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Tail bounds for sums of independent two-sided exponential random variables

Jiawei Li, Tomasz Tkocz

TL;DR

This work extends Janson's tail bounds for sums of independent exponentials to the symmetric two-sided (Laplace) setting by establishing tight upper and lower bounds for S = ∑ a_i X_i with X_i ~ Laplace and a_i > 0. The upper bound uses a Chernoff argument with the mgf leading to a bound of the form exp{- (α^2/2) h(2t/α)} where α = sqrt{Var(S)}/a_*, while the lower bound leverages a Gaussian-mixture representation of X_i to produce matching leading-order behavior e^{-α t} (up to a 1/√t factor). The paper also presents a general nonnegative-distribution framework that recovers and extends Janson-type bounds for exponential and gamma cases, discusses moments, S-inequalities, and the impact of heavy tails, and shows the main result fits into a broader mixture-based paradigm. These results deepen understanding of how the tail of a weighted sum with dominant weight compares to the tail of a single large summand and provide tools for moment and tail analysis in related settings.

Abstract

We establish upper and lower bounds with matching leading terms for tails of weighted sums of two-sided exponential random variables. This extends Janson's recent results for one-sided exponentials.

Tail bounds for sums of independent two-sided exponential random variables

TL;DR

This work extends Janson's tail bounds for sums of independent exponentials to the symmetric two-sided (Laplace) setting by establishing tight upper and lower bounds for S = ∑ a_i X_i with X_i ~ Laplace and a_i > 0. The upper bound uses a Chernoff argument with the mgf leading to a bound of the form exp{- (α^2/2) h(2t/α)} where α = sqrt{Var(S)}/a_*, while the lower bound leverages a Gaussian-mixture representation of X_i to produce matching leading-order behavior e^{-α t} (up to a 1/√t factor). The paper also presents a general nonnegative-distribution framework that recovers and extends Janson-type bounds for exponential and gamma cases, discusses moments, S-inequalities, and the impact of heavy tails, and shows the main result fits into a broader mixture-based paradigm. These results deepen understanding of how the tail of a weighted sum with dominant weight compares to the tail of a single large summand and provide tools for moment and tail analysis in related settings.

Abstract

We establish upper and lower bounds with matching leading terms for tails of weighted sums of two-sided exponential random variables. This extends Janson's recent results for one-sided exponentials.

Paper Structure

This paper contains 11 sections, 6 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:sym-exp']}
  3. Generalisations
  4. Examples
  5. Proof of Theorem \ref{['thm:gene']}: the upper bound
  6. Proof of Theorem \ref{['thm:gene']}: the lower bound
  7. Further remarks
  8. Moments
  9. Upper bounds on upper tails from S-inequalities
  10. Heavy-tailed distributions
  11. Theorem \ref{['thm:sym-exp']} in a more general framework

Key Result

Theorem 1

Let $X_1, \ldots, X_n$ be independent standard two-sided exponential random variables (i.e. with density $\frac{1}{2}e^{-|x|}$, $x \in \mathbb{R}$). Let $S = \sum_{i=1}^n a_iX_i$ with $a_1,\ldots,a_n$ positive. For every $t > 1$, where $\alpha = \frac{\sqrt{\mathop{\mathrm{Var}}\nolimits(S)}}{\max_{i \leq n} a_i} = \frac{\sqrt{2\sum_{i=1}^n a_i^2}}{\max_{i \leq n} a_i}$, $h(u) = \sqrt{1+u^2}-1-\

Theorems & Definitions (10)

  • Theorem 1
  • Proposition 2
  • Lemma 3
  • proof
  • Theorem 4
  • Lemma 5
  • proof
  • Theorem 6: Gluskin and Kwapień, GK
  • proof
  • Remark 7