Tail inequalities for restricted classes of discrete random variables

Abstract

Let $X$ be an integrable discrete random variable over $\{0,1,2,\dots \}$ with $\mathbb{P}(X=i+1)\le \mathbb{P}(X=i)$ for all $i$. Then for any integer $a\ge 1$, $\mathbb{P}(X\le a)\le \mathbb{E}[X]/(2a-1)$. Let $W$ be an discrete random variable over $\{\dots ,-2,-1,0,1,2,\dots \}$ with finite second moment where the $\mathbb{P}(W=i)$ values are unimodal. Then $\mathbb{P}(|W-\mathbb{E}[W]|\ge a)\le (\mathbb{V}(W)+1/12)/(2(a-1/2{)}^2)$.

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