8 problems

Let $X_1,\ldots,X_{n+1}$ be independent and identically distributed random variables taking values in a finite subset of a torsion-free group, and let $H$ denote discrete Shannon e…

Concavity conjecture. The function $f$ is concave on $[0,1]$. This conjecture proposes a stronger form of regularity for entropy under normalized sums of independent identically di…

Let $(X,Y)$ be a symmetric log-concave random vector in $\mathbb{R}^2$; in particular, $X$ and $Y$ may be independent symmetric real random variables. Write $N^{\frac12}$ for the R…

Let $(X,Y)$ be a symmetric $\kappa$-concave random vector in $\mathbb{R}^2$, and suppose that $X$ and $Y$ have equal $p$-Rényi entropy. Convexity conjecture for Rényi entropy. The…

Recall that a probability measure on $\mathbb{R}^n$ is $\kappa$-concave if it satisfies the Brunn–Minkowski-type inequality associated with $\kappa$; a random vector is $\kappa$-co…

Let ${\boldsymbol X} \equiv \{X_i\}$ be i.i.d. arbitrarily distributed random variables with mean $\lambda$, and let … The preceding central limit theorem states that the p.m.f. of…

Auxiliary-variable conjecture. There exists a random variable $V$ such that $Y_1\to Y_0\to V$ and

Sharp two-summand Rényi entropy power conjecture.