This paper investigates the capacity of general multiple-input single-output (MISO)  optical intensity channels (OICs) under per-antenna peak- and average-intensity constraints. We first consider the MISO equal-cost constrained OIC (EC-OIC), where, apart from the peak-intensity constraint, average intensities of inputs are equal to arbitrarily preassigned constants. The second model of our interest is the  MISO bounded-cost constrained  OIC (BC-OIC), where, as compared with the EC-OIC, average intensities of inputs are no larger than arbitrarily preassigned constants.
By introducing quantile functions, stop-loss transform and convex ordering of nonnegative random variables, we prove two decomposition theorems for bounded and nonnegative random variables, based on which we equivalently transform both the EC-OIC and the BC-OIC into single-input single-output channels under a peak-intensity and several stop-loss mean constraints. Lower and upper capacity bounds for both channels are established, based on which the asymptotic capacity at high and low signal-to-noise-ratio are determined.

§ INTRODUCTION

Optical wireless communication (OWC) is a promising technique for future wireless communication due to its abundant bandwidth, license-free deployment, and absence of interference with existing radio frequency (RF) systems <cit.>. From the consideration of implementation complexity, a particularly appealing transmission scheme in current OWC systems is intensity-modulation direct-detection (IM/DD). In such a scheme, the transmitter modulates the intensity of optical signals coming from light emitting diodes (LEDs), and the receiver applies photodetectors to measure incoming optical intensities <cit.>. As a consequence of this operation, the transmit signal is real and nonnegative, which is fundamentally different from that of its RF counterpart. Furthermore, considering safety reasons and hardware limitations, the peak and the average intensities of transmit signals typically have to be restricted.

The optical intensity channel (OIC) is the most commonly used model for linear IM/DD systems impaired by additive white Gaussian noise (AWGN), just like the AWGN channel in RF communications. During the last few decades, OICs have been extensively studied in the literature from two closely related aspects <cit.>. One aspect is on capacity analysis, which describes fundamental limits of OICs for reliable communication.  While exact characterization of channel capacity under a first-order moment or an amplitude constraint is still an open problem <cit.>, many capacity bounds and asymptotic results have been derived; see, e.g., <cit.> for a comprehensive survey. For single-input single-output (SISO) OICs under a peak- and/or an average-intensity constraint, capacity bounds and asymptotic results have been established <cit.>. In <cit.>, capacity results of parallel SISO OICs with a total average-intensity constraint have been presented, as well as a low-complexity intensity allocation algorithm. Various capacity slopes at low signal-to-noise-ratios (SNRs) have been characterized in <cit.> for general MIMO OICs, while high-SNR asymptotic capacities have been investigated in <cit.> for MIMO OICs of full column rank and under a total average-intensity constraint, or per-antenna peak-intensity constraints, or both.

For general multiple-input single-output (MISO) and multiple-input multiple-output (MIMO) OICs with per-antenna peak-intensity constraints and a total average-intensity constraint, the optimal signaling strategy and capacity results have also been derived in <cit.> and <cit.>, respectively. The other aspect is on modulation and coding, related to system designs for optimizing various performance metrics, such as bit-error rates, block-error rates and the minimum Euclidean distance of constellations.  Modulation schemes including on-off keying,  color-shift keying <cit.>, pulse-position modulation <cit.>, different variants of unipolar orthogonal frequency-division multiplexing <cit.> have been studied, and code constructions described in <cit.>.

This paper mainly focuses on the capacity analysis of general MISO OICs under per-antenna peak- and average-intensity constraints.
We first consider the MISO equal-cost constrained OIC (EC-OIC), where, apart from the peak constraints,  expectations (i.e., average intensity) of inputs need to be arbitrarily preassigned constants. The prime motivation for considering EC-OICs is arising requirements of more granular illumination adjustment and chromaticity control in visible light communications <cit.>. Then we extend the results to the MISO bounded-cost constrained OIC (BC-OIC), where the average-intensity constraints are relaxed to be no larger than arbitrarily preassigned constants as compared to the EC-OIC.
An important motivation for the BC-OIC is the increasing attention to the distributed design of active antennas <cit.>, where each antenna has its own amplifier with limited power. Furthermore, it will be shown that the per-antenna average-intensity inequality constraints are no longer equivalent to equality constraints for MISO OICs in terms of channel capacity as those in the SISO OIC with the ratio of the maximum allowed average intensity to the maximum allowed peak intensity being no larger than ${1}/{2}$. For this reason, theoretical investigation on MISO OICs under such two types of intensity constraints is meaningful. The results concerning the above channels may also shed light on performance analysis of OWC networks, in which users usually have individual intensity constraints <cit.>.

In previous works, the average-intensity constraint in MISO or MIMO channels is mostly imposed on the total average intensity of all transmit antennas. In those scenarios intensity allocation among all transmit antennas is allowed, and by using the optimal signaling scheme, capacity analysis can be accomplished <cit.>. Per-antenna intensity constraints spontaneously make the total average intensity of inputs limited and hence impose a much stronger restriction on the choice of intensity allocation. Consequently, the optimal signaling scheme in the total average-intensity constrained scenario may no longer be admissible to the per-antenna intensity-constrained scenario. To the best of our knowledge, the existing relevant capacity results concerning multi-transmitter OICs under per-antenna intensity constraints are only found in  <cit.>, which mainly focuses on MIMO OICs at low SNR, and shows that the low-SNR capacity slope can be achieved by a certain maximally-correlated input.

As we will see, the mathematical techniques involved in analyzing a multi-transmitter system with per-antenna constraints are related to decomposing a random variable into several possibly dependent random variables with constraints on their supports and expectations. As opposite to the usually studied problem of analyzing behaviors of the sum of random variables, its inverse problem, i.e., decomposition of a random variable, is less studied. Moreover, most relevant mathematical results are limited to the case of summands being independent random variables <cit.>.
To address the challenges brought by the support constraint, individual first-moment constraints and statistical dependence on the inputs, we introduce concepts of the stop-loss transform and convex ordering, which have been commonly used in economics and insurance mathematics. Our main technical contribution is necessary and sufficient conditions for two types of decomposability of a bounded and nonnegative random variable.

Based on the proposed decomposition theorems, we present several capacity results in general MISO channel under per-antenna intensity constraints. The main results of this paper are summarized as follows:

* Decomposition results: We establish necessary and sufficient conditions for two types of decomposition of a bounded and nonnegative random variable into a convex summation in a particular form; see Theorems <ref> and <ref>. In the constructive proof of Theorem <ref>, a novel approach, called greedy decomposition, to mapping the sum random variable onto its corresponding components is presented; see Sec. <ref>. Furthermore, an iteration-based and a partition-based algorithm for the greedy decomposition are proposed; see Algorithms <ref> and <ref>.

* Equivalent capacity expression:  Based on two decomposition theorems, we prove that either the MISO EC-OIC and the MISO BC-OIC is equivalent to some SISO channel with an amplitude constraint and several moment constraints (defined by the stop-loss transform of the input at certain points); see Corollaries <ref> and <ref>.

* Capacity bounds: For the EC-OIC and the BC-OIC, lower capacity bounds are derived by maximizing the differential entropy of the input for the equivalent SISO channel and applying Entropy Power Inequality (EPI); see Theorems <ref> and <ref>. One type of upper bounds is derived by maximizing the variance of the equivalent input and  applying the “Gaussian maximizes differential entropy” argument; see Theorems <ref> and <ref>. Another type of upper bounds is obtained by using duality-bounding technique; see Theorems <ref> and <ref>.

* Asymptotic capacity: Low-SNR capacity slopes of both the EC-OIC and the BC-OIC are determined via a maximum-variance argument; see Theorem <ref>. High-SNR asymptotic capacities of both channels are also specified by showing that derived upper and lower capacity bounds are asymptotically tight; see Theorem <ref>.

The paper is organized as follows. We end the introduction with Table <ref> that summarizes notational conventions in the paper.  Sec. <ref> introduces  two types of channel, formulates and simplifies the problem of interest, and provides a reader-friendly guidance for results in the paper.. Sec. <ref> gives preliminaries in this paper. In Sec. <ref>, results on two types of decomposition of a bounded and nonnegative random variable and consequent equivalent capacity expressions for our considered channels are presented. Sec. <ref> presents lower bounds, upper bounds, and asymptotic results on capacities. Derivation of results on decomposition is postponed into Sec. <ref>. Sec. <ref> focuses on the decomposition algorithms and signaling issues. The paper is concluded in Sec. <ref>.

Summary of notations used throughout the paper.

$\mathbb{P}\{\cdot\}$   Probability of an event

$\mathbb{E}\{\cdot\}$   Expectation operator

$\mathcal{N}(0,\sigma^2)$   Gaussian distribution with zero mean and variance $\sigma^2$

$\mathcal{Q}(\cdot)$   Gaussian Q-function

$\mathsf{supp}\, X$   Support of the random variable $X$

$\mathscr{P}_{\! X}$   Probability measure of $X$

$Q_{\! X}(\cdot)$   Quantile function of $X$

$\hh(X)$   Differential entropy of $X$

$\II(X;Y)$   Mutual information between $X$ and $Y$

$\bm{1}_{n}$   $n$-dimensional all-ones column vector

$\left[n\right]$   Index set $\{ 1,2,\cdots,n\}$ for a positive integer $n$

$\mathcal{A}^n$   $n$-fold Cartesian product of the set $\mathcal{A}$

$|\mathcal{A}|$   Cardinality of the set $\mathcal{A}$

$\min \mathcal{A}$, $\max \mathcal{A}$   Minimum and maximum of a totally ordered set $\mathcal{A}$

$(a)_+$    $\max \{a,0\}$ for a real number $a$

2*$\max\{\mathbf{a},\mathbf{b}\}$   Element-wise maximum $\trans{(\max\{a_1,b_1\}, \cdots, \max\{a_n,b_n\})}$

for $n$-dimensional column vectors $\mathbf{a}$ and $\mathbf{b}$

$\mathbf{a} \preccurlyeq \mathbf{b}$    Difference $\mathbf{b}-\mathbf{a}$ is nonnegative

2*$\mathbf{a}_{\mathcal{J}}$    $\trans{(a_{j_1},\cdots,a_{j_{|\mathcal{J}|}})}$, where $j_1,\cdots,j_{{|\mathcal{J}|}}$ are all elements of the set $\mathcal{J}\subseteq \left[n\right]$

in the ascending order and $\mathbf{a}$ is an $n$-dimensional column vector

§ CHANNEL MODEL, PROBLEM FORMULATION AND SUMMARY OF RESULTS

§.§ Channel Model

Consider an $\nt \times 1$ ($\nt\ge2$)  MISO OIC

\begin{equation}\label{eqn:model1}
\end{equation}

where the $\nt$-dimensional vector $\mathbf{X}=\trans{\left(X_1,\cdots,X_{\nt}\right)}$ denotes the channel input, whose entries denote optical intensities emitted from LED transmitters, and hence are nonnegative:

$${\color{black}X_k} \ge 0, \quad \forall   k\in {\color{black}\left[\nt\right]};$$

where the $\nt$-dimensional vector ${\mathbf{h}}=\trans{\left(h_1,\cdots,h_{\nt}\right)}$ denotes the constant channel gain vector with positive entries, taking into account LED conversion factors, path loss, and the photodetector responsivity, etc., the sum of which is normalized to be unit, i.e.,

$$\sum_{i=1}^{\nt} h_i=1;$$

where $Z$ denotes the overall effect of the ambient shot noise induced by background radiations and the thermal noise of electronic devices, which is approximated to follow the Gaussian distribution with zero mean and variance $\sigma^2$, i.e.,

$$Z \sim \mathcal{N}(0,\sigma^2),$$

and is independent of $\mathbf{X}$; and where $Y$ denotes the photocurrent output. It should be noted that the output $Y$ can be negative even if the input ${\color{black}\mathbf{X}}$ is always nonnegative.

In this paper, we assume that full channel knowledge is available at both the transmitter and the receiver side. Considering the limited dynamic range of LED devices and the requirement of illumination quality or energy consumption, the input $\vect{X}$ is subject to one of the following two types of per-antenna intensity constraints:

* Equal-cost constraints: the channel input is subject to per-antenna peak-intensity constraints and average-intensity equality constraints, i.e.,

\begin{equation}\label{eqn:ecc}
\begin{aligned}
&{\color{black}0 \le X_k \le 1},\\
&\mathbb{E}\left[X_k\right] = \alpha_k, \quad \forall   k\in {\color{black}\left[\nt\right]},
\end{aligned}
\end{equation}

where the constant $\alpha_k\in \left[0,1\right]$ denotes the ratio of the average intensity to the maximum allowed peak intensity of the $k$-th transmitter.

* Bounded-cost constraints: the channel input is subject to per-antenna peak-intensity constraints and average-intensity inequality constraints, i.e.,

\begin{equation}\label{eqn:bcc}
\begin{aligned}
&{\color{black}0 \le X_k \le 1},\\
&\mathbb{E}\left[X_k\right]\le \alpha_k,~\quad \forall   k\in {\color{black}\left[\nt\right]},
\end{aligned}
\end{equation}

where the constant $\alpha_k\in\left[0,1\right]$ is the ratio of the maximum allowed average intensity to the maximum allowed peak intensity of the $k$-th transmitter.

For brevity of expression, we let ${\bm{\alpha}} = \trans{\left(\alpha_1,\alpha_2,\ldots,\alpha_{\nt}\right)}$. Without loss of generality, we assume the entries of ${\bm{\alpha}}$ are ordered as $\alpha_1\ge \alpha_2 \ge \cdots \ge \alpha_{\nt}$. In the rest of the paper, we call the OIC under input constraints (<ref>) as the EC-OIC, and under constraints (<ref>) as the BC-OIC.

We equivalently rewrite the MISO channel (<ref>) as

\begin{equation}\label{model:siso}
Y= S +Z,
\end{equation}

where the equivalent input is

\begin{eqnarray}\label{eqn:model2}
\begin{aligned}
S&\triangleq \trans{\mathbf{h}}\mathbf{X}=\sum_{k=1}^{\nt}h_kX_k.\\
\end{aligned}
\end{eqnarray}

Then $S$ must satisfy

0 ≤S ≤1,

for the EC-OIC or

for the BC-OIC.

§.§ Problem Formulation and Simplification

In this paper, we are concerned with capacity results on the MISO OIC (<ref>) under above two different per-antenna intensity constraints (<ref>) or (<ref>). The single-letter capacity expression for the EC-OIC is given by

𝖢_E(h,α,σ) = sup_𝒫_𝐗  satisfying   (<ref>) (𝐗;Y).

Similarly, for the BC-OIC,

𝖢_B(h,α,σ) = sup_𝒫_𝐗  satisfying   (<ref>) (𝐗;Y).

By flipping the input as $\bm{1}_{\nt}-\mathbf{X}$, we have

𝖢_E (h,α,σ) = 𝖢_E  (h,   1_-α,σ) , ∀  0 ≼α  ≼1_,

for the EC-OIC. Note that all coordinates of ${\bm{1}}_{\nt}-\bm{\alpha}$ are no larger than $\frac{1}{2}$ if $\alpha_{\nt} \geq \frac{1}{2}$.

For the BC-OIC, by following the proof of <cit.>, we can easily show that

𝖢_B  (h,α,σ) = 𝖢_B(h, 1/21_,σ), if α_ ≥1/2.

Moreover, the following proposition shows that we can further restrict our attention to the case where entries in $\bm{\alpha}$ follow a strictly decreasing order, i.e., $\alpha_1>\cdots>\alpha_{\nt}>0$.

Given an $\nt\times 1$ EC-OIC (or BC-OIC) with a channel gain vector $\mathbf{h}$ and an average-intensity constraint vector $\bm{\alpha}$, if there exists an index $ i \in {\color{black}\left[\nt-1\right]}$ satisfying ${\alpha}_i={\alpha}_{i+1}$, then

𝖢_E (or B)(𝐡,α,σ)=𝖢_E (or B)(𝐡',α_[]∖{i},σ),

with the $(\nt-1)$-dimensional vectors
$$\mathbf{h}' =\trans{ (h_1,h_2,\ldots,h_{i-1},h_i+h_{i+1},h_{i+2},\ldots,h_{\nt})} $$

$$\bm{\alpha}_{{\color{black}\left[\nt\right]}\setminus \{i\}} =\trans{ (\alpha_1,\alpha_2,\ldots,\alpha_{i-1},\alpha_{i+1},\alpha_{i+2},\ldots,\alpha_{\nt})}.$$

Let $\mathbf{X}$ be an arbitrary feasible input.[By the convention of optimization theory, we use the term “feasible” to mean that the input $\mathbf{X}$ satisfies the constraints (<ref>) or (<ref>) for the EC-OIC or the BC-OIC, respectively.] Note that the equivalent input $S=\trans{\mathbf{h}}\mathbf{X}$ is a function of the channel input $\mathbf{X}$, and $\mathbf{X} \rightarrow S \rightarrow Y$ forms a Markov chain. Then $\II \left(\mathbf{X};Y\right)=\II \left(S;Y\right)$.

Construct a new input $\widetilde{\mathbf{X}}$ by letting  $\widetilde{X}_i=\widetilde{X}_{i+1}=\frac{h_iX_i+h_{i+1}X_{i+1}}{h_{i}+h_{i+1}}$ and $\widetilde{X}_j=X_j, \, \forall j \in {\color{black}\left[\nt\right]}\backslash\{i,i+1\}$. It is obvious that the constructed input $\widetilde{\mathbf{X}}$ is also feasible, and satisfies $\trans{\mathbf{h}}\widetilde{\mathbf{X}}=\trans{\mathbf{h}}\mathbf{X}$. Hence for any feasible input, we have

$$\II \left(\widetilde{\mathbf{X}};\trans{\mathbf{h}}\widetilde{\mathbf{X}}+Z\right)=\II \left(\mathbf{X};\trans{\mathbf{h}}\mathbf{X}+Z\right).$$

Since $\trans{\mathbf{h}}\widetilde{\mathbf{X}}$ is also a feasible equivalent input (induced by the input $\widetilde{\mathbf{X}}_{{\color{black}\left[\nt\right]}\setminus \{i\}}$) to the $(\nt-1)\times 1$ channel with $\mathbf{h}'$ and $\bm{\alpha}_{{\color{black}\left[\nt\right]}\setminus \{i\}}$,  we have

$$\mathsf{C}_{\text{E (or B)}}\left(\mathbf{h},\bm{\alpha},\sigma\right)\le\mathsf{C}_{\text{E (or B)}}\left(\mathbf{h}',\bm{\alpha}_{{\color{black}\left[\nt\right]}\setminus \{i\}},\sigma\right).$$

In the reverse direction, by restricting $X_i=X_{i+1}$, the original $\nt \times 1$ EC-OIC (or BC-OIC) is degenerated to a $(\nt-1)\times 1$ channel with $\mathbf{h}'$ and $\bm{\alpha}_{{\color{black}\left[\nt\right]}\setminus \{i\}}$, which implies

$$\mathsf{C}_{\text{E (or B)}}\left(\mathbf{h},\bm{\alpha},\sigma\right)\ge\mathsf{C}_{\text{E (or B)}}\left(\mathbf{h}',\bm{\alpha}_{{\color{black}\left[\nt\right]}\setminus \{i\}},\sigma\right).$$

The proof is completed.

Proposition <ref> shows the mergeability of transmitters with identical normalized per-antenna intensity constraints for both the EC-OIC and the BC-OIC, and spatial repetition across those transmitters is optimal in the sense of channel capacity. Specifically, if the average intensities of all transmitters are constrained by an identical ratio, i.e., $\bm{\alpha}=\alpha\bm{1}$, simply sending identical signals and treating the considered MISO OIC as a SISO one do not induce any loss. For example, according to (<ref>) the optimal intensity allocation for the BC-OIC with $\alpha_{\nt}\ge \frac{1}{2}$ is exactly $\frac{1}{2}\bm{1}_{\nt}$, which implies such the channel can be regarded as a peak-limited SISO OIC with an inactive average intensity constraint.

For above reasons, in the rest of paper, we assume $\alpha_1>\cdots>\alpha_{\nt}>0$ and $\alpha_{\nt} \leq \frac{1}{2}$  without loss of generality.

In terms of capacity, the above model applies to the channel with unnormalized parameters. For a general MISO OIC with per-antenna maximum allowed peak intensity $ {\boldsymbol{\amp} } = \trans{(\amp_1,\amp_2,\ldots,\amp_{\nt})}$, average intensity (or maximum allowed average intensity) ${ \boldsymbol{\const{P}}}  = \trans{ (\alpha_1\amp_1,\alpha_2\amp_2,\ldots,\alpha_{\nt}\amp_{\nt})}$, unnormalized channel gain vector ${\widetilde{\mathbf{h}}} =\trans{(\tilde{h}_1,\tilde{h}_2,\ldots,\tilde{h}_{\nt})} $, and the AWGN with a standard deviation $\tilde{\sigma}$, its capacity can be easily shown to be
$  \mathsf{C}_{\textnormal{E (or B)}}\left(\vect{h},\bm{\alpha},\sigma\right)$ via parameter normalization, where $\sigma=\tilde{\sigma}/\left(\sum_{i=1}^{\nt}\tilde{h}_i \amp_i\right)$ and $\vect{h}$ is a normalized vector with entries $h_k = \tilde{h}_k \amp_k/\left(\sum_{i=1}^{\nt}\tilde{h}_i \amp_i\right), \, \forall k\in {\color{black}[\nt]}$.

§.§ Summary of Results

Herein, without introducing any new terminology, we provide a concise summary of results in the paper, which will help the reader grasp our basic idea.

* Stage 1: We first show that it is sufficient to consider the capacity problem in the case where $\alpha_1 > \alpha_2 > \cdots > \alpha_{\nt}$ and $\alpha_{\nt}\le \frac{1}{2}$; see Sec. <ref>;

* Stage 2: Then we turn to the problem of for what distributions can a random variable $S$, satisfying $0 \le S \le 1$ and $\mathbb{E}[S]=\trans{\mathbf{h}}\bm{\alpha}$, be decomposed as $S=\sum_{k=1}^{\nt}h_k X_k$ with $0 \le X_k \le 1$ and $\mathbb{E}[X_k]=\alpha_k$. This problem is tackled in two steps.

* Necessity: If $S$ can be decomposed in the above form, then $ \mathbb{E}\left[\left(S- \left(\sum_{k\in \mathcal{J}}h_k \right) \right)_+\right] \le \sum_{k\in\mathcal{J}^{\rm c}} h_k\alpha_k$ for all $2^{\nt}$ subsets $\mathcal{J}\subseteq [\nt]$.

* Sufficiency: If the above expectation condition is satisfied for $\nt-1$ subsets $\mathcal{J}=[1]$, $[2]$, $\cdots$, and $[\nt-1]$, then $S$ indeed has such a decomposition. For this, we give a constructive proof, where a procedure for generating the desired decomposition is presented.

* Stage 3:  Next, we answer the similar problem of under which conditions the random variable $S$, satisfying $0 \le S \le 1 $ and $\mathbb{E}[S]\le\trans{\mathbf{h}}\bm{\alpha}$, can be decomposed as $S=\sum_{k=1}^{\nt}h_k X_k$ with $0 \le X_k \le 1$ and $\mathbb{E}[X_k]\le \alpha_k$. It is shown that if sufficient conditions in Stage 2 are fulfilled, there exists such type of decomposition for $S$ as well.

* Stage 4: With the above results on decomposition, alternative capacity expressions for the MISO EC-OIC and the MISO BC-OIC are derived, which significantly simplify the subsequent capacity computation because
the input distributions over which one needs to optimize now are under much simpler constraints.

* Stage 5: By relaxing some of the input conditions to be satisfied and using techniques in the existing literature, several upper and lower capacity bounds are derived and then asymptotic capacities are obtained by showing tightness of some bounds.

§ PRELIMINARIES

This section is devoted to presenting some definitions and propositions that we will use.

§.§ Distribution and Quantile Functions

Unless otherwise stated, the supports of random variables under our consideration are subsets of the interval $[0,1]$.

The cumulative distribution function $F_{\!X}(x)$ of a random variable $X$ with $\mathsf{supp} \, X \subseteq [0,1]$ is defined as

\begin{equation}
F_{\!X}(x)=\mathbb{P}\left\{X\le x\right\},
\end{equation}

which implies that $F_{ \!X}(x)$ is non-decreasing and right-continuous, and has at most countable discontinuous points on the interval $x\in [0,1]$.

The generalized inverse function of $F_{\!X}(x)$, namely, quantile function, is defined

Q_ X(p) ≜inf{x∈[0,1]:  p≤F_ X (x) },

for $0 < p \le 1$,[Since the random variable $X$ is bounded above, we define $p$ on $\left( 0,1\right]$ instead of the conventional open interval $\left( 0,1\right)$.] which correspondingly is non-decreasing and left-continuous. The following well-known property of quantile functions can be immediately derived by the definition (<ref>).

The quantile function $Q_{\! X}(p)$ satisfies the Galois inequality:

\begin{eqnarray}\label{eq:Galois}
F_{\! X}(x)\ge p~\textnormal{if and only if}~  x\ge Q_{\! X}(p),
\end{eqnarray}

for any $p\in(0,1]$.

A consequence of the Galois inequality is the following proposition.

Let $U$ be a random variable uniformly distributed on the interval $\left(0,1\right]$. For any random variable $X$, we have

Xd= Q_ X(U),

and furthermore,

𝔼[X]=∫_0^1 Q_X(p) p,

where $\overset{\text{d}}=$ stands for “equality in cumulative distribution”. Conversely, if a random variable $X\overset{\text{d}}= f(U)$, where the function $f(u)$ is nondecreasing, left-continuous and satisfies $0  \le f(u) \le 1$ for $u\in (0,1]$, then

Q_ X(p)=f(p), p∈(0,1].

Note that (<ref>) is an immediate consequence of the Galois inequality (<ref>) and (<ref>) follows (<ref>). If $X\overset{\text{d}}= f(U)$, we have

F_ X(x)   =ℙ{f(U)≤x}

=sup{p∈(0,1]: f(p)≤x } ,

where (<ref>) follows from the fact that $f(p)$ is nondecreasing and left-continuous. It is easy to show that the function $f(u)$ satisfies the Galois inequality (<ref>) as well. Since $F_{\! X} (x)$ is nondecreasing and right-continuous, we have $f(p)=\inf \left\{x\in [0,1]:\, p\le F_{\! X} (x) \right\}=Q_{\! X}(p)$.

§.§ Stop-Loss Transform

Since a monotonic function on a closed interval must be Riemann integrable, we can define the stop-loss transform (SLT) of a random variable as follows.

The stop-loss transform of a random variable $X$ with $\mathsf{supp}\,X\subseteq [0,1]$ is

π_ X (t)=∫_t^1 ( 1-F_ X(x) ) x

for $t\in\left[0,1\right]$.

The SLT is a well-known concept in actuarial science for ordering risks <cit.>.
Several properties of SLT are listed as follows without proof; see, e.g., <cit.> for more details.

(P1)  $\pi_{\scriptscriptstyle \! X} (t)$ is nonnegative and nonincreasing;

(P2) $\pi_{\scriptscriptstyle \! X} (t)$ is convex;

(P3) $0\le \pi_{\scriptscriptstyle \! X} (t) \le 1-t$ and thus $\pi_{\scriptscriptstyle \! X} (1)=0$;

(P4) $\pi_{\scriptscriptstyle \! X} (t)=\mathbb{E}\left[ (X-t)_+ \right]$ and thus $\pi_{\scriptscriptstyle \! X} (0)=\mathbb{E}\left[X\right]$.

Here we present an alternative geometrical interpretation of the SLT, which connects it with the quantile $Q_{\!X}(p)$.

For a random variable $X$ with $\mathsf{supp}\, X \subseteq [0,1]$, let the region $\mathcal{R}^{  \scriptscriptstyle \! X}_t \subseteq \mathbb{R}^2$ be

ℛ^   X_t= { (p,y)∈ℝ^2: t ≤y < Q_X(p), p∈(0,1] }, t∈[0,1].

Then the area of $\mathcal{R}^{  \scriptscriptstyle \! X}_t$ satisfies

Area  ℛ^   X_t
=π_ X (t), t∈[0,1].

The contrapositive of Proposition <ref> reveals that

ℛ^   X_t= { (p,y)∈ℝ^2: y≥t,  0 ≤1-p< 1- F_ X(y) }.

Area  ℛ^   X_t

=    Area  { (y,p̅)∈ℝ^2: y≥t,  0 ≤p̅< 1- F_ X(y) }

=   ∫_t^+∞ (1- F_ X(y)) y

=   ∫_t^1
( 1- F_ X(y))

=   π_ X (t)

for $t\in\left[0,1\right]$.

§.§ Convex Ordering

The convex order is a classical stochastic order that compares two random variables with an equal mean and commonly known in economics and insurance mathematics <cit.>.

Given two nonnegative random variables $X$ and $Y$ with finite mean, $X$ is said to precede $Y$ in the convex order sense, denoted as $X\le_{\text{cx}}Y$, if
either one of the following two equivalent propositions holds:

* $
\mathbb{E}\left[f(X)\right]\le \mathbb{E}\left[f(Y)\right]
$ for all convex functions $f$ such that $\mathbb{E}\left[f(X)\right]$ and $\mathbb{E}\left[f(Y)\right]$ exist;

* $
\pi_{\scriptscriptstyle \! X}(t)\le \pi_{\scriptscriptstyle \! Y}(t)~\text{for all } t\ge 0,~\text{and}~\pi_{\scriptscriptstyle \! X}(0)=\pi_{\scriptscriptstyle \! Y}(0)

Due to $\pi_{\scriptscriptstyle \! X}(0)=\mathbb{E}\left[X\right]$, the relation $X\le_{\text{cx}}Y$ leads to $\mathbb{E}\left[X\right]=\mathbb{E}\left[Y\right]$. Furthermore, by letting $f(x)=(x-\mathbb{E}\left[X\right])^2$, we attain the following corollary, which reveals that the convex order implies the variance order.

If $X\le_{\text{cx}}Y$, then the following variance order holds

𝔼[ (X-𝔼[X])^2]≤𝔼[ (Y-𝔼[Y])^2] .

§.§ Comonotonic Distribution

Research on comonotonicity arises naturally in risk analysis since the upper bound (in the sense of convex ordering) of the sum of several random variables with given marginals is achieved by a comonotonic distribution <cit.>. In the following, we introduce two relevant definitions.

A set $\mathcal{X}\subseteq \mathbb{R}^n$ is comonotonic if $\mathbf{x} \preccurlyeq \mathbf{x}^{\prime}$ or $\mathbf{x}^{\prime} \preccurlyeq \mathbf{x}$ holds for any $\mathbf{x},\mathbf{x}^\prime \in \mathcal{X}$.

A random vector $\mathbf{X}=\trans{\left(X_1,\cdots,X_n\right)}$ is comonotonic if either one of following equivalent propositions holds:

* there is a comonotonic set $\mathcal{X}\subseteq \mathbb{R}^n$ such that $\mathbb{P}\left\{\mathbf{X}\in \mathcal{X} \right\}=1$;

* $\mathbf{X}\overset{\text{d}}= \trans{{\color{black}\left(Q_{\! X_{\! 1}}(U),\cdots,Q_{\! X_{\! n}}(U)\right)}}$, where the random variable $U$ is uniformly distributed on $\left(0,1\right]$ and $Q_{\! X_{\! k}}(\cdot)$ is the quantile function of $X_k$ for $k\in {\color{black}\left[n\right]}$.

As a special statistical dependence structure, comonotonicity is used to describe the phenomenon that several random variables always move in the same direction, i.e., simultaneously non-decreasing or non-increasing.

§.§ $\left(\mathbf{h},\bm{\alpha}\right)$-Decomposability

We first introduce the following notation. For any index set $\mathcal{J}\subseteq {\color{black}\left[\nt\right]}$, denote the partial sum of channel gains as

and hence the cumulative sum of channel gains can be expressed as

𝖧_[k]≜∑_i=1^kh_i, ∀  k∈[],

with $\mathsf{H}_{{\color{black}[0]}}=0$.

Furthermore, denote the weighted average intensity of the last $\nt-k$ transmitters as

\begin{eqnarray}\label{eqn:constraint3}
\bar{\alpha}_k=\frac{\sum_{i={k+1}}^{\nt}h_i\alpha_i}{1-\mathsf{H}_{\color{black}[k]}},~\forall k\in {\color{black}\left[\nt-1\right]},
\end{eqnarray}

with $\bar{\alpha}_{\nt}=0$. Clearly, we have $\bar{\alpha}_0 > \bar{\alpha}_1 > \cdots > \bar{\alpha}_{\nt-1}=a_{\nt}>\bar{\alpha}_{\nt}$.

Next, we define the $\left(\mathbf{h},\bm{\alpha}\right)$-decomposability of a random variable, which is a crucial concept in this paper.

A random variable $S$ is said to be $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable if there exists a random vector $\mathbf{W}=\trans{\left( W_1,\cdots,W_{\nt}\right)}$ satisfying  $\mathsf{supp}\, \mathbf{W} \subseteq [0,1]^{\nt}$ and $\mathbb{E}\left[\mathbf{W}\right]=\bm{\alpha}$ such that
$S= \trans{\mathbf{h}}\mathbf{W}$.

It is clear that any equivalent input signal $S$ (given by (<ref>)) of the EC-OIC must be $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable. Conversely, if a random variable $S$ is $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable, there exists a channel input $\mathbf{W}$ feasible to the EC-OIC such that $S$ is the corresponding equivalent input. Thus, we conclude that a random variable $S$ is a feasible equivalent input for the EC-OIC if and only if $S$ is $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable. We remind the reader that the above statement does not necessarily hold for the BC-OIC. Instead, for the BC-OIC, $S$ is a feasible equivalent input if and only if there exists a vector $\mathbf{a}$ satisfying $\bm{0} \preccurlyeq \mathbf{a} \preccurlyeq \bm{\alpha}$ such that $S$ is $\left(\mathbf{h},\mathbf{a}\right)$-decomposable.

A particular subclass of $\left(\mathbf{h},\bm{\alpha}\right)$-decomposability defined as follows will be also useful in our subsequent analysis.

A random variable $S$ is said to be comonotonically $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable if there exists a random vector $\mathbf{X}^{\rm c}=\trans{ \left(Q_{\! X_{\! 1}}\left(U\right),\cdots,Q_{\! X_{\! \nt}}\left(U\right)\right)}$, satisfying  $\mathsf{supp}\, \mathbf{X}^{\rm c} \subseteq [0,1]^{\nt}$ and $\mathbb{E}\left[\mathbf{X}^{\rm c}\right]=\bm{\alpha}$, such that

S=𝐡𝐗^c= ∑_k=1^ h_k Q_ X_ k(U),

where $U$ is a random variable uniformly distributed on $\left(0,1\right]$ and $Q_{\! X_{\! k}}(\cdot)$ is the quantile function of $X_k$ for $k\in {\color{black}\left[\nt\right]}$.

§.§ Maximally Convex Distribution

We develop the definition and properties of maximally convex distributions, which will be very useful throughout this paper.

A discrete random variable $\bar{S}_{\mathbf{h},\bm{\alpha}}$ is said to be a maximally convex distribution

if $\mathsf{supp}\,\bar{S}_{\mathbf{h},\bm{\alpha}} = \left\{ 0, \mathsf{H}_{\color{black}[1]},\, \cdots, \mathsf{H}_{\color{black}[\nt-1]},\, 1\right\}$ and its corresponding probability masses are $\left\{ 1-\alpha_1,\, \alpha_1-\alpha_2,\, \cdots, \, \alpha_{\nt-1}-\alpha_{\nt},\,\alpha_{\nt}\right\}$.

The following properties of the maximally convex distribution can be easily verified.

The SLT of $\bar{S}_{\mathbf{h},\bm{\alpha}}$, denoted by $\bar{\pi}_{\mathbf{h},\bm{\alpha}}(t)$, is the piecewise linear function successively joining $\nt+1$ breakpoints $ \left(\mathsf{H}_{\color{black}[i]},\left(1-\mathsf{H}_{\color{black}[i]}\right)\bar{\alpha}_i\right)$ for all $i\in\{0\}\cup {\color{black}\left[\nt\right]} $.

$\bar{S}_{\mathbf{h},\bm{\alpha}}$ is comonotonically $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable.

Let $R_0=\bar{S}_{\mathbf{h},\bm{\alpha}}$ and

= (S̅_𝐡,α-𝖧_[k])_+

for $k\in {\color{black}\left[\nt\right]}$.

It can be checked that

𝔼[R_k]=∑_m=k+1^ h_mα_m

for $k\in {\color{black}\left[\nt-1\right]}$.
Then we let

X_k   =R_k-1-R_k/h_k

for $k\in {\color{black}\left[\nt\right]}$. Notice that the function $\left(s-\mathsf{H}_{\color{black}[k-1]}\right)_+-\left(s-\mathsf{H}_{\color{black}[k]}\right)_+$ is monotonically increasing with $s$. Hence, the so-constructed random vector $\bar{\mathbf{X}}_{\bm{\alpha}}=\trans{\left(X_1,\cdots,X_{\nt}\right)}$ is comonotonic and satisfies $\mathsf{supp}\, \bar{\mathbf{X}}_{\bm{\alpha}} \subseteq [0,1]^{\nt}$, $\mathbb{E}\left[\bar{\mathbf{X}}_{\bm{\alpha}}\right]=\bm{\alpha}$ and $\bar{S}_{\mathbf{h},\bm{\alpha}}= \trans{\mathbf{h}}\bar{\mathbf{X}}_{\bm{\alpha}}$.

It can be checked that the probability mass function of $\bar{\mathbf{X}}_{\bm{\alpha}}$ is given by $\mathbb{P}\left\{\bar{\mathbf{X}}_{\bm{\alpha}}=\bm{0} \right\}=1-\alpha_1$, $\mathbb{P}\left\{\bar{\mathbf{X}}_{\bm{\alpha}}=\bm{1} \right\}=\alpha_{\nt}$, and $\mathbb{P}\left\{\bar{\mathbf{X}}_{\bm{\alpha}}=\sum_{i=1}^k \mathbf{e}_i \right\}=\alpha_{k}-\alpha_{k+1}$ for $k\in [\nt-1]$, where $\mathbf{e}_i$ denotes the $i$-th column vector of the $\nt\times \nt$ identity matrix. The constructed random vector $\bar{\mathbf{X}}_{\bm{\alpha}}$ is indeed a maximally correlated $\nt$-variate binary distribution developed in <cit.>, which is solely determined by $\bm{\alpha}$ and has the largest variance over all probability laws feasible to the EC-OIC.

§ CONDITIONS FOR DECOMPOSABILITY AND EQUIVALENT CAPACITY EXPRESSIONS

This section is devoted to main results in this paper, which consist of two decomposition theorems and consequent capacity expressions for the EC-OIC and the BC-OIC.

§.§ Decomposition Results

We first present necessary and sufficient conditions for $(\mathbf{h},\bm{\alpha})$-decomposability.

Let $S$ be a random variable satisfying $\mathsf{supp}\, S\subseteq [0,1]$ and $\mathbb{E}[S]=\trans{\mathbf{h}}\bm{\alpha}$. Then the following claims are equivalent:

* $S$ is $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable;

* For each $k\in {\color{black}\left[\nt-1\right]}$,

π_ S(𝖧_[k])≤(1-𝖧_[k])α̅_k;

* $ S \le_{\textnormal{cx}}  \bar{S}_{\mathbf{h},\bm{\alpha}} $;

* $S$ is comonotonically $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable;

See Sec. <ref>.

Theorem <ref> reveals the equivalence between $(\mathbf{h},\bm{\alpha})$-decomposability and comonotonic $(\mathbf{h},\bm{\alpha})$-decomposability and hence an equivalent input feasible to the EC-OIC must be comonotonically $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable as well. This fact suggests that it is sufficient to take comonotonic random vectors into account when we seek for the capacity-achieving input for the EC-OIC with an arbitrary SNR.

Theorem <ref> also demonstrates that, among all $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable random variables, the maximally convex distribution $\bar{S}_{\mathbf{h},\bm{\alpha}}$ is the largest one in the sense of convex order. Hence, the best upper bound on the SLT over all equivalent inputs feasible to the EC-OIC is exactly the SLT of $\bar{S}_{\mathbf{h},\bm{\alpha}}$, i.e., $\bar{\pi}_{\mathbf{h},\bm{\alpha}}(t)$. In Figure <ref>, we plot the SLT of the maximally convex distribution and the maximum-entropy distribution of a $3\times 1$ EC-OIC with $\mathbf{h}=\trans{\left(0.4,0.2,0.4\right)}$ and $\bm{\alpha}=\trans{\left(0.8,0.3,0.1\right)}$, which indeed follow the convex ordering as stated in Theorem <ref>. We also plot the SLT of the maximum-entropy distribution of the corresponding amplitude-limited SISO OIC (<ref>) with a relaxed intensity constraint (<ref>), which is not always below $\bar{\pi}_{\mathbf{h},\bm{\alpha}}(t)$ and hence not feasible to the above-mentioned EC-OIC. As a consequence of Corollary <ref> $\bar{S}_{\mathbf{h},\bm{\alpha}}$ maximizes the variance of the equivalent input for the EC-OIC. This is consistent with the low-SNR result in <cit.>.

SLT functions of the maximally convex distribution and the maximum-entropy distributions for a $3\times 1$ EC-OIC channel.

In the constructive proof of Theorem <ref>, a decomposition method (<ref>), called greedy decomposition, is provided to generate a comonotonic random vector $\mathbf{X}^{\rm c}$ satisfying $\mathbf{X}^{\rm c}\in [0,1]^{\nt}$, $\mathbb{E}\left[\mathbf{X}^{\rm c}\right]=\bm{\alpha}$ and $S=\trans{\mathbf{h}}\mathbf{X}^{\rm c}$. A more detailed description of decomposition algorithms as well as signaling issues are postponed into Sec. <ref>. The lack of knowledge of the greedy decomposition will not impede the reader in understanding our capacity results.

We also point out that the greedy decomposition is just one type of the comonotonic decomposition. For an $(\mathbf{h},\bm{\alpha})$-decomposable random variable, there may exist other decompositions. Suppose the inputs $X_1,\cdots,X_{\nt}$ are independent, log-concave and satisfy intensity constraints (<ref>), e.g., each $X_k$ being a truncated exponential distribution. Then we can rewrite the equivalent input $S=\trans{\mathbf{h}}\mathbf{X}$ as $S=\trans{\mathbf{h}}\mathbf{W}$ with $W_k=g_k(S)$ for $k\in {\color{black}\left[\nt\right]}$, where the functions $g_k(s)$ are given by

Due to the law of total expectation, so-constructed random variables $W_k$ must satisfy the constraints (<ref>) as well. Furthermore, based on Efron’s monotonicity theorem <cit.>, $(W_1,\cdots,W_{\nt})$ is comonotonic.[Note that the decomposition (<ref>) needs the distributions of $\nt$ inputs, while the greedy decomposition only needs the distribution of the equivalent input.]

Based on Theorem <ref> and stochastic ordering, the necessary and sufficient conditions for another type of decomposition is given as follows.

Let $S$ be a random variable satisfying $\mathsf{supp}\, S\subseteq [0,1]$ and $\mathbb{E}[S]\le\trans{\mathbf{h}}\bm{\alpha}$. Then the following claims are equivalent:

* There exists a vector $\mathbf{a}$ satisfying $\bm{0} \preccurlyeq \mathbf{a} \preccurlyeq \bm{\alpha}$ such that $S$ is $\left(\mathbf{h},\mathbf{a}\right)$-decomposable;

* For each $k\in \left[\nt-1\right]$,

π_ S(𝖧_[k])≤(1-𝖧_[k])α̅_k;

* $S$ is $\left(\mathbf{h},\mathbf{a}^{\dagger} \right)$-decomposable, where the vector  $	\mathbf{a}^{\dagger} =   \min\left\{\beta \bm{1},\bm{\alpha}\right\} $ with some $\beta \in \left[0,\alpha_1 \right]$ such that  $\trans{\mathbf{h}}\mathbf{a}^{\dagger}=\mathbb{E}[S]$.

See Appendix <ref>.

As illustrated in Sec. <ref>, when $\alpha_{\nt}\ge \frac{1}{2}$ the capacity-achieving average intensity of the BC-OIC is $\mathbf{a}^{\star}=\frac{1}{2}\bm{1}$ so that the BC-OIC is degenerated to a SISO intensity channel under a peak- and an average-intensity constraint without inducing any capacity loss.

In the case of $\alpha_{\nt}< \frac{1}{2}$, it follows from Theorem <ref> that the capacity-achieving average intensity $\mathbf{a}^\star$ belongs to the set $\left\{   \min\{\beta \bm{1},\bm{\alpha}\}:\beta\in [0,\alpha_1]
\right\}$.

Hence the capacity of the BC-OIC can be also formulated as a maximization over the capacities of a specific class of EC-OICs, i.e.,

𝖢_B(𝐡,α,σ)=max_β∈[0,α_1]   𝖢_E(𝐡, min{β1,α},σ).

Note that when $\beta\in [0,\alpha_{\nt}]$ the average intensity vector $\min\{\beta \bm{1},\bm{\alpha}\}$ is exactly $\beta\bm{1}$ so that the induced EC-OIC is equivalent to a SISO OIC under a peak and an average-intensity constraints as well. Hence, we have $\mathsf{C}_{\textnormal{E}}\left(\mathbf{h}, \min\{\beta \bm{1},\bm{\alpha}\},\sigma \right)=\mathsf{C}_{\textnormal{E}}\left(1,\beta,\sigma\right)$ for $\beta\in [0,\alpha_{\nt}]$.
Due to the monotonicity of the SISO capacity  $\mathsf{C}_{\textnormal{E}}\left(1,\beta,\sigma\right)$ <cit.>, the capacity-achieving average intensity can be further restricted to $\mathbf{a}^\star \in \left\{   \min\{\beta \bm{1},\bm{\alpha}\}:\beta\in [\alpha_{\nt},\alpha_1]
\right\} $.

§.§ Equivalent Capacity Expressions

Clearly, the above two theorems actually state necessary and sufficient conditions under which a random variable $S$ is a feasible equivalent input for the EC-OIC or the BC-OIC, respectively. This naturally leads to alternative expressions of the capacities $\mathsf{C}_{\textnormal{E}}\left(\mathbf{h},\bm{\alpha},\sigma\right)$ and $\mathsf{C}_{\textnormal{B}}\left(\mathbf{h},\bm{\alpha},\sigma\right)$ as follows.

The capacity $\mathsf{C}_{\textnormal{E}}\left(\mathbf{h},\bm{\alpha},\sigma\right)$ can be equivalently expressed as

𝖢_E(𝐡,α,σ)=sup_𝒫_  S (S;S+Z),

where the supremum is over all probability laws ${\color{black}\mathscr{P}_{ \! S}}$ on $S$ with $\mathsf{supp}\, S\subseteq [0,1]$ and under stop-loss mean constraints:

\begin{align}
&\mathbb{E}\left[\left(S-\mathsf{H}_{\color{black}[k]}\right)_+\right]\le \left(1-\mathsf{H}_{\color{black}[k]}\right) \bar{\alpha}_k, ~\forall\, k\in {\color{black}\left[\nt-1\right]}.\label{eq:equiv_ecmc2}
\end{align}

Theorem <ref> implies that an input $\mathbf{X}$ is feasible to the EC-OIC if and only if the corresponding equivalent input $S=\trans{\mathbf{h}}\mathbf{X}$ satisfies $\mathsf{supp}\, S\subseteq [0,1]$, $\mathbb{E}[S]=\trans{\mathbf{h}}\bm{\alpha}$ and $\pi_{\scriptscriptstyle \! S}(\mathsf{H}_{\color{black}[k]})\le \left(1-\mathsf{H}_{\color{black}[k]}\right)\bar{\alpha}_k$ for $k\in {\color{black}\left[\nt-1\right]}$. Corollary <ref> thus immediately follows.

For the BC-OIC, we exhibit a similar expression of $\mathsf{C}_{\textnormal{B}}\left(\mathbf{h},\bm{\alpha},\sigma\right)$ in the following.

The  capacity $\mathsf{C}_{\textnormal{B}}\left(\mathbf{h},\bm{\alpha},\sigma\right)$ can be equivalently written as

𝖢_B(𝐡,α,σ)=max_𝒫_  S  (S;S+Z),

where the supremum is over all probability laws ${\color{black}\mathscr{P}_{\!  S}}$ on $S$ with $\mathsf{supp}\, S\subseteq [0,1]$ and under stop-loss mean inequality constraints:

𝔼[(S-𝖧_[k])_+]≤(1-𝖧_[k]) α̅_k, k∈{0}∪[-1].

Accroding to Theorem <ref>, an input $\mathbf{X}$ is feasible to the BC-OIC if and only if the corresponding equivalent input $S=\trans{\mathbf{h}}\mathbf{X}$ satisfies $\mathsf{supp}\, S\subseteq [0,1]$, $\mathbb{E}[S]\le\trans{\mathbf{h}}\bm{\alpha}$ and $\pi_{\scriptscriptstyle \! S}(\mathsf{H}_{\color{black}[k]})\le \left(1-\mathsf{H}_{\color{black}[k]}\right)\bar{\alpha}_k$ for $k\in \left[\nt-1\right]$, form which Corollary <ref> follows.

Corollaries <ref> and <ref> enlighten the fact that either a MISO EC-OIC or a MISO BC-OIC is equivalent to a SISO channel under an amplitude constraint and $\nt$ stop-loss mean constraints, which significantly simplifies subsequent capacity analysis.

§ CAPACITY BOUNDS AND ASYMPTOTIC CHARACTERISTICS

In this section, we present capacity results for the EC-OIC and the BC-OIC based on the derived capacity expressions.

§.§ Lower Bounds

The following capacity lower bounds are obtained by applying the EPI, combined with maximizing the differential entropy over all feasible inputs. We first present a lower bound for the EC-OIC.

The capacity of the EC-OIC is lower-bounded as

𝖢_E(𝐡,α,σ)≥1/2log(1+exp(2 	γ_ E)/2πe σ^2),

γ_ E =    min_ν_0,λ_0 ∈ℝ λ_1,⋯,λ_-   1≥0 ∑_i=0^-1λ_i(1-𝖧_[i])α̅_i-1-ν_0

+exp(ν_0){∑_k=1^h_k exp[-λ_0𝖧_[k-1]+∑_i=1^k-2λ_i(𝖧_[i]-𝖧_[k-1]) ]ζ[ (-∑_i=0^k-1λ_i )h_k]  }

with $\zeta \left(x\right)=\frac{e^x-1}{x}$.[By letting derivatives $\zeta^{(j)} \left(0\right)=\frac{1}{j+1}$ for $j\in \mathbb{N}$, $\zeta \left(x\right)$ is continuously differentiable and strictly convex over the real line.]

See Appendix <ref>.

Similarly, we have the following lower bound for the BC-OIC.

The capacity of the BC-OIC is lower-bounded as

𝖢_B(𝐡,α,σ)≥1/2log(1+exp(2 γ_B)/2πe σ^2),

γ_B ≜   min_ν_0 ∈ℝ λ_0,⋯,λ_-1≥0 ∑_i=0^-1λ_i(1-𝖧_[i])α̅_i-1-ν_0

+exp(ν_0){∑_k=1^h_k exp[-λ_0𝖧_[k-1]+∑_i=1^k-2λ_i(𝖧_[i]-𝖧_[k-1]) ]ζ[ (-∑_i=0^k-1λ_i )h_k]  }.

See Appendix <ref>.

The above two lower bounds are derived by choosing an equivalent input distribution so as to maximize the differential entropy $\hh (S)$. Since either the EC-OIC or the BC-OIC can be transformed into a SISO channel with stop-loss mean constraints, we can use convex programming to obtain the input distribution with maximized differential entropy. The maximum-entropy distribution of the equivalent input $S$ for the EC-OIC (or BC-OIC) is given by

p_ S^*(s) = exp(ν_0^*-λ_0^* s-( ∑_i=1^-1 λ_i^* (s-𝖧_[i] )_+ )),  s∈[0,1],

where $\nu_0^*$, $\lambda_0^*$, and $\lambda_1^*,\cdots,\lambda_{\nt -1}^*$ is the optimal solution to (<ref>) (or (<ref>)).

§.§ Upper Bounds

In this subsection, we present several upper bounds on capacities of the EC-OIC and the BC-OIC.

§.§.§ Upper Bounds by SISO and MISO Capacity Expression

Capacities of both the EC-OIC and the BC-OIC can be upper-bounded by utilizing SISO or MISO capacity expression in the existing literature.

The capacity of the EC-OIC is upper-bounded as

𝖢_E(𝐡,α,σ)≤𝖢_E(1,min{𝐡α,1-𝐡α },σ),

while the capacity of the BC-OIC is upper-bounded as

𝖢_B(𝐡,α,σ)≤𝖢_B(1,min{ 𝐡α,1/2},σ).

We sketch the proof as follows. The upper bound (<ref>) for the EC-OIC is derived by first relaxing the intensity constraint (<ref>) on the equivalent input to (<ref>) (i.e., ignoring the stop-loss mean inequality constraints (<ref>)) and then using the symmetry of the EC-OIC capacity (shown in (<ref>)). Along similar lines and using (<ref>), (<ref>) can be easily obtained for the BC-OIC.

In <cit.>, the authors have studied the capacity of the peak-limited MISO OIC with an inequality constraint on the total average intensity. We denote the capacity in this scenario as $\mathsf{C}_{\textnormal{B-TA}}(\mathbf{h},\alpha,\sigma^2)$, where $\alpha$ denotes the ratio between the maximum allowed total average intensity and
the maximum allowed peak intensity, and here it can serve as a upper bound as well.

The capacity of the EC-OIC is upper-bounded as

𝖢_E(𝐡,α,σ)≤𝖢_B-TA(h,min{ α_1,-α_1,/2},σ),

while the capacity of the BC-OIC is upper-bounded as

𝖢_B(𝐡,α,σ)≤𝖢_B-TA(h,min{ α_1,/2},σ).

The upper bound

For the BC-OIC, we let the intensity constraint (<ref>) be relaxed to (<ref>) and note that $\mathsf{C}_{\textnormal{B-TA}}(\mathbf{h},\alpha,\sigma^2)$ is maximized at $\alpha=\nt/2$. Then (<ref>) immediately follows. (<ref>) is derived by $\mathsf{C}_{\textnormal{E}}\left(\mathbf{h},\bm{\alpha},\sigma\right) \le \mathsf{C}_{\textnormal{B}}\left(\mathbf{h},\bm{\alpha},\sigma\right)$, (<ref>), and (<ref>).

§.§.§ Upper Bounds by Maximum-Variance Argument

The following upper bounds are obtained by maximizing the variance of the equivalent input. In <cit.> the authors proved that the maximally correlated $\nt$-variate binary distribution has the maximum variance for the EC-OIC, which immediately leads to Theorem <ref>. Here we present an alternative proof from the perspective of stochastic ordering, which can also be directly extended to the BC-OIC.

The capacity of the EC-OIC is upper-bounded as

where $\const{V}_{\textnormal{max}}^{\textnormal{E}}\left(\mathbf{h},\bm{\alpha}\right)$ denotes the maximum variance of $S$, and

V_max^E(𝐡,α)=∑_i=1^∑_j=1^ h_ih_j (min{α_i,α_j} -α_iα_j).

See Appendix <ref> for an alternative proof.

The capacity of the BC-OIC is upper-bounded as

V_max^B(𝐡,α)   =𝖧^2_[k_β^*]β^* (1-β^*)+2𝖧_[k_β^*](1-𝖧_[k_β^*])α̅_k_β^*(1-β^*)

+∑_i=k_β^*+1^∑_j=k_β^*+1^ h_ih_j (min{α_i,α_j} -α_i α_j),

with $\beta^*=\inf \left\{ \beta\in\left(\alpha_{\nt},\alpha_1\right]:\mathsf{H}_{\color{black}[{k_{\beta}}]} \left(1-2\beta\right)-2\left(1-\mathsf{H}_{\color{black}[{k_{\beta}}]}\right)\bar{\alpha}_{k_{\beta}} \le 0 \right\}$ and $k_\beta=\{k\in \left[\nt\right]:\alpha_k\ge \beta\}$.

See Appendix <ref>.

§.§.§ Upper Bounds by Duality Capacity Expression

We present the following upper bounds by using the duality upper-bounding technique. These bounds are later proved to be tight at high SNR in Sec. <ref>.

The capacity of the EC-OIC is upper-bounded as

≤log(1+  𝖯/√(2πe) σ) +  ∑_i=0^-1λ_i (1-𝖧_[i])α̅_i
+ ∑_i=0^-1λ_iσ/√(2π)(1-e^-(1+δ)^2/2σ^2)

for real numbers $\lambda_0,\cdots,\lambda_{\nt-1}\ge 0$

≤    log(1+ 𝖯/√(2πe) σ) +  ∑_i=0^-1λ_i (1-𝖧_[i])α̅_i
+ ∑_i=1^-1λ_iσ/√(2π)(1-e^-(1+δ)^2/2σ^2)

+λ_0 σ( ϕ(1/σ)-ϕ(δ/σ))
-λ_0  (𝒬(1/σ)+𝒬(δ/σ))

for real numbers $\lambda_0\le 0$ and $\lambda_1,\cdots,\lambda_{\nt-1}\ge0$,

𝖯=	∫_0^1+δexp( -λ_0 y-∑_i=1^-1 λ_i (y-𝖧_[i] )_+ ) y,

$\mathcal{Q}(\cdot)$ denotes Gaussian Q-function, and $\delta>0$ is a free parameter.

See Appendix <ref>.

Similarly, we get the following upper bound for the BC-OIC.

The capacity of the BC-OIC is upper-bounded as

≤log(1+  𝖯/√(2πe) σ)
+   ∑_i=0^-1λ_i (1-𝖧_[i])α̅_i  	+   ∑_i=0^-1λ_iσ/√(2π)(1-e^-(1+δ)^2/2σ^2)

for real numbers $\lambda_0,\cdots,\lambda_{\nt-1}\ge 0$, where $\mathsf{P}$ is defined in (<ref>)
and $\delta>0$ is a free parameter.

See Appendix <ref>.

§.§ Asymptotic Capacity Expressions

In this subsection we present low- and high-SNR asymptotic results.

§.§.§ Low-SNR Capacity Slope

Based on <cit.>, the capacities of interest can be lower-bounded as
𝖢_E (or B)( 𝐡,α,σ) ≥V_max^E (or B)(𝐡,α)/2σ^2
+ o(1/σ^2).

For the reverse direction, upper bounds in terms of maximum variance in Theorems <ref> and <ref> imply that
𝖢_E (or B)( 𝐡,α,σ) ≤V_max^E (or B)(𝐡,α)/2σ^2

Combining (<ref>) and (<ref>), we immediately obtain the following low-SNR capacity characterization.

The low-SNR  capacity slope of the EC-OIC (or BC-OIC) is

lim_σ↑∞ { σ^2𝖢_E (or B)(𝐡,α,σ) } = V_max^E (or B)(𝐡,α)/2,

where $\const{V}_{\textnormal{max}}^{\textnormal{E}}\left(\mathbf{h},\bm{\alpha}\right)$ and $\const{V}_{\textnormal{max}}^{\textnormal{B}}\left(\mathbf{h},\bm{\alpha}\right)$ are defined in (<ref>) and (<ref>), respectively.

We would like to remind the reader that maximized variances  ${\const{V}_{\textnormal{max}}^{\textnormal{E}}\left(\mathbf{h},\bm{\alpha}\right)}$ and ${\const{V}_{\textnormal{max}}^{\textnormal{B}}\left(\mathbf{h},\bm{\alpha}\right)}$ for the EC-OIC and the BC-OIC can be achieved by maximally convex distributions $\bar{S}_{\mathbf{h},\bm{\alpha}}$ and $\bar{S}_{\mathbf{h},\mathbf{a}^\dagger}$ , respectively, where the vector $\mathbf{a}^\dagger$ is defined in Theorem <ref>.

§.§.§ High-SNR Asymptotic Capacity

By showing that the afore-mentioned EPI-based lower bounds match with duality upper bounds at high SNRs, we obtain the following result.

The high-SNR asymptotic capacity of the EC-OIC (or BC-OIC) is

lim_σ↓0^+	𝖢_E (or B)(𝐡,α,σ)-log1/σ = -1/2log2πe+γ_E (or B),

where $\gamma_{\textnormal{E}}$ and $\gamma_{\textnormal{B}}$ are defined in (<ref>) and (<ref>), respectively.

See Appendix <ref>.

§.§ Numerical Results

In this subsection, we present numerical evaluations for our derived bounds, where involved duality bounds (<ref>), (<ref>) and (<ref>) are numerically minimized over the allowed values of $\lambda_0,\cdots,\lambda_{\nt-1}$ and $\delta$.

Figure <ref> depicts the derived lower and
upper bounds for a $3 \times 1$ EC-OIC with ${\mathbf{h}}=\trans{\left(0.3,0.1,0.6\right)}$ and $\bm{\alpha}=\trans{\left(0.8,0.3,0.1\right)}$, while Figure <ref> depicts the lower and
upper bounds for a $3 \times 1$ BC-OIC with $\mathbf{h}=\trans{\left(0.3,0.2,0.5\right)}$ and $\bm{\alpha}=\trans{\left(0.5,0.3,0.2\right)}$.
In both figures, we present capacity upper bounds obtained by combining Propositions <ref> and <ref> and existing upper bounds for the SISO OIC under a peak- and an average-intensity constraint <cit.> and the MISO OIC with a total average-intensity and per-antenna peak-intensity constraints <cit.>.
Compared with existing capacity bounds, the duality-based upper bound gives better approximation on capacity at high SNR, and
matches with the EPI-based lower bound asymptotically as SNR tends to infinity. A noteworthy observation in Figure <ref> is that the upper bound obtained by the SISO capacity (Proposition <ref>) coincides with our duality upper bound (<ref>) at high SNR. This may be due to the fact that there exists some $\beta\in [\alpha_{\nt},\alpha_1]	$ such that the maximum-entropy distribution of the relaxed SISO channel is $(\vect{h},\min\{\beta \bm{1},\bm{\alpha}\})$-decomposable, i.e., feasible to the considered BC-OIC as well.

To better verify our derived asymptotic results at low and high SNRs, we further investigate $2 \times 1$ EC-OICs and BC-OICs with equal channel gains ${\vect{h}} = \trans{(\frac{1}{2},\frac{1}{2})}$ in Figure 5, whose results exhibit the symmetry property in terms of changing the order of $\bm{\alpha}$. As we see, the asymptotic capacities at both low and high SNRs  decrease as $\bm{\alpha}$ tends to corner, while the results of the BC-OIC are Schur convex.

Capacity bounds of a $3\times1$ EC-OIC with ${\mathbf{h}}=\trans{\left(0.3,0.1,0.6\right)}$ and ${\bm{\alpha}}=\trans{\left(0.8,0.3,0.1\right)}$. The maximum gap between upper and lower bounds is about $0.35$ nats.

Capacity bounds of a $3\times 1$ BC-OIC with ${\mathbf{h}}=\trans{\left(0.3,0.2,0.5\right)}$ and ${\bm{\alpha}}=\trans{\left(0.5,0.3,0.2\right)}$. The maximum gap between upper and lower bounds is about $0.47$ nats.

Asymptotic capacity results of $2\times 1$ EC-OICs (above) and BC-OICs (below) with equal channel gains $h_1=h_2=0.5$.

§ DERIVATION OF DECOMPOSITION RESULTS

§.§ Greedily-Constructed Quantile

A crucial part in the proof of Theorem <ref> is a class of parametric functions, named greedily-constructed quantiles, which will be introduced in a heuristic manner as follows.

We begin with a short literature review. As shown in <cit.>, the capacity of a MISO OIC under a peak-intensity and a total average-intensity constraint can be achieved by a minimum-energy signaling strategy, which prioritizes the input with a larger channel coefficient and enables it to grow first due to the nature of linear programming. For the EC-OIC, a maximally correlated multivariate binary input (see <cit.> ) has been shown to maximize the variance of the equivalent input in <cit.>, which lets the transmitter with the maximum average intensity grow prior to the others. We remark that both the minimum-energy signaling and the maximally correlated multivariate binary input can be regarded as special classes of comonotonic distributions. For the above reasons, it is natural to investigate the behaviors of comonotonic inputs in MISO OICs of our interest.

Quantile decomposition of a random variable $S$ into comonotonic inputs.

Given a comonotonic input $\mathbf{X}$ with mean $\bm{\alpha}$, the equivalent input $S=\trans{\mathbf{h}}\mathbf{X}$ must be comonotonically $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable. According to Definition <ref>, the quantile $Q_{\!  S}(p)$ of $S$ is a mixture of quantiles as $Q_{\!  S}(p)=\sum_{k=1}^{\nt} h_k Q_{\! X_{\! k}}(p)$. Hence, a random variable $S$ with the quantile $Q_{\! S}(p)$ is comonotonically $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable if and only if there exist $\nt$ quantiles $Q_{\! X_{\! k}}(p)$ ($k\in {\color{black}\left[\nt\right]}$) satisfying the following three conditions:

* ${\color{black}Q_{\! X_{\! k}}}\!:\!(0,1]\!\to\![0,1]$ is non-decreasing and left-continuous;

* $\int_{0}^1 {\color{black}Q_{\! X_{\! k}}}(p) \dd p=\alpha_k$;

* $Q_{\! S} (p )=\sum_{k=1}^{\nt} h_k {\color{black}Q_{\! X_{\! k}}}(p)$.

Thus, the core problem in comonotonic decomposition of a random variable is how to decompose its quantile function $Q_{\! S}(p)$. In Figure <ref>, we present a heuristic geometrical interpretation of the strategy for constructing required quantiles. Suppose that $\sum_{k=1}^{m} h_k {\color{black}Q_{\! X_{\! k}}}(p)$ is known and is the quantile of $R_m=\sum_{k=1}^{m}h_k {\color{black}Q_{\! X_{\! k}}(U)}$. The curve of quantile sum $\sum_{k=1}^{m-1} h_k{\color{black}Q_{\! X_{\! k}}}(p)$ must lie in the green region due to boundedness of quantiles, i.e., larger than zero, less than $\sum_{k=1}^{m} h_k {\color{black}Q_{\! X_{\! k}}}(p)$ and $\sum_{k=1}^{m-1} h_k$, and within a curved strip of width $h_{m}$ below $\sum_{k=1}^{m} h_k {\color{black}Q_{\! X_{\! k}}}(p)$. In addition, to meet the integral requirement of ${\color{black}Q_{\! X_{\! m}}}(p)$, the sum quantile $\sum_{k=1}^{m} h_k{\color{black}Q_{\! X_{\! k}}}(p)$ must satisfy $\textnormal{Area}~ \mathcal{A}_1 \le h_{m}\alpha_{m}$ and $\textnormal{Area}~ \mathcal{A}_1+ \textnormal{Area}~\mathcal{A}_2 \ge h_{m}\alpha_{m}$. Note that the magenta region $\mathcal{A}_1=\mathcal{R}^{  \scriptscriptstyle R_m}_{\mathsf{H}_{\color{black}[m-1]}}$ (the right-hand side is defined in Proposition <ref>) and the yellow region $\mathcal{A}_3$ is a vertical translate of $\mathcal{R}^{  \scriptscriptstyle R_m}_{h_{m}}$. Then we have

Area  𝒜_1+ Area 𝒜_2

=∫_0^1 (∑_k=1^m h_kQ_ X_ k(p)) p -Area 𝒜_3

= ∑_k=1^m h_kα_k  -Area 𝒜_3

= ∑_k=1^m h_kα_k  -  Area  ℛ^  R_m_h_m.

Thus, to guarantee the existence of $Q_{m}(p)$ with a required definite integration value, the most greedy approach in previously constructing $\sum_{k=1}^{m}h_k{\color{black}Q_{\! X_{\! k}}}(p)$ is to let the areas of $\mathcal{R}^{  \scriptscriptstyle R_m}_{h_{m}}$ and $\mathcal{R}^{  \scriptscriptstyle R_m}_{\mathsf{H}_{\color{black}[m-1]}}$ as small as possible, i.e., the area of $\sum_{k=1}^{m}h_k{\color{black}Q_{\! X_{\! k}}}(p)$ should be preempted at the bottom. From above considerations, in Proposition <ref> we construct and analyze the “most greedy” class of quantile functions, i.e., so-called greedily-constructed quantiles, which coincide with $Q_{\! S} (p )$ before a threshold and otherwise coincide with a downward translate of $	Q_{\! S} (p )$.

Let $S$ be a random variable on $[0,1]$ and define the greedily-constructed quantile as the following nonnegative parametric function

ψ_S ( p;v,z ) ≜

Q_ S (p ), ;

v,  ;

Q_ S (p  )-z, ,

where the variable $p\in(0,1]$, and parameters  $v \in [0,1]$ and $z\in\left[0,1-v\right]$. Then

* $\psi_{\scriptscriptstyle \!S} ( p;v,z)$ is the quantile function of $R=\varphi \left( S;v,z\right)$, where
φ( s;v,z) =

s, ,

v, ,

s-z, .

* The SLT of $R$ is

π_ R (t) =

π_ S(t)-π_ S(v)+π_ S(v+z), ,

π_ S(t+z), .

We first rewrite

R   =φ( S;v,z)

=φ( Q_ S(U);v,z)

Q_ S(U),

Q_ S(U)-z,

Q_ S(U),

Q_ S(U)-z,

=ψ_S ( U;v,z),

where (<ref>) follows from the Galois inequality (<ref>).

Since $Q_{\! S} (p )$ is the quantile of $S$, the function $\psi_{\scriptscriptstyle \!S} ( p;v,z )$ is left-continuous on the interval $p\in(0,1]$ and piecewise non-decreasing on the intervals $\left(0,F_{\! S}(v )\right]$, $ \left(F_{\! S}(v ) ,F_{\! S}(v+z)\right]$ and $\left(F_{\! S}(v+z),1\right]$, respectively. Recall that $Q_{\! S}\left(F_{\! S}(v)\right)=\inf \left\{
s: F_{\! S}(s)\ge F_{\! S}(v) \right\}
$ and hence $Q_{\! S}\left(F_{\! S}(v)\right)\le v$.

Next, based on the contrapositive of Proposition <ref>, we have $Q_{\! S}(p)>v+z$ for any real number $p\in \left(F_{\! S}(v+z),1\right]$. Hence, the parametric function $\psi_{\scriptscriptstyle \!S} ( p;v,z )$ is non-decreasing and left-continuous on the interval $p\in(0,1]$. Proposition <ref> shows that $\psi_{\scriptscriptstyle \!S} ( U;v,z)$ is the quantile function of $R$.

It is clear that $0 \le R \le S$, and hence $\mathsf{supp}\,R \subseteq [0,1]$. We notice that if $t>v$, $\left(\varphi \left( s;v,z\right)-t\right)_+=\left(s-t-z\right)_+$. Then

π_ R (t)   =𝔼[ (R-t)_+]

=𝔼[ (φ( S;v,z)-t)_+]

=𝔼[ (S-t-z)_+]

=π_ S(t+z), t>v.

The other case is concluded similarly by noticing that the following equality

(φ( s;v,z)-t)_+

=(s-t)_+-((s-v)_++v-t) +((s-v-z)_++v-t)

holds for $t\le v$.

With a slight abuse of terminology, we also refer to the expression (<ref>) as the greedily-constructed quantile. Furthermore, in the remainder of this paper, we work with the expression (<ref>) rather than (<ref>), which frees us from the redundant discussion about quantiles.
We will see later that, components of a decomposable random variable can be constructed via the greedily-constructed quantiles.

§.§ Proof of Thereom <ref>

We first prove a necessary condition for the $\left(\mathbf{h},\bm{\alpha}\right)$-decomposability as follows.

An $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable random variable $S$ must satisfy:

* $\mathsf{supp}\, S\subseteq [0,1]$;

* Its SLT $\pi_{\scriptscriptstyle \! S}(t)$ satisfies

π_ S (𝖧_𝒥)≤∑_k∈𝒥^c h_kα_k

for any index set $\mathcal{J}\subseteq {\color{black}\left[\nt\right]}$.

From the definition of $\left(\mathbf{h},\bm{\alpha}\right)$-decomposability, we can rewrite $S$ as
$S= \trans{\mathbf{h}}\mathbf{W}$, where the random vector $\mathbf{W}=\trans{\left( W_1,\cdots,W_{\nt}\right)}$ with  $\mathsf{supp}\, \mathbf{W} \subseteq [0,1]^{\nt}$ and $\mathbb{E}\left[\mathbf{W}\right]=\bm{\alpha}$. Due to nonnegativity of $\mathbf{h}$ and $\trans{\bm{1}}\mathbf{h}=1$, we immediately know $\mathsf{supp}\, S\subseteq [0,1]$. Then note that the function with respect to $s\in[0,1]$ satisfies the following inequality for any index set $\mathcal{J}\subseteq  {\color{black}\left[\nt\right]}$

𝔼[∑_k∈𝒥^ch_kW_k|S=s]≥(s-𝖧_𝒥 )_+,

due to $\mathsf{supp}\, \mathbf{W} \subseteq [0,1]^{\nt}$. By using the law of total expectation, we get

π_ S (𝖧_𝒥)   = 𝔼[(S-𝖧_𝒥 )_+]

≤𝔼_S[𝔼[∑_k∈𝒥^c h_kW_k|S]]

= 𝔼[∑_k∈𝒥^ch_kW_k]

=∑_k∈𝒥^c h_kα_k.

This is the complete proof of Lemma <ref>.

Then we show the equivalence among the four statements of Theorem <ref> by proving $1\ \Longrightarrow \ 2$ ,   $2 \ \Longrightarrow \ 3$, $3 \ \Longrightarrow \ 4$, and $4 \ \Longrightarrow \ 1$ all hold.

The definition of comonotonic $(\mathbf{h},\bm{\alpha})$-decomposability immediately leads to $4 \ \Longrightarrow \ 1$.

By Lemma <ref>, we can directly get 1 $\Longrightarrow$ 2.

Note that $\bar{\pi}_{\mathbf{h},\bm{\alpha}}(0)=\pi_{\scriptscriptstyle \! S}\left( 0 \right)$, $\bar{\pi}_{\mathbf{h},\bm{\alpha}}(1)=\pi_{\scriptscriptstyle \! S}\left( 1 \right)=0$, and

\begin{equation}
\bar{\pi}_{\mathbf{h},\bm{\alpha}}(\mathsf{H}_{\color{black}[k]})=\left(1-\mathsf{H}_{\color{black}[k]}\right)\bar{\alpha}_k\ge \pi_{\scriptscriptstyle \! S}\left(\mathsf{H}_{\color{black}[k]}\right).
\end{equation}

Furthermore, due to the convexity of the SLT and the piecewise linearity of $\bar{\pi}_{\mathbf{h},\bm{\alpha}}(t)$, we have

\begin{equation}
\pi_{\scriptscriptstyle \! S}\left(t\right)\le\bar{\pi}_{\mathbf{h},\bm{\alpha}}\left(t\right), \ \forall t\in[0,1].
\end{equation}

Hence $2 \ \Longrightarrow \ 3$.

A bit involved part is to show  $3 \ \Longrightarrow \ 4$. Let $R_{\nt}^*=S$, $v_1^*=0$, and the $\nt$-dimensional real vector $\hat{\bm{\alpha}}^{(i)}$ be determined by $\hat{\alpha}_{k}^{(i)}=\alpha_k$ for $1 \le k \le i$ and others zero. We now use an iterative decomposition method to prove this.  It is sufficient to show

for each $i\in\{\nt,\cdots,2\}$, if $R_{i}^*$ satisfies  $\mathsf{supp}\, R_{i}^* \subseteq [0,\mathsf{H}_{\color{black}[i]}] $ and $R_{i}^* \le_{\textnormal{cx}} \bar{S}_{\mathbf{h},\hat{\bm{\alpha}}^{(i)}}$, then

* there exists a number $v_i^*\in\left[0,\mathsf{H}_{\color{black}[i-1]}\right]$ such that $R_{i-1}^*=\varphi\left(R_{i}^*;v_i^*,h_{i}\right)$ has the expectation $\mathbb{E}\left[R_{i-1}^*\right]=\sum_{m=1}^{i-1}h_m\alpha_m$;

* so-constructed  $R_{i-1}^*$ satisfies $\mathsf{supp}\, R_{i-1}^* \subseteq [0,\mathsf{H}_{\color{black}[i-1]}] $ and $R_{i-1}^* \le_{\textnormal{cx}} \bar{S}_{\mathbf{h},\hat{\bm{\alpha}}^{(i-1)}}$;

Now we prove the above two statements. Denote the SLT of $R_{i}^*$ by $\pi_{\scriptscriptstyle\! {\color{black}R_{i}^*}}(t)$. Using Proposition  <ref>, we obtain

𝔼[φ(R_i^*,v,h_i)]=π_ R_i^*(0)-π_ R_i^*(v)+π_ R_i^*(v+h_i)

which is continuous and nondecreasing on the closed interval $v\in\left[0,\mathsf{H}_{\color{black}[i-1]}\right]$.

Noting that $R_{i}^* \le_{\textnormal{cx}} \bar{S}_{\mathbf{h},\hat{\bm{\alpha}}^{(i)}}$, we have

π_ R_i^* (𝖧_𝒥)≤π̅_𝐡,α̂^(i) (𝖧_𝒥) ≤∑_k∈𝒥^ch_k α̂_k^(i)

for any index set $\mathcal{J}\subseteq {\color{black}\left[\nt\right]}$,
where the second inequality follows from the $(\mathbf{h},\hat{\bm{\alpha}}^{(i)})$-decomposability of $\bar{S}_{\mathbf{h},\hat{\bm{\alpha}}^{(i)}}$.
Substituting $v=0$ into (<ref>), we have

=π_ R_i^*(h_i)

=∑_m=1^i-1h_m α_m ,

where (<ref>) follows from the inequality (<ref>) by letting $\mathcal{J}=\{i\}$ and (<ref>) follows from the definition of $\hat{\bm{\alpha}}^{(i)}$. In the same way, we have

=π_ R_i^*(0)-π_ R_i^*(𝖧_[i-1])+π_ R_i^*(𝖧_[i])

=π_ R_i^*(0)-π_ R_i^*(𝖧_[i-1])

≥∑_m=1^ih_mα_m-h_i α_i

= ∑_m=1^i-1h_mα_m,

where (<ref>) follows from $\pi_{\scriptscriptstyle\! R_{i}^*}(\mathsf{H}_{\color{black}[i]})=0$ implied by $\mathsf{supp}\, R_{i}^* \subseteq [0,\mathsf{H}_{\color{black}[i]}]$. Hence, by the continuity of $\mathbb{E}\left[\varphi\left(R_{i}^*,v,h_{i}\right)\right]$ on the closed interval $\left[0,\mathsf{H}_{\color{black}[i-1]}\right]$, there must exist a solution $v_i^*\in\left[0,\mathsf{H}_{\color{black}[i-1]}\right]$ (possibly non-unique) to the following equation

which is equivalent to

π_ R_i^*(v)-π_ R_i^*(v+h_i)=h_iα_i.

Then we let $R_{i-1}^*=\varphi\left(R_{i}^*;v_i^*,h_{i}\right)$ and the SLT of  $R_{i-1}^*$ is given by Proposition <ref> as follows

π_ R_i-1^*(t)=

π_ R_i^*(t)-h_iα_i , ,

π_ R_i^*(t+h_i), ,

where the first equality follows from $ \pi_{\scriptscriptstyle\! R_{i}^*}(v_i^*)-\pi_{\scriptscriptstyle\! R_{i}^*}(v_i^*+h_{i})=h_{i}\alpha_{i}$.

Clearly, there exists a unique $\tau\in\{ 1,2,\cdots, i-1\}$ such that $\mathsf{H}_{\color{black}[{\tau-1}]}<v_i^{*} \le \mathsf{H}_{\color{black}[{\tau}]}$.
To prove that so-constructed $R_{i-1}^*$ satisfies $\mathsf{supp}\, R_{i-1}^* \subseteq [0,\mathsf{H}_{\color{black}[{i-1}]}] $ and $R_{i-1}^* \le_{\textnormal{cx}} \bar{S}_{\mathbf{h},\hat{\bm{\alpha}}^{(i-1)}}$, we will alternatively provide a case-by-case proof of $\pi_{\scriptscriptstyle\! R_{i-1}^*}\left(\mathsf{H}_{\color{black}[{k}]}\right)\le \bar{\pi}_{\mathbf{h},\hat{\bm{\alpha}}^{(i-1)}} \left(\mathsf{H}_{\color{black}[{k}]}\right)$ for all $k\in {\color{black}\left[\nt\right]}$ as follows.

* Case 1: $k\ge \tau$. Note that  $\mathsf{H}_{\color{black}[{k}]}\ge \mathsf{H}_{\color{black}[{\tau}]}\ge v_i^*$. If $k\ge i-1$, then by plugging $t=\mathsf{H}_{\color{black}[{k}]}$ into (<ref>) and utilizing the monotonicity of SLT we get

π_ R_i-1^*(𝖧_[k])   =π_ R_i^*(𝖧_[k]+h_i)

≤π_ R_i^*(𝖧_[i])

Due to the nonnegativity of SLT, we have $\pi_{\scriptscriptstyle\! R_{i-1}^*}\left(\mathsf{H}_{\color{black}[{k}]}\right)=0$ if $k\ge i-1$.
Otherwise, we have

π_ R_i-1^*(𝖧_[k])   =π_ R_i^*(𝖧_[k]+h_i)

where (<ref>) follows from (<ref>) by letting $\mathcal{J}=\left\{1,2,\cdots,k,i \right\}$.

* Case 2: $k\le \tau-1$. We have $\mathsf{H}_{\color{black}[{k}]}< v_i^*$ and hence

π_ R_i-1^*(𝖧_[k])   =π_ R_i^*(𝖧_[k])-h_iα_i

where (<ref>) follows from the fact that $k+1\le \tau \le i-1$.

In summary, we have derived that

π_ R_i-1^*(0)= π̅_𝐡,α̂^(i-1)(0)=∑_m=1^i-1h_iα_i,

π_ R_i-1^*(1)= π̅_𝐡,α̂^(i-1)(1)=0,

π_ R_i-1^*(𝖧_[k])≤π̅_𝐡,α̂^(i-1)(𝖧_[k]), ∀ k∈[].

Then, in view of the convexity of SLT and piecewise linearity of $\bar{\pi}_{\mathbf{h},\hat{\bm{\alpha}}^{(i-1)}}\left(t\right)$, we have

π_ R_i-1^*(t)≤π̅_𝐡,α̂^(i-1)(t), t∈[0,1],

and thus $R_{i-1}^* \le_{\textnormal{cx}} \bar{S}_{\mathbf{h},\hat{\bm{\alpha}}^{(i-1)}}$. Considering that $\pi_{\scriptscriptstyle\! R_{i-1}^*}(\mathsf{H}_{\color{black}[{i-1}]})\le \pi_{\mathbf{h},\hat{\bm{\alpha}}^{(i-1)}}\left(\mathsf{H}_{\color{black}[{i-1}]}\right)=0$ and $R_{i-1}^*\ge 0$, we have $\mathsf{supp}\,R_{i-1}^*\subseteq [0,\mathsf{H}_{\color{black}[{i-1}]}]$.

Conduct the above iterative construction of $R_{i-1}^*$ from $i=\nt$ to $i=2$ and  let

X_k^c   ≜R_k^*-R_k-1^*/h_k

for $k\in {\color{black}\left[\nt\right]}$.

Considering the definition of $\varphi \left( x;v,z\right)$ and the fact $\mathsf{supp}\,R_{i}^*\subseteq [0,\mathsf{H}_{\color{black}[{i}]}]$, we have $\mathsf{supp}\, X_k^{\rm c} \subseteq [0,1]$ and $\mathbb{E}\left[X_k^{\rm c}\right]=\alpha_k$ for all $k\in {\color{black}\left[\nt\right]}$. Furthermore, we note that: 1) (<ref>) implies the comonotonicity between $X_k^{\rm c}$ and $R_{k}^*$; 2) $R_{k-1}^*=\varphi\left(R_{k}^*;v_k^*,h_{k}\right)$ implies the comonotonicity between $R_{k}^*$ and $R_{k-1}^*$. Thus, we conclude that $\mathbf{X}^{\rm c}=\trans{\left(X_1^{\rm c},\cdots,X_{\nt}^{\rm c}\right)}$ is comonotonic and $S=\sum_{k=1}^{\nt} h_kX_k^{\rm c}$ is comonotonically $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable.

§.§ Proof of Thereom <ref>

We first show
$1\ \Longrightarrow \ 2$. Let $\mathbf{a}$ be an arbitrary vector satisfying $\bm{0} \preccurlyeq \mathbf{a} \preccurlyeq  \bm{\alpha}$ and $S$ can be decomposed as $S=\sum_{k=1}^{\nt}h_kX_k$ with $\mathsf{supp}\, X_k \subseteq [0,1]$ and $\mathbb{E}[X_k]=a_i$ for each $k\in [\nt]$. Then we have

𝔼[(S-𝖧_[k])_+]   =
π_ S (𝖧_[k])

≤∑_j=k+1^ h_j𝔼[X_j]

≤∑_j=k+1^ h_jα_j

=(1-𝖧_[k]) α̅_k,  k∈{0} ∪[-1],

where (<ref>) follows from Theorem <ref>.

Next, we show $2\ \Longrightarrow \ 3$. For any $S$ with $\mathsf{supp}\, S\subseteq [0,1]$ and satisfying constraints (<ref>), we let $\mathbb{E}\left[S\right]{\color{black}=\pi_{\scriptscriptstyle \! S}(0)}=\varepsilon$, $\varepsilon_0=0$ and
$${\color{black}\varepsilon_j=\left( \mathsf{H}_{\color{black}[{\nt-j+1}]} \right) \times \left( \alpha_{\nt-j+1} \right)
+\left(1-\mathsf{H}_{\color{black}[{\nt-j+1}]}\right) \times
\left( \bar{\alpha}_{\nt-j+1}\right)
,~\forall \,} j\in {\color{black}\left[\nt\right]}.$$

Then there must exist a unique integer $ \tau \in {\color{black}\left[\nt\right]}$ such that $\varepsilon \in \left(\varepsilon_{\tau-1},\varepsilon_{\tau}\right]$.

𝐚^ε= (a_1^ε,⋯,a_^ε)=ε1, if τ=1;

and otherwise

α_m    ,

(ε- ∑_m=-τ+2^h_ma^ε_m)/𝖧_[-τ+1]     .

Note that the so-constructed $\mathbf{a}^{\varepsilon}$ satisfies $a_1^{\varepsilon} \ge a_2^{\varepsilon} \ge \cdots a^{\varepsilon}_{\nt}$ and $\bm{0} \preccurlyeq \mathbf{a}^{\varepsilon} \preccurlyeq  \bm{\alpha}$. For $k\ge \nt-\tau+1$, the inequalities (<ref>) imply

π_ S(𝖧_[k])    ≤∑_j=k+1^ h_jα_j= ∑_j=k+1^ h_j a_j^ε,

while for $k\le \nt-\tau$ we have

π_ S(𝖧_[k])     ≤(1-𝖧_[k]/𝖧_[-τ+1])π_ S(0)
+𝖧_[k]/𝖧_[-τ+1]π_ S(𝖧_[-τ+1])

≤(1-𝖧_[k]/𝖧_[-τ+1])π_ S(0)

where (<ref>) follows from the convexity of the SLT. Theorem <ref> immediately leads to the $\left(\mathbf{h},\mathbf{a}^{\varepsilon}\right)$-decomposability of $S$. It is clear that the set $\left\{ \mathbf{a}^\varepsilon: \varepsilon \in \left[ 0 ,  \trans{\mathbf{h}}\bm{\alpha} \right] \right\} $ can be reparameterized as

{ 𝐚^ε: 0 ≤ε≤𝐡α}
={   min{β1,α}:β∈[0,α_1]

Therefore, $S$ must be also $\left(\mathbf{h},\mathbf{a}^{\dagger} \right)$-decomposable, where $	\mathbf{a}^{\dagger} =   \min\left\{\beta \bm{1},\bm{\alpha}\right\} $ with some $\beta \in \left[0,\alpha_1 \right]$ such that  $\trans{\mathbf{h}}\mathbf{a}^{\dagger}=\mathbb{E}[S]$.

The proof is concluded by noticing $\bm{0} \preccurlyeq \mathbf{a}^{\dagger} \preccurlyeq \bm{\alpha}$, which implies $3\ \Longrightarrow  1$.

§ SIGNALING FOR EC-OIC AND BC-OIC

In this section, we discuss the signaling problem for a given equivalent input $S$ feasible to the EC-OIC or the BC-OIC. The optimization of equivalent inputs or channel inputs should be paid particular attention and are not studied here. The readers can refer to <cit.>, where an optimal spatial constellation in term of the received minimum Euclidean distance is proposed for the case with or without CSIT.

We have seen that our capacity results can be derived once we prove the equivalence between a MISO OIC with per-antenna intensity constraints and a SISO channel with an amplitude constraint and several stop-loss mean constraints. However, in practical signaling, it is vital for each transmitter to know its corresponding component obtained by decomposing an $(\mathbf{h},\bm{\alpha})$-decomposable random variable $S$ rather than $S$ itself. Hence, an algorithm to efficiently decompose $S$ into a feasible channel input $\mathbf{X}^{\rm c}$ will be useful.

In the proof of Theorem <ref> (see Sec. <ref>), a constructive method, namely greedy decomposition, iteratively computes $\nt$ comonotonic components of a $(\mathbf{h},\bm{\alpha})$-decomposable random variable, which relies on greedily-constructed quantile functions with $\nt-1$ parameters $v_{\nt}^{*}$, $v_{\nt-1}^{*}$, $\cdots$, and $v_2^*$ to be determined and a known parameter $v_1^*=0$. To assist the reader, we excerpt this iterative algorithm from Sec. <ref> as follows.
[Iteration-based greedy decomposition]

Conditioned on $S=s$ for any given $s\in[0,1]$, the greedy decomposition performs

for $k\in \left[\nt\right] \setminus \{1\}$.

We remind that calculation of parameters $v_{\nt}^{*}$, $v_{\nt-1}^{*}$, $\cdots$, and $v_2^*$ is accomplished by solving equations (<ref>), which needs the SLT $\pi_{\scriptscriptstyle\! R_{i-1}^*}(t)$ (given in (<ref>)) of the remainder part $R_{i-1}^*$ in each step.
In what follows, we present an alternative perspective on the greedy decomposition via partitioning the interval $[0,1)$, and calculation of desired parameters is only based on the SLT of $S$.[By convention, we adopt a left-closed-right-open interval $[0,1)$ for a unified representation of sets obtained by finite union and set difference in this section.]

Let $S$ be an $(\mathbf{h},\bm{\alpha})$-decomposable random variable, $\kappa_{\nt}^*=v_{\nt}^*$, $\eta_{\nt}(\kappa)=\kappa+h_{\nt}$ and the interval $\mathcal{P}_{\nt}=\left[\kappa_{\nt}^*,\eta_{\nt}(\kappa_{\nt}^*)\right)$. For $i\in\{\nt,\cdots,2\}$, $\kappa_{i-1}^*$ is a solution to the equation

π_ S(κ)-π_ S(η_i-1(κ))
=∑_m=i-1^i-1+N_i-1(κ)h_mα_m, κ∈[0,min{κ_i^*,𝖧_[i-1]}],

where $\kappa+h_{i-1} \le \eta_{i-1}(\kappa) \le 1$ is uniquely determined by the Lebesgue measure

μ([κ,η_i-1(κ) ]∖⋃_m=i^𝒫_i  )=h_i-1,

the Borel sets

𝒫_j=[κ_j^*,η_j(κ_j^*) )∖⋃_m=j+1^𝒫_m, j≤-1,

N_i-1(κ)=|{j:κ_j^*≤η_i-1(κ), i-1< j≤}|.

Then, for each $k\in {\color{black}\left[\nt\right]}$, $\kappa_k^*$ is also a solution to (<ref>).

For any given $(\mathbf{h},\bm{\alpha})$-decomposable $S$, we will assign the $k$-th one of the above $\nt$ disjoint sets, i.e., $\mathcal{P}_k$ of length $h_k$, to the $k$-th transmitter. Recall that we let $R_{k-1}^*=\varphi\left(R_{k}^*;\kappa_k^*,h_{k}\right)$ and $X_k^{\rm c}
=\left(R_{k}^*-R_{k-1}^*\right)/{h_{k}}$ in the greedy decomposition. Then the definition of the parametric function $\varphi\left(s;v,z\right)$ immediately implies that when $\kappa_k^* \le R_{k}^* <\kappa_k^*+h_{k} $ the variation of $R_{k}^*$ can be regarded as solely changing $X_k^{\rm c}$ and keeping $R_{k-1}^*$ invariant. Motivated by this fact, via mathematical induction  we can prove that the continuous variation of the random variable $S$ in the set $\mathcal{P}_k$ is solely determined by the $k$-th transmitter for each $k\in {\color{black}\left[\nt\right]}$.[We omit the induction-based proof for brevity since its involved techniques are similar to those in Appendix <ref>.] Hence, we obtain the following decomposition algorithm, which implements the greedy decomposition as well.

[Partition-based greedy decomposition]

Conditioned on $S=s$ for any given $s\in[0,1]$, the greedy decomposition performs $x_{k}^c=0$ if $s\le  \kappa_k^*$, and otherwise

μ{p∈𝒫_k:p≤s }/h_k,  k∈[].

From a geometric perspective, the signal sent by the $k$-th transmitter is exactly a scaled length of the part of $\mathcal{P}_k$ with elements no larger than the realization $s$. Since each set $\mathcal{P}_k$ is a finite (at most $\nt-k+1$) union of disjoint intervals, the computational complexity involved in Algorithm <ref> is low.

Note that the sets $\mathcal{P}_k$ obtained in Proposition <ref> depend on the distribution of the equivalent input $S$. For a MISO OIC with inputs under a total average-intensity constraint and peak-intensity constraints <cit.>, given a feasible equivalent input $S=\trans{\mathbf{h}}\mathbf{X}$ and sorting all transmitters in descending order $h_1\ge \cdots \ge h_{n_t}$, the inputs can be directly obtained by $X_k=(S-\mathsf{H}_{\color{black}[k]})_+/h_k$ for $k\in {\color{black}\left[\nt\right]}$. Those piecewise linear functions can be regarded as a special case of (<ref>) with a simple and deterministic interval partition $\mathcal{P}_k=\left[\mathsf{H}_{\color{black}[k-1]},\mathsf{H}_{\color{black}[k]}\right)$ for $k\in\mathcal{S}$, regardless of the distribution of $S$. A similar phenomenon arises in its MIMO counterpart <cit.>, where the inputs can be computed according to a deterministic partition of the zonotope generated by all column vectors of the channel matrix.[A zonotope (also named “zonohedra”) generated by $n$ vectors $\mathbf{a}_1,~\mathbf{a}_2,\cdots,~\mathbf{a}_{n}\in \mathbb{R}^{m}$ is the Minkowski sum $\left\{\sum_{i=1}^{n}x_i \mathbf{a}_i: 0 \le x_i \le 1 \right\}$ <cit.>.] This observation reflects a fundamental difference between OICs with per-antenna average-intensity constraints and those with a total average-intensity constraint.

Clearly, Algorithms 1 and 2 carry out the greedy decomposition method and compute comonotonic components  of a $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable random variable $S$. The difference between the two algorithms is that the former outputs in an iterative manner while the latter in an independent manner. Both algorithms can be applied directly to the EC-OIC. For the BC-OIC, the signaling problem can be easily addressed after an additional step. Given a random variable $S$ feasible to the BC-OIC, we calculate the auxiliary average-intensity vector  $	\mathbf{a}^{\dagger} =   \min\left\{\beta \bm{1},\bm{\alpha}\right\} $ with some $\beta \in \left[0,\alpha_1 \right]$ such that  $\trans{\mathbf{h}}\mathbf{a}^{\dagger}=\mathbb{E}[S]$. As shown in Theorem <ref>, the inequality $\bm{0} \preccurlyeq \mathbf{a}^{\dagger} \preccurlyeq  \bm{\alpha}$ holds and  $S$ is $\left(\mathbf{h},\mathbf{a}^{\dagger}\right)$-decomposable so that Algorithms 1 and 2 can be applied as well.

In what follows, we present several examples to illustrate signaling procedures for the EC-OIC and the BC-OIC.

[Signaling for the maximally convex distribution] By solving (<ref>) for $\bar{S}_{\mathbf{h},\bm{\alpha}}$, its partition is simply given by
$\mathcal{P}_k=\left[\mathsf{H}_{\color{black}[k-1]},\mathsf{H}_{\color{black}[k]}\right),~k\in {\color{black}\left[\nt\right]}$. Algorithm 1 outputs the maximally correlated $\nt$-variate binary distributions as well, i.e., the same with the decomposition results in Proposition <ref>.

[Signaling for the maximum-entropy distribution]
Consider a $3\times 1$ EC-OIC with normalized channel gains $\mathbf{h}=\trans{\left(0.4,0.2,0.4\right)}$ and required average intensities ${\color{black}\bm{\alpha}}=\trans{\left(0.8,0.3,0.1\right)}$. The maximum-entropy distribution is numerically computed as $p_{\scriptscriptstyle \! S}^*(s)=\exp ({-0.4286+2.9176s-6.5987(s-0.4)_+})$.
By solving (<ref>), we obtain relevant parameters as  $\kappa_3^*=0.564$, $\kappa_2^*=0.4$, and $\kappa_1^*=0$. Thus, $
\mathcal{P}_3= \left[0.564,0.964\right),~
\mathcal{P}_2= \left[0.4,0.564\right) \cup \left[(0.964,1\right)$ and $
\mathcal{P}_1= \left[0,0.4\right)$, which are plotted in the bottom of Figure <ref>. Moreover, in the top of Figure <ref>, we align the $s$-axis and plot the outputs of Algorithm 1 and Algorithm 2, which completely match with each other.

Interval partition for a maximum-entropy distribution.

[Signaling for a mergeable $3 \times1 $ BC-OIC]
Consider a $3\times 1$ BC-OIC with unnormalized parameters: the channel gains  $\widetilde{\mathbf{h}}=\trans{\left(4\times 10^{-6} ,1.5\times 10^{-6},3\times 10^{-6}\right)}$, the maximum allowed peak intensities $\bm{\amp}=\trans{\left(2 ,3,2.5 \right)}$, ratios of the maximum allowed average intensity to corresponding maximum allowed peak intensity  $\widetilde{\bm{\alpha}}=\trans{\left(0.4,0.1,0.1\right)}$, and the AWGN standard deviation $\tilde{\sigma}$.

We first compute the normalized channel gains as $h_1'=0.4$, $h_2'=0.225$ and $h_3'=0.375$ by $h_k' = \tilde{h}_k \amp_k/\left(\sum_{i=1}^{\nt}\tilde{h}_i \amp_i\right)$ for all $ k\in \{1,2,3\}$. Correspondingly, the normalized standard deviation of AWGN is $\sigma=\tilde{\sigma}/\left(\sum_{i=1}^{\nt}\tilde{h}_i \amp_i\right)$.

Next, note that $\alpha_2=\alpha_3$ and thus we can let the second and the third transmitter send identical normalized signals. Based on Proposition <ref>, the considered channel is equivalent to a $2\times 1$ BC-OIC with normalized channel gains $\mathbf{h}=\trans{\left(0.4,0.6\right)}$ and maximum allowed average intensities $\bm{\alpha}=\trans{\left(0.4,0.1\right)}$.
In what follows, we consider a continuous and two finite-alphabet distributions of the equivalent input, respectively.

(1) Maximum-entropy distribution:
$p_{\scriptscriptstyle \! S}^*(s) = \exp \left(1.466-4.270s\right),~ s\in [0,1]$. We compute the expectation as $\mathbb{E}\left[S\right]=\trans{\mathbf{h}}\bm{\alpha}$ and thus $S$ is $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable.

Applying (<ref>), we obtain $\kappa_2^*=0.272$, $\mathcal{P}_2= \left[0.272,0.872\right)$ and $
\mathcal{P}_1= \left[0,0.272\right)\cup \left[0.872,1\right)$.

Given a realization $s=0.2$ generated by $p_{\scriptscriptstyle \! S}^*(s)$, the first transmitter sends the unnormalized signal $x_1=\frac{s}{h_1}\times \amp_1=1$ and other two transmitters send zero.

Given a realization $s=0.5$, the first transmitter sends $x_1=\frac{0.272}{h_1}\amp_1=1.36$, and other two transmitters send $x_2=\frac{s-0.272}{h_2}\amp_2=1.14$ and $x_3=\frac{s-0.272}{h_2}\amp_3=0.95$.

Given a realization $s=0.9\ge \sup \mathcal{P}_2$, the second and the third transmitters send their peak values, i.e., $x_2=\amp_2=3$ and $x_3=\amp_3=2.5$, and the first transmitter sends $x_1=\frac{0.272+s-0.872}{h_1}\amp_1=1.5$.

(2) Nonequiprobable OOK: $\mathcal{S}_{\textnormal{OOK}}=\{0,1\}$ with $\mathbb{P}\left\{S=1\right\}=0.1$. Note that the expectation $\mathbb{E}\left[S\right]=0.15{\color{black}\,\,=\trans{\mathbf{h}}\mathbf{a}^{\dagger}  }$, where $\mathbf{a}^{\dagger}=\trans{\left(0.1,0.1\right)}$. Here $S$ can be regarded as a feasible input to a mergeable $2\times 1$ EC-OIC with parameters $\mathbf{h}$ and $\mathbf{a}^{\dagger}$ and thus spatial repetition can be used such that the signaling for $\mathcal{S}_{\textnormal{OOK}}$ is simplified as
$X_1=\amp_1 S$, $x_2=\amp_2 S$ and $x_3=\amp_3 S$.

(3) Equally spaced and equiprobable $8$ASK: $\mathcal{S}_{8\textnormal{ASK}}=\{0,\Delta,2\Delta,\cdots,7\Delta\}$, where the minimum Euclidean distance $\Delta$ is maximized as $\Delta=0.0629$. Calculate the expectation $\mathbb{E}\left[S\right]=\trans{\mathbf{h}}\bm{\alpha}$ and hence $S$ is $\left(\mathbf{h},\bm{\alpha}\right)$-decomposable. Applying (<ref>), we obtain $\kappa_2^*=0.226$, $\mathcal{P}_2= \left[0.226,0.826\right)$ and $
\mathcal{P}_1= \left[0,0.226\right)\cup \left[0.826,1\right)$. Given $s=i\Delta$, the decomposition can be further simplified as
$x_1=\frac{i\Delta}{h_1}\amp_1$ and $x_2=x_3=0$ if $i\le 3$, and otherwise $x_1=\frac{\kappa_2^*}{h_1}\amp_1$, $x_2=\frac{(i\Delta-\kappa_2^*)}{h_2}\amp_2$ and $x_3=\frac{(i\Delta-\kappa_2^*)}{h_2}\amp_3$.[For a specific distribution, a carefully designed decomposition algorithm may be more suitable than the greedy decomposition from the considerations of practical signaling. For example, we can alternatively let
$X_1=\frac{\alpha_1}{h_1\alpha_1+h_2\alpha_2}\amp_1S$, $X_2=\frac{\alpha_2}{h_1\alpha_1+h_2\alpha_2}\amp_2S$ and $X_3=\frac{\alpha_2}{h_1\alpha_1+h_2\alpha_2}\amp_3S$ for $\mathcal{S}_{8\textnormal{ASK}}$.]

Inspired by the above examples, in order to design a good spatial constellation we can let our proposed algorithms concatenated with a well-designed constellation of the equivalent input in terms of Euclidean distance. This transmitter architecture may deserve special attention.

§ CONCLUSION

In this paper, we consider two types of MISO OICs under per-antenna intensity constraints.
We first express their capacities as a maximization problem
over distributions of the sum random variable $S = \trans{\vect{h}} \vect{X}$. The main
challenge is to equivalently transform per-antenna average-intensity constraints on
$\vect{X}$ to constraints on $S$, since the mapping from $\vect{x}$
onto $s=\trans{\mathbf{h}}\vect{x}$ is not injective. This problem is solved by finding the necessary and sufficient conditions for two types of decomposition of a bounded and nonnegative random variable. Specifically, we show that both types of original MISO OICs with per-antenna intensity constraints can be transformed into
a SISO one with an amplitude constraint and several stop-loss mean constraints.
Then we obtain several capacity bounds for our considered MISO OICs by alternatively bounding capacities of their corresponding equivalent SISO OICs. Using EPI and duality upper-bounding techniques, we derive new lower and upper bounds on the capacity, and they asymptotically match at high SNR.
In the low-SNR regime, the capacity slope is shown to be proportional
to the maximum variance of the equivalent input $S$ under stop-loss mean

§ ACKNOWLEDGMENT

The authors wish to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions that improved the quality of this paper.

§ PROOF OF THEOREMS <REF> AND <REF>

We will prove Theorem <ref> for the EC-OIC, and the involved techniques are applied to the case of the BC-OIC as well.
Rewrite the original entropy-maximizing problem under stop-loss equality and inequality constraints (<ref>) as follows:

\begin{equation}\label{prob:original}
\begin{aligned}
&\textnormal{minimize} ~~&&\int_0^1 p (s)\log p (s)\dd s\\
&\textnormal{subject to}~~ && \quad ~~ p(s)\ge 0\\
& ~&& \int_0^1   p (s) \dd s  =1\\
& ~&& \int_0^1 s p (s) \dd s =\trans{\mathbf{h}}\bm{\alpha}\\
& ~&&
\int_0^1  \left(s-\mathsf{H}_{\color{black}[k]}\right)_+
p (s) \dd s
\le \left(1-\mathsf{H}_{\color{black}[k]}\right) \bar{\alpha}_k, ~\forall\, k\in {\color{black}\left[\nt-1\right]}
\end{aligned}
\end{equation}

Then we consider the Lagrangian functional[Here we let a Lagrange multiplier be $-({\color{black}\nu}_0+1)$ instead of the conventional ${\color{black}\nu}_0$.]

L(p,ν_0,λ)=     ∫_0^1 p  logp s-(ν_0+1)(∫_0^1 p s-1)
+λ_0(∫_0^1 sp s-𝐡α)

( ∫_0^1 (s-𝖧_[k])_+
p s
-(1-𝖧_[k]) α̅_k

where numbers ${\color{black}\nu}_0,\lambda_0\in \mathbb{R}$ and $\lambda_1,\cdots,\lambda_{\nt-1}\ge 0$. By using the Euler-Lagrange equation, we obtain the following extremum condition for the functional $L\left(p,{\color{black}\nu}_0,\bm{\lambda}\right)$ from the perspective of calculus of variations:

∂L/∂p= logp-ν_0+λ_0s+∑_i=1^-1λ_i(s-𝖧_[k])_+ =0,

whose solution is

p (s) = exp(ν_0 -λ_0  s-( ∑_i=1^-1 λ_i  (s-𝖧_[i] )_+ ))>0,  s∈[0,1].

Substituting (<ref>) into (<ref>), we obtain the dual function as

g( ν_0,λ)=     - ∑_i=1^-1λ_i(1-𝖧_[i])α̅_i

-∫_0^1 exp(ν_0-λ_0 s-( ∑_i=1^-1 λ_i (s-𝖧_[i] )_+ )) s .

Then we consider the Lagrange dual problem

γ_ E =     min_ν_0,λ_0 ∈ℝ λ_1,⋯,λ_-1 ≥0 -g( ν_0,λ).

Since the duality gap between the primary problem (<ref>) and its Lagrange dual (<ref>) is zero, i.e., strong duality holds, the maximum differential entropy equals to the solution to the dual problem, i.e., $\gamma_{ \textnormal{E}} $.

The proof is concluded by calculating the involved integration in (<ref>) and using EPI to obtain the lower bound on the differential entropy $\hh(Y)$ of the  output $Y$ <cit.>.

§ PROOF OF THEOREMS <REF> AND <REF>

By the maximum entropy argument, we have
(Y) ≤1/2log(2πe (σ^2+ V_max^E  (or B)(𝐡,α) )),

where $\const{V}_{\max}^{\textnormal{E}}(\vect{h},\bm{\alpha})$ and $\const{V}_{\max}^{\textnormal{B}}(\vect{h},\bm{\alpha})$ denote the maximum variances over all feasible equivalent input $S$ for the EC-OIC and the BC-OIC, respectively. Hence we only need to show $\const{V}_{\textnormal{max}}^{\textnormal{E }}\left(\mathbf{h},\bm{\alpha}\right)$ and $\const{V}_{\textnormal{max}}^{\textnormal{B}}\left(\mathbf{h},\bm{\alpha}\right)$ are exactly the RHS of (<ref>) and (<ref>), respectively.

In the EC-OIC, for any feasible input $\mathbf{X}$, we have $S \le_{\textnormal{cx}}\bar{S}_{\mathbf{h},\bm{\alpha}}$. Then by Corollary <ref> and the fact
Var(S̅_𝐡,α )=∑_i=1^∑_j=1^ h_ih_j (min{α_i,α_j} -α_iα_j),

we have $\const{V}_{\textnormal{max}}^{\textnormal{E }}\left(\mathbf{h},\bm{\alpha}\right)=\textnormal{Var}\left(\bar{S}_{\mathbf{h},\bm{\alpha}} \right)$, and hence the proof of Theorem <ref> is completed.

We now turn to the BC-OIC case. Under bounded-cost constraints (<ref>), finding the maximum variance can be formulated as the following intensity optimization problem with respect to the variables $\mathbf{a}$

\begin{equation}\label{prob:low}
\begin{aligned}
&\textnormal{maximize} ~~&&\const{V}_{\textnormal{max}}^{\textnormal{E}}\left(\mathbf{h},\mathbf{a}\right)\\
&\textnormal{subject to}~~ && \bm{0} \preccurlyeq \mathbf{a} \preccurlyeq \bm{\alpha}
\end{aligned}
\end{equation}

i.e., maximization over all feasible average intensity for the BC-OIC.

Next, we show the solution to the optimization problem (<ref>) is $\mathbf{a}^{*}=\min\{\beta^{*} \bm{1},\bm{\alpha}\}$, and the maximum variance is the RHS of (<ref>).

Following the same arguments as in the proof of Theorem <ref>, we can show for any feasible input $\mathbf{X}$ with the average intensity $\mathbf{a}$ satisfying  $\bm{0} \preccurlyeq \mathbf{a} \preccurlyeq \bm{\alpha}$, its equivalent input signal $S$ must be $(\mathbf{h},{\color{black}\mathbf{a}^\dagger})$-decomposable,  where the vector  $	\mathbf{a}^{\dagger} =   \min\left\{\beta \bm{1},\bm{\alpha}\right\} $ with some $\beta \in \left[0,\alpha_1 \right]$ such that  $\trans{\mathbf{h}}\mathbf{a}^{\dagger}=\trans{\mathbf{h}}\mathbf{a}$. Following Theorem <ref>, we have  $\bar{S}_{\mathbf{h},{\color{black}\mathbf{a}^\dagger}}\ge_{\textnormal{cx}} S$. Hence, the following order relation holds

Var(S̅_𝐡,𝐚^† )≥Var(S ),

which implies the variance-maximized average-intensity vector $\mathbf{a}^{\star}$ can be restricted to $\mathbf{a}^{\star}=\min\{\beta \bm{1},\bm{\alpha}\}$ for $\beta\in [0,\alpha_1]$ as well (similar to the discussion in Remark <ref>).

We only need to consider the nontrivial case $\alpha_{\nt} < \frac{1}{2}$. Let $\nu(\beta)=\const{V}_{\textnormal{max}}^{\textnormal{E}}\left(\mathbf{h},\min\{\beta \bm{1},\bm{\alpha}\}\right)$. Note that the function $\nu(\beta)$ is continuous in the interval $\beta\in[0,\alpha_{\nt}]$, and piecewise smooth. It is clear that $\nu(\beta)=\beta(1-\beta)$ for $\beta\in[0,\alpha_{\nt}]$, which is maximized at $\beta=\alpha_{\nt}$.
Letting $k_\beta=\{k\in \left[\nt\right]:\alpha_k\ge \beta\}$, then $\beta\in \left(\alpha_{{k_\beta}+1},\alpha_{k_\beta} \right]$ holds. We rewrite $\nu(\beta)$ as

ν(β)=   𝖧^2_[k_β]β(1-β)+2𝖧_[k_β]∑_m=k_β+1^nh_iα_i(1-β)

+  ∑_i=k_β+1^∑_j=k_β+1^ h_ih_j (min{α_i,α_j} -α_i α_j)_𝖪

where the quantity $\mathsf{K}$ is a constant in the interval $\beta \in\left(\alpha_{{k_\beta}+1},\alpha_{k_\beta} \right]$.
Then we obtain

ν^'(β)=𝖧^2_[k_β] (1-2β)-2𝖧_[k_β](1-𝖧_[k_β])α̅_k_β, β∈(α_k_β+1,α_k_β ].

Noting that $k_{\beta_1}\ge k_{\beta_2}\ge 1$, so $\mathsf{H}_{\color{black}[{k_{\beta_1}}]}\ge \mathsf{H}_{\color{black}[{k_{\beta_2}}]}>0$ and $\bar{\alpha}_{k_{\beta_1}}\le\bar{\alpha}_{k_{\beta_2}}$. We have

=𝖧_[k_β_2](𝖧_[k_β_2] (1-2β_2)-2(1-𝖧_[k_β_2])α̅_k_β_2)

≤𝖧_[k_β_2](𝖧_[k_β_1] (1-2β_1)-2(1-𝖧_[k_β_1])α̅_k_β_1)

=𝖧_[k_β_2]/𝖧_[k_β_1]	ν^'(β_1)

Hence for an arbitrary pair $(\beta_1,\beta_2)$ satisfying $ \alpha_{\nt}<\beta_1\le \beta_2$, if $\nu^\prime(\beta_1)\le 0$, we have $\nu^\prime(\beta_2)\le 0$. Thus, the optimal average intensity satisfies

β^*=inf{ β∈(α_,α_1]:𝖧_[k_β] (1-2β)-2(1-𝖧_[k_β])α̅_k_β ≤0 }.

Substituting (<ref>) into (<ref>), the proof of Theorem <ref> is completed.

§ PROOF OF THEOREMS <REF> AND <REF>

We only prove Theorem <ref> for the EC-OIC, and the proof of Theorem <ref> follows essentially the same arguments.

We evaluate the duality-based upper bound

\begin{align}\label{E101}
\mathsf{C}_{\textnormal{E}}\left(\mathbf{h},\bm{\alpha},\sigma\right) \leq \sup_{{\color{black}\mathscr{P}_{\! S}} }\mathbb{E}_{{\color{black}S\sim \mathscr{P}_{\! S}}}[D(W(\cdot|S)||R(\cdot))]
\end{align}

for the auxiliary density

R(y) =

if  y ∈(-∞, 0),

(1-β)/𝖯 ·e^ -∑_i=0^-1 λ_i (y-𝖧_[i] )_+     if   y∈[0,1+δ],

if  y ∈(1+δ, ∞),

with free parameters $\delta>0$,  $\lambda_0\in \mathbb{R}$, $\lambda_1,\cdots,\lambda_{\nt-1}\ge 0$, and $\beta\in(0,1)$ which will be specified later.

We notice that
-∫_-∞^0 W(y|s)logR(y)   y

=      -∫_-∞^0
(logβ/√(2π)σ -
y^2/2σ^2)   y

=     -log(β/√(2π)σ)
+ 1/2𝒬(s/σ)
+ 1/2(s/σ)^2
- s/2σ

≤     -(logβ/√(2π)σ-

=      - logβ/√(2πe)σ

ϕ(s) = 1/√(2π) e^-s^2/2,

$\mathcal{Q}(\cdot)$ denotes Gaussian Q-function, and (<ref>) holds because of
0 ≤ϕ(η)-	η𝒬(η) ≤1/√(2π),   η≥0.

-∫_1+δ^∞ W(y|s)logR(y)
≤      - logβ/√(2πe)σ

-∫_0^1+δ W(y|s)logR(y)  y

=     -∫_0^1+δ
log(1-β/𝖯  )

(∑_i=0^-1 λ_i (y-𝖧_[i] )_+
) y.

Rewrite the first term in the RHS of (<ref>) as
) y

=     -log(1-β/𝖯 )
(1-𝒬(s/σ) -
𝒬(1+δ-s/σ) ).

The second term in the RHS of (<ref>) can further be
upper-bounded by
(∑_i=0^-1 λ_i (y-𝖧_[i] )_+
) y

=      ∑_i=0^-1λ_iσ(
- ϕ(1+δ-s/σ) )

+ ∑_i=0^-1λ_i (s-𝖧_[i]) (1 - 𝒬(s-𝖧_[i]/σ)
- 𝒬(1+δ-s/σ) )

≤     ∑_i=1^-1λ_iσ(
- ϕ(1+δ/σ) )
+ ∑_i=1^-1λ_i (s-𝖧_[i])_+ +λ_0 s

+λ_0 σ( ϕ(s/σ)-ϕ(1+δ-s/σ))
-λ_0 s (𝒬(s/σ)+𝒬(1+δ-s/σ))

≤    ∑_i=0^-1λ_i (s-𝖧_[i])_+ + ∑_i=0^-1λ_iσ/√(2π)(1-e^-(1+δ)^2/2σ^2) ,  if λ_0≥0,

(∑_i=0^-1 λ_i (y-𝖧_[i] )_+
) y

≤    ∑_i=0^-1λ_i (s-𝖧_[i])_+  + ∑_i=1^-1λ_iσ/√(2π)(1-e^-(1+δ)^2/2σ^2)

+λ_0 σ( ϕ(1/σ)-ϕ(δ/σ))
-λ_0  (𝒬(1/σ)+𝒬(δ/σ)),  if λ_0≤0.

Here (<ref>) follows from (<ref>)  and the fact that, for $s \in[0,1]$,
1 - 𝒬(s-𝖧_[i]/σ)
- 𝒬(1+δ-s/σ)
≥1 - 𝒬(s-1/σ)
- 𝒬(1+δ-s/σ)
>   0,

and (<ref>) and (<ref>) follow from the fact that
\sigma \left( \phi(\frac{s}{\sigma})-\phi\left(\frac{1+\delta-s}{\sigma}\right)\right)
-s \left(\mathcal{Q}\left(\frac{s}{\sigma}\right)+\mathcal{Q}\left(\frac{1+\delta-s}{\sigma}\right)\right)$ is strictly decreasing with $s\in [0,1]$.

By choosing
β= √(2πe )σ/ 𝖯+√(2πe)σ

and combining (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we conclude the proof.

§ PROOF OF THEOREM <REF>

We only prove Theorem <ref> for the EC-OIC, and the involved techniques can be applied directly to the BC-OIC.

The maximum-entropy distribution $p_{\scriptscriptstyle \! S}^*(s)$ of the equivalent input for the EC-OIC is defined in (<ref>). For the case of $\lambda_0^*\ge 0$, by substituting a suboptimal choice $\delta=\sqrt{\sigma}$ and $\lambda_i=\lambda_i^*$ for $i\in\{0,\cdots,\nt-1\}$ into (<ref>) in Theorem <ref>, we get a upper bound as

≤log(1+  𝖯(σ)/√(2πe) σ) +  ∑_i=0^-1λ_i^* (1-𝖧_[i] )α̅_i
+ ∑_i=0^-1λ_iσ/√(2π)(1-e^-(1+δ)^2/2σ^2) .

Note that

lim_σ↓0^+ 𝖯(σ)   =lim_σ↓0^+	∫_0^1+√(σ)exp( -λ_0^* y-∑_i=1^-1 λ_i^* (y-𝖧_[i] )_+ ) y

=	∫_0^1exp( -λ_0^* y-∑_i=1^-1 λ_i^* (y-𝖧_[i] )_+ )y

=e^-ν_0^*	∫_0^1p_ S^*(y) y

0 ≤(1-e^-(1+δ)^2/2σ^2)  ≤1.

Then we have

lim sup_σ↓0^+     𝖢_E(𝐡,α,σ)-log1/σ

≤    lim_σ↓0^+   log(1+ e^-ν_0^*/√(2πe) σ) -log1/σ +  ∑_i=0^-1λ_i^* (1-𝖧_[i])α̅_i

=    lim_σ↓0^+   log(σ+ e^-ν_0^*/√(2πe) )+  ∑_i=0^-1λ_i^* (1-𝖧_[i])α̅_i

=    -1/2log2πe-ν_0^* +  ∑_i=0^-1λ_i^* (1-𝖧_[i])α̅_i

=    -1/2log2πe+γ_E,

where (<ref>) follows from the continuity of a logarithmic function and (<ref>) follows from the fact that

γ_ E    =  ∑_i=0^-1λ_i^*(1-𝖧_[i])α̅_i
-1-ν_0^*+∫_0^1 p^*_ S (s) s

= ∑_i=0^-1λ_i^*(1-𝖧_[i])α̅_i
-ν_0^* .

By using the lower bound in Theorem <ref>, the reverse part of the proposition can be obtained as follows

lim inf_σ↓0^+     𝖢_E(𝐡,α,σ)-log1/σ

≥    lim_σ↓0^+1/2log(1+exp(2 	γ_ E)/2πe σ^2)-log1/σ

≥    -1/2log2πe+γ_E.

Due to the boundedness of $ \phi(\frac{1}{\sigma})-\phi\left(\frac{\delta}{\sigma}\right)$ and
\lim_{x\to +\infty}\mathcal{Q}(x)=0
the proof for the case of  $\lambda_0^*\le 0$ can be accomplished similarly.

§ PROOF OF PROPOSITION <REF>

Before proving Proposition <ref>, we first need the following lemmas:

Suppose both $v_i^*$ and $\tilde{v}_i^*\in \left[0,\mathsf{H}_{\color{black}[i-1]}\right]$ are solutions to the equation (<ref>) and let $R_{i-1}^*=\varphi\left(R_{i}^*;v_i^*,h_{i}\right)$ and $\tilde{R}_{i-1}^*=\varphi\left(R_{i}^*;\tilde{v}_i^*,h_{i}\right)$. Then $\pi_{\scriptscriptstyle\! R_{i-1\!}^*}(t)=\pi_{\scriptscriptstyle\! \tilde{R}_{ i-1\! }^*}(t)$.

Without loss of generality, we assume that	$v_i^* \le \tilde{v}_i^*$. Applying (<ref>), it is clear that $\pi_{\scriptscriptstyle\! R_{i-1\!}^*}(t)=\pi_{\scriptscriptstyle\! \tilde{R}_{ i-1\! }^*}(t)$ for $t\in \left[ 0,v_i^*\right]$ and $t\in \left[ \tilde{v}_i^*,1\right]$, respectively.

For $t\in \left[v_i^*,\tilde{v}_i^*\right]$, we notice that

π_ R_i-1^*(t)-π_ R̃_ i-1 ^*(t) =π_ R_i^*(t+h_i)-π_ R_i^*(t)+h_iα_i,

which is continuous and nonincreasing with $t$. Note that

π_ R_i-1^*(v_i^*)-π_ R̃_ i-1 ^*(ṽ_i^*)
=π_ R_i^*(v_i^*+h_i)-π_ R_i^*(v_i^*)+h_iα_i

=π_ R_i^*(ṽ_i^*+h_i)-π_ R_i^*(ṽ_i^*)+h_iα_i

where (<ref>) follows from the assumption that $v_i^*$ and $\tilde{v}_i^*$ are solutions to (<ref>). Thus, we conclude that $\pi_{\scriptscriptstyle\! R_{i-1\!}^*}(t)-\pi_{\scriptscriptstyle\! \tilde{R}_{ i-1\! }^*}(t) = 0$ for all $t\in[0,1]$.

$v_{i}^*\ge v_{i-1}^*$  for $i\in\{n,\cdots,2\}$.

Notice that $v_2^*\ge 0$ and $v_1^*=0$. We will prove $v_{i}^*\ge v_{i-1}^*$ for any $i \in \{\nt,\cdots,3\}$ (when $\nt \ge 3$) by contradiction.

Assuming $v_{i-1}^*>v_i^*$, we get $v_i^* \le \mathsf{H}_{\color{black}[i-2]}$. Substituting $v=v_{i}^*$ into $\varphi\left(R_{i-1}^*,v,h_{i-1}\right)$, we have

which follows from the definition of $\varphi \left( x;v,z\right)$.
Then we consider

=π_ R_i^*(0)-π_ R_i^*(v_i^*)+π_ R_i^*(v_i^*+h_i+h_i-1)

=π_ R_i-1^*(0)-π_ R_i^*(v_i^*+h_i)+π_ R_i^*(v_i^*+h_i+h_i-1),

where (<ref>) follows from $\pi_{\scriptscriptstyle\! R_{i-1}^*}(0)=\pi_{\scriptscriptstyle\! R_{i}^*}(0)-\pi_{\scriptscriptstyle\! R_{i}^*}(v_{i}^*)+\pi_{\scriptscriptstyle\! R_{i}^*}(v_{i}^*+h_{i})$.
Thus, we have

=π_ R_i^*(v_i^*+h_i)-π_ R_i^*(v_i^*+h_i+h_i-1)/h_i-1

=1/h_i-1∫_v_i^*+h_i^v_i^*+h_i+h_i-1 ( 1-F_ R_i^*(x) ) x

≤1-F_ R_i^*(v_i^*+h_i)

≤1/h_i∫_v_i^*^v_i^*+h_i (1-F_ R_i^*(x)) x

=π_ R_i^*(v_i^*)-π_ R_i^*(v_i^*+h_i)/h_i

= α_i

< α_i-1,

where $F_{\scriptscriptstyle\! R_{i}^*}(x)$ denotes the cumulative distribution function of $R_{i}^*$.
Hence we know

which implies $v_{i}^*\ge v_{i-1}^*$ and contradicts the assumption  due to the monotonicity of  $\mathbb{E}\left[\varphi\left(R_{i}^*,v,h_{{\color{black}i}}\right)\right]$.

Now we prove Proposition <ref> via induction on $k$. The existence of the solution to (<ref>) can be verified by letting $\kappa=v_{k-1}^*$, which is omitted here.

* The proposition in the base case $k=\nt$ is clearly true.

* Suppose $\kappa_k^*$ is the solution to

π_ R_k^*(κ_k^*)-π_ R_k^*(κ_k^*+h_k)=h_kα_k

for each $k\ge i$ ($2\le i\le \nt$) and consider the case of $k=i-1$.

Due to the monotonicity of the solutions $v_k^*$, we have $\kappa_i^*\le \cdots \le \kappa_{\nt}^*$. Then using (<ref>), for $\kappa \le \kappa_{i}$ we have

π_ S(κ)   =π_ R_-1^*(κ)+h_α_

=π_ R_i-1^*(κ)+∑_m=i^h_mα_m,

where (<ref>) follows from (<ref>).

It is clear that the Borel sets $\mathcal{P}_i,\cdots, \mathcal{P}_{\nt}$ are disjoint and satisfy $\mu\left( \mathcal{P}_j\right)=h_j$ for $i \le j\le \nt$. In addition, we have

η_j(κ_j^*) ≤η_i-1(κ_i-1^*),  if κ_j^* < η_i-1(κ_i-1^*),

which follows from the following equality

μ([κ_i-1^*,η_i-1(κ_i-1^*))∖⋃_m=i^𝒫_i  )=h_i-1.

We let $\mathcal{J}= \left\{j:\kappa_j^* < \eta_{i-1}(\kappa_{i-1}^*) , i-1< j\le \nt \right\}$.

If $\mathcal{J}=\emptyset$, then we have $\eta_{i-1}(\kappa_{i-1}^*)\le \kappa_i^*$ and similarly obtain

π_ S(η_i-1(κ_i-1^*))   =π_ R_-1^*(η_i-1(κ_i-1^*))+h_α_

=π_ R_i-1^*(η_i-1(κ_i-1^*))+∑_m=i^h_mα_m.

If $\mathcal{J} \neq \emptyset$, we denote $M=\max \mathcal{J}$. Due to $\kappa_i^*\le \cdots \le \kappa_{\nt}^*$, we have $\kappa_j^* \ge \eta_{i-1}(\kappa_{i-1}^*)$ if and only if $j>M$. Using (<ref>), we conclude that the disjoint sets $\mathcal{P}_j$ satisfy

\begin{align}
&\mathcal{P}_j \subseteq	\left[\kappa_{i-1}^*,\eta_{i-1}(\kappa_{i-1}^*) \right),~i\le j \le M,\\
&\mathcal{P}_j   \cap 	\left[\kappa_{i-1}^*,\eta_{i-1}(\kappa_{i-1}^*)\right)=\emptyset,~j>M.
\end{align}

Applying the equality (<ref>), we immediately have

η_i-1(κ_i-1^*)=κ_i-1^*+∑_ j=i^M h_jα_j

Then we rewrite $\pi_{\scriptscriptstyle \! S}\left(\eta_{i-1}(\kappa_{i-1}^*)\right)$ as

π_ S(η_i-1(κ_i-1^*))   =π_ R_ M^*(κ)+∑_M<m≤h_mα_m

=π_ R_ M-1^*(η_i-1(κ_i-1^*)-h_M)+∑_M<m≤h_mα_m

=π_ R_ i-1^*(η_i-1(κ_i-1^*)-∑_j=i^Mh_j)+∑_M<m≤h_mα_m

=π_ R_ i-1^*(κ_i-1^*+h_i-1)+∑_M<m≤h_mα_m.

Hence we have

π_ S(κ_i-1^*)-π_ S(η_i-1(κ_i-1^*))=∑_m=i-1^i-1+N_i-1(κ_i-1^*)h_mα_m,

which implies

π_ R_i-1^*(κ_i-1^*)-π_ R_i-1^*(κ_i-1^*+h_i-1)=h_i-1α_i-1.

The proof is completed.