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Partial progress: here is a proof for the analogous "Gaussian case" (i.e., $Z\sim N(\mathbf{0}_d, \mathbf{I}_d)$) and $n=2$. … gamma^2}{2}\lVert X+Y \rVert_2^2} } }\right] $$ (the denominator changes because it comes from $\mathbb{E}_Z[e^{\gamma \langle\sum_{j=1}^n X^{(j)},Z\rangle}]$, and that expression changes between the Gaussian …

Suppose $X_1,\dots,X_n$ are independent Gaussians, where $X_k \sim N(\mu_k,1)$. I am interested in $$ Z \stackrel{\rm def}{=} \max_{1\leq k\leq n} X_k $$ and specifically on the asymptotics of $\mathb …

j)},Z\rangle)]^2}{\prod_{i=1}^d \cosh(\gamma\sum _{j=1}^nX^{(j)}_i)}\right]\tag{1}$$ where $\gamma\in(0,1]$, $d\gg 1$, $X=(X^{(j)})_{1\leq j\leq n}$ is a collection of i.i.d. standard $d$-dimensional Gaussian …

Let $X$ be a (sub)gaussian r.v. on $\mathbb{R}^d$; say $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$; and let $a\colon\mathbb{R}^d\to [0,1]$ be a function with $\mathbb{E}[a(X)] > 0$. …