Tail inequalities for restricted classes of discrete random variables

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

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