Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables

TL;DR

This paper derives exact upper bounds for the expected values of nonincreasing functions of sums of independent nonnegative random variables, leading to improved bounds on left-tail probabilities using binomial, Poisson, and normal distributions.

Contribution

It introduces new, sharper bounds on tail probabilities and exponential moments for sums of nonnegative independent variables, extending previous results with broader function classes and settings.

Findings

New bounds are tighter than previous estimates.

Bounds are expressed via specific distributions like binomial, Poisson, and normal.

Applicable to various fixed and variable parameter settings.

Abstract

Let $X_1,\dots,X_n$ be independent nonnegative random variables (r.v.'s), with $S_n:=X_1+\dots+X_n$ and finite values of $s_i:=E X_i^2$ and $m_i:=E X_i>0$. Exact upper bounds on $E f(S_n)$ for all functions $f$ in a certain class $\mathcal{F}$ of nonincreasing functions are obtained, in each of the following settings: (i) $n,m_1,\dots,m_n,s_1,\dots,s_n$ are fixed; (ii) $n$, $m:=m_1+\dots+m_n$, and $s:=s_1+\dots+s_n$ are fixed; (iii)~only $m$ and $s$ are fixed. These upper bounds are of the form $E f(\eta)$ for a certain r.v. $\eta$. The r.v. $\eta$ and the class $\mathcal{F}$ depend on the choice of one of the three settings. In particular, $(m/s)\eta$ has the binomial distribution with parameters $n$ and $p:=m^2/(ns)$ in setting (ii) and the Poisson distribution with parameter $\lambda:=m^2/s$ in setting (iii). One can also let $\eta$ have the normal distribution with mean $m$ and variance $s$ in any of these three settings. In each of the settings, the class $\mathcal{F}$ contains, and is much wider than, the class of all decreasing exponential functions. As corollaries of these results, optimal in a certain sense upper bounds on the left-tail probabilities $P(S_n\le x)$ are presented, for any real $x$. In fact, more general settings than the ones described above are considered. Exact upper bounds on the exponential moments $E\exp\{hS_n\}$ for $h<0$, as well as the corresponding exponential bounds on the left-tail probabilities, were previously obtained by Pinelis and Utev. It is shown that the new bounds on the tails are substantially better.