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

Valentas Kurauskas

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 $Q(X_i) \le α_i \in (0,1]$ for each $i \in \{1, \dots, n\}$. Juškevičius (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 $Q(Y_i) =α_i$ that has the smallest variance, i.e. the distribution of $Y_i$ has probabilities $α_i, \dots, α_i, β_i$ or probabilities $β_i, α_i, \dots, α_i$ on some interval of integers, where $0 \le β_i < α_i$. We prove this conjecture asymptotically: i.e., we show that for each $δ> 0$ there is $V_0 = V_0(δ)$ such that if ${\mathrm Var} (\sum Y_i) \ge V_0$ then $Q(\sum X_i) \le (1+δ) Q(\sum Y_i)$. 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.

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

Abstract

For a random variable define . Let be independent integer random variables. Suppose for each . Juškevičius (2023) conjectured that where are independent and is a random integer variable with that has the smallest variance, i.e. the distribution of has probabilities or probabilities on some interval of integers, where . We prove this conjecture asymptotically: i.e., we show that for each there is such that if then . This implies an analogous asymptotically optimal inequality for concentration at a point when , , 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.
Paper Structure (14 sections, 33 theorems, 270 equations)

This paper contains 14 sections, 33 theorems, 270 equations.

Table of Contents

  1. Introduction
  2. Part I: Main proof
  3. Definitions
  4. Useful lemmas
  5. ε-dominated concentration functions
  6. Combining the proofs for the balanced case
  7. Entry point: Proof of Theorem \ref{['thm.tse']}
  8. Part II: Proving Lemma \ref{['lem.less_peaked_medium']}
  9. Approximation by discretized multivariate normal distribution
  10. Projections of a smooth multivariate distribution
  11. Applying inverse Littlewood--Offord
  12. A theorem of Odlyzko and Richmond
  13. Proof of Lemma \ref{['lem.less_peaked_medium']}
  14. Appendix

Key Result

Theorem 1.1

For any $\delta > 0$ there exists a number $V_0(\delta)$ such that the following holds. Let $X_1, \dots, X_n$ be a sequence of independent integer random variables such that $\mathcal{Q}(X_i) \le \alpha_i$, $\alpha_i \in (0,1]$ and let $S_n = \sum_{i=1}^n X_i$. Let $Y_1, \dots, Y_n$ be independent, then for some $a_1, \dots, a_n \in \{-1, 1\}$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Corollary 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7: aravindabobkovmarsigliettimelbourne
  • ...and 30 more