{2}, \frac{1}{2\sigma_{1,1}^4}\left(\sqrt{\frac{\tilde{\kappa}+\kappa+j}{2k}} - \frac{b}{2}\right)^2\right)}{\Gamma\left(\frac{n_{1,1}}{2}\right)}, 1 - \frac{\Gamma\left(\frac{n_{1,2}}{2}, \frac{1}{2\sigma_{1,2}^4}\left(\sqrt{\frac{\tilde{\kappa}+\kappa+j}{2k}} - \frac{\tilde{b}}{2}\right)^2\right)}{\Gamma\left(\frac{n_{1,2}}{2}\right)} \right\} \quad (174) $$

$$ = \max \left\{ \frac{\Gamma\left(\frac{n_{1,1}}{2}, \frac{1}{2\sigma_{1,1}^4}\left(\sqrt{\frac{\tilde{\kappa}+\kappa+j}{2k}} - \frac{b}{2}\right)^2\right)}{\Gamma\left(\frac{n_{1,1}}{2}\right)}, \frac{\Gamma\left(\frac{n_{1,2}}{2}, \frac{1}{2\sigma_{1,2}^4}\left(\sqrt{\frac{\tilde{\kappa}+\kappa+j}{2k}} - \frac{\tilde{b}}{2}\right)^2\right)}{\Gamma\left(\frac{n_{1,2}}{2}\right)} \right\} \quad (175) $$

$$ \le \max \left\{ \frac{(\sigma_{1,1}^2 \sqrt{2})^{2-n_{1,1}} \zeta}{\Gamma(\frac{n_{1,1}}{2})} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2k}} - \frac{b}{2} \right)^{n_{1,1}-2} e^{-\frac{1}{2\sigma_{1,1}^4} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2k}} - \frac{b}{2} \right)^2}, \right. \\ \left. \frac{(\sigma_{1,2}^2 \sqrt{2})^{2-n_{1,2}} \zeta}{\Gamma(\frac{n_{1,2}}{2})} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2\tilde{k}}} - \frac{\tilde{b}}{2} \right)^{n_{1,2}-2} e^{-\frac{1}{2\sigma_{1,2}^4} \left( \sqrt{\frac{\tilde{\kappa} + \kappa + j}{2\tilde{k}}} - \frac{\tilde{b}}{2} \right)^2} \right\}, \quad (176) $$

where in (i) we use the fact that $||Z_{1,1}||/\sigma_{1,1}^2$ and $||Z_{1,2}||/\sigma_{1,2}^2$ follow a central chi-distribution of degree $n_{1,1}$ and $n_{1,2}$, respectively. In the last inequality, we use the following bound [36]:

$$ \Gamma(a, x) < \zeta x^{a-1} e^{-x}, \quad (177) $$

for $a > 1, \zeta > 1, x > \zeta/(\zeta - 1)(a - 1)$. This concludes the proof.

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