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Let $S_n=\tau_1+\cdots+\tau_n$ be a sum of independent Bernoulli random variables such that $\mathbb{P}(\tau_i=1)=p_i$. Is it true that the mode of $S_n$ is either its mean rounded up or rounded down? …

Let $X,Y$ be two centered Gaussian random variables each with variance at most $1$. Note that we do not assume independence. I would like to minimize $$\mathbb{P}(|X|\leq 1, |Y|\leq 1).$$ Is it true t …

The celebrated Berry-Esseen inequality tells us that the rate of convergence in the univariate CLT is of magnitude $\frac{1}{\sqrt{n}}$ for sums $S_n=X_1+\cdots+X_n$ of independent random variables $X …

Let $Z_1,\ldots, Z_n$ be independent standard gaussian random variables. Is it true that $X=\max\{Z_1,\ldots,Z_n\}$ has a log-concave distribution function?

Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in …

Let $Z_1,\ldots, Z_n$ be standardized Gaussian random variables and denote $\rho_{ij}=\mathbb{E}Z_iZ_j$. Can one give an asymptotically sharp bound for $$\mathbb{P}\,(\max_{1\leq i\leq n}Z_i>x), \quad …

Let $S$ be the class of all $2$ by $2$ matrices of the form $$\begin{bmatrix} 1 & a \\ a & 1 \end{bmatrix},\, |a|\leq 1.$$ Is there a single matrix $M\in S$ such that for any $N\ …

Let $X_1,\ldots,X_n$ be independent Bernoulli random variables such that $\mathbb{P}(X_i=\pm 1)=1/2$ and consider two collections of real numbers $a_1,\ldots,a_n, b_1,\ldots, b_n$. For the moment let …