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Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of …

I am faced with an information-theoretic upper bound, such as \begin{align} \sqrt{\alpha'}2^{I_\alpha(X;Y)}, \end{align} where $I_\alpha(X;Y)$ is the Rényi mutual information with parameter $\alpha>1$ …

Assuming the von Mises-Fisher distribution as $$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$ where $\kappa \ge 0$ …

Assume that we have heavy-tailed distribution $F(x)$ such that \begin{align} F(x)=\mathbb{P}[X\geq x]=x^{-0.5}. \end{align} Then, we produce $N$ independent samples $X_1,X_2,\ldots,X_N$ from this dist …

Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\s …

Let $Y$ denote a Gaussian random variable characterized by a mean $\mu$ and a variance $\sigma^2$. Consider $N$ independent and identically distributed (i.i.d.) copies of $Y$, denoted as $Y_1, Y_2, \l …

If we have a real random variable $X$ such that $a\leq X\leq b$ almost surely, we can establish the following inequality: \begin{align} \mathbb{E}\left[\exp\Big(t(X-\mathbb{E}[X])\Big)\right]\leq\exp\ …

Let $X$ be a random variable and $Y=f(X)$ where $f$ is a deterministic function. Moreover, assume that there exists a deterministic function $g(.)$ such that the following probability is small. \begin …

Suppose $X$, $Y$, $X'$ and $Y'$ are random variables whose probability density follows the following relations. \begin{align} \|p_X-p_{X'}\|_{\mathrm{TV}}&\leq\epsilon_1,\\ \|p_Y-p_{Y'}\|_{\mathrm{TV} …

Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability dist …

Suppose $I(X;Y)$ denotes mutual information and on the other hand there is a relationship as follows. \begin{align} |p(y)-p(y|x)|<\delta p(y),\qquad\forall x,y. \end{align} Then we can say about mutua …

Let $\mathbf{U}$ and $\mathbf{V}$ be random unitary matrices independent of random input vector $\mathbf{x}$. Moreover, $\mathbf{z}$ be random iid complex Gaussian vector with zero mean and identity c …

Let $\mathbf{U}$ be a random unitary matrix and $\mathbf{z}$ be a random i.i.d complex Gaussian vector (unitary invariant). Assume that the following relation is satisfied: \begin{align} \mathbf{y}=\m …

Let $\mathbf{Z}$ be a $m\times n$ matrix with zero-mean unit-variance i.i.d complex Gaussian entries. What is the maximum value of mutual information $I(\mathbf{X}_1,\mathbf{X}_2;\mathbf{Y})$ such tha …

Let $X$ be a random variable which takes values in $\mathcal{X}$. Assume that we pass $X$ through two independent conditional pdf $p_{X_1|X}$ and $p_{X_2|X}$ and choose $X_1$ with probability $\lambda …