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Let $\left\{V_1,\ldots,V_n\right\}$ be a set of (possibly dependent) identically distributed Bernoulli random variables, with $$ p = \mathbb{P}\left(V_i=1\right) = 1-\mathbb{P}\left(V_i=0\right),\quad \forall \; i=1,\ldots, n.$$ Furthermore, define $\overline{V}_i = V_i-p$ for any $i=1,\ldots, n$. We also know that $p<\frac{1}{2}$. Is there a strictly positive scalar $C$ (possibly depends on $n$, e.g. $2^n$) such that

$$\mathbb{E}\left(\Pi^n_{i=1}\overline{V}_i\right) \leq C \mathbb{E}\left(\Pi^n_{i=1}V_i\right)?$$

Here $\mathbb{E}$ stands for the expected value operator.

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  • $\begingroup$ In general no, since for example it's possible that $\Pi^n_{i=1}V_i$ is $0$ with probability $1$ and also $\Pi^n_{i=1}\overline{V}_i$ is strictly positive with probability $1$. For this it's enough that with probability $1$, the number of $i$ with $V_i=0$ is always strictly positive and even. $\endgroup$
    – James Martin
    Commented Jun 30, 2019 at 23:37

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