An asymptotically optimal bound for the concentration function of a sum of independent integer random variables

Abstract

For a random variable $X$ define $Q(X) = \sup_{x \in \mathbb{R}} \mathbb{P}(X=x)$. Let $X_1, \dots, X_n$ be independent integer random variables. Suppose $Q(X_i) \le \alpha_i \in (0,1]$ for each $i \in \{1, \dots, n\}$. Ju\v{s}kevi\v{c}ius (2023) conjectured that $Q(X_1 + \dots +X_n) \le Q(Y_1 + \dots+ Y_n)$ where $Y_1, \dots, Y_n$ are independent and $Y_i$ is a random integer variable with $Q(Y_i) =\alpha_i$ that has the smallest variance, i.e. the distribution of $Y_i$ has probabilities $\alpha_i, \dots, \alpha_i, \beta_i$ or probabilities $\beta_i, \alpha_i, \dots, \alpha_i$ on some interval of integers, where $0 \le \beta_i < \alpha_i$. We prove this conjecture asymptotically: i.e., we show that for each $\delta > 0$ there is $V_0 = V_0(\delta)$ such that if ${\mathrm Var} (\sum Y_i) \ge V_0$ then $Q(\sum X_i) \le (1+\delta) Q(\sum Y_i)$. This implies an analogous asymptotically optimal inequality for concentration at a point when $X_1$, $\dots$, $X_n$ take values in a separable Hilbert space. Our long and technical argument relies on several non-trivial previous results including an inverse Littlewood--Offord theorem and an approximation in total variation distance of sums of multivariate lattice random vectors by a discretized Gaussian distribution.