[Paper Review] On Asymptotically Tight Tail Bounds for Sums of Geometric and Exponential Random Variables

Yaonan Jin, Yingkai Li+2 more|arXiv (Cornell University)|Feb 7, 2019

Probability and Risk Models4 references4 citations

TL;DR

This paper establishes asymptotically tight exponential tail bounds for sums of independent geometric and exponential random variables using Chernoff bounds and Stirling's approximation. It proves that the derived bounds are optimal in the large-n limit, with the exponent matching the exact logarithmic tail probability up to o(n) terms.

ABSTRACT

In this note we prove bounds on the upper and lower probability tails of sums of independent geometric or exponentially distributed random variables. We also prove negative results showing that our established tail bounds are asymptotically tight.

Motivation & Objective

• To derive sharp upper and lower tail bounds for normalized sums of i.i.d. geometric and exponential random variables.
• To establish the asymptotic tightness of these bounds as the number of variables n tends to infinity.
• To provide simplified approximations of the rate function H(λ,μ) for practical use and comparison with prior work.
• To demonstrate that the leading-order exponent in the tail bounds cannot be improved, confirming optimality.

Proposed method

• Derives tail bounds using the Chernoff bound technique applied to sums of i.i.d. geometric and exponential random variables.
• Introduces the rate function H(λ,μ) = μλlnλ − (1+μλ)ln((1+μλ)/(1+μ)) for geometric sums and G(λ) = λ−1−lnλ for exponential sums.
• Applies Stirling’s approximation to exact binomial and Poisson tail probabilities to derive asymptotic expressions.
• Uses Poisson and binomial counting process characterizations to express tail probabilities exactly before asymptotic analysis.
• Compares the derived bounds with existing results from [Jan18] and [AAGZ17a], showing tighter or more practical forms.
• Proves the tightness of bounds by showing the limit of −lnPr / (n×rate) → 1 as n→∞, confirming optimality of the exponent.

Experimental results

Research questions

• RQ1 Are the derived tail bounds for sums of geometric and exponential random variables asymptotically tight?
• RQ2 How do the rate functions H(λ,μ) and G(λ) compare to existing bounds in terms of tightness and practicality?
• RQ3 Can the exponent in the tail bound be improved beyond the derived H(λ,μ) or G(λ) for large n?
• RQ4 What are the leading-order asymptotic behaviors of the tail probabilities for these sums?
• RQ5 How do the approximations of H(λ,μ) simplify the use of the bounds in applications?

Key findings

• The upper and lower tail probabilities for geometric sums decay as exp{−n·H(λ,μ)} with H(λ,μ) defined as μλlnλ − (1+μλ)ln((1+μλ)/(1+μ)).
• For exponential sums, the tail probabilities decay as exp{−n·G(λ)} with G(λ) = λ−1−lnλ.
• The limit limₙ→∞ [−lnPr / (n·H(λ,μ))] = 1 for both upper and lower tails of geometric sums, proving asymptotic tightness.
• Similarly, limₙ→∞ [−lnPr / (n·G(λ))] = 1 for exponential sums, confirming the optimality of the exponent.
• The remainder terms in the asymptotic expansion of the tail probabilities are O(log n), which are negligible compared to the O(n) leading term.
• Several simplified approximations of H(λ,μ) are provided, such as H(λ,μ) ≥ (μ/2(1+μ))(λ−1)² for λ∈(0,1], showing practical utility.