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

TL;DR

This paper proves an asymptotic bound for the maximum point probability of sums of independent integer random variables, confirming a conjecture about minimal concentration under variance constraints, with implications for Hilbert space-valued variables.

Contribution

It establishes an asymptotic version of Juškevičius's conjecture, showing the minimal concentration bound holds when the sum's variance is sufficiently large.

Findings

Proves the conjecture asymptotically for large variance sums.

Extends the result to Hilbert space-valued random variables.

Uses advanced probabilistic and combinatorial tools like inverse Littlewood–Offord theorems.

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.