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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 ...

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 ...