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Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...

We are given two correlated random variables $V$ and $X$ supported over a finite alphabets $\mathcal{V}$ and $\mathcal{X}$. Suppose the marginal $P_V$ and conditional distribution $P_{X|V}$ are ...

Let $P\in\mathbb{R}^{n\times n}$ be a doubly-stochastic matrix. That is: $$P(x,y)\geq 0,\quad \sum_xP(x,y)=1,\quad \sum_yP(x,y)=1.$$ I would like to know if lower and upper bounds on the sample ...

A definition of the mixing time for Markov chains is given by \begin{equation} \tau_{\text{mix}}\equiv\inf{\{t>0: \sup_i\left\vert \frac{\boldsymbol{p}(t|p_j(0)=\delta_{ij})}{\boldsymbol{\pi}}-\...

Let $(A_1,B_1)$ and $(A_2,B_2)$ be two random variables with the joint distributions $p_{A_1B_1}$ and $p_{A_2B_2}$, respectively. Moreover, we have $$\mathbb{P}[(A_1,B_1)\neq (A_2,B_2)]=\alpha.$$ Then,...