An asymptotically optimal bound for the concentration function of a sum of independent integer random variables
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| Format: | Preprint |
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2026
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| _version_ | 1866917333220458496 |
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| author | Kurauskas, Valentas |
| author_facet | Kurauskas, Valentas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11043 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An asymptotically optimal bound for the concentration function of a sum of independent integer random variables Kurauskas, Valentas Probability Combinatorics 60C05 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. |
| title | An asymptotically optimal bound for the concentration function of a sum of independent integer random variables |
| topic | Probability Combinatorics 60C05 |
| url | https://arxiv.org/abs/2603.11043 |