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Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function, and assume that it grows at most linearly at infinity for simplicity. Denote by $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$, …

Let $p \ge 1$ be an integer. Does there exist a constant $C_p$ such that for every random variable $X \ge 0$, $$ \mathbb{E} \left[ \left(X - \mathbb{E} \left[ X \right] \right)^{2p} \right] \le C_p \m …

The answer is yes. Let $Y$ be an independent random variable distributed as $X$. We have $$ \|X - \mathbb{E}[X]\|_{L^{2p}} = \|\mathbb{E}\left[X - Y \, | \, X\right]\|_{L^{2p}} \le \|X - Y\|_{L^{2p} …

Let $(x_i)_{i \in \mathbb{N}}$ be a family of independent $\pm 1$ centered Bernoulli random variables, and let $p, q > 1$. There exists a constant C such that for every (finite) linear combination $Y$ …