A Method For Bounding Tail Probabilities

Abstract

We present a method for upper and lower bounding the right and the left tail probabilities of continuous random variables (RVs). For the right tail probability of RV $X$ with probability density function $f (x)$, this method requires first setting a continuous, positive, and strictly decreasing function $g (x)$ such that $-f (x)/g' (x)$ is a decreasing and increasing function, $\forall x>x_0$, which results in upper and lower bounds, respectively, given in the form $-f (x) g (x)/g' (x)$, $\forall x>x_0$, where $x_0$ is some point. Similarly, for the upper and lower bounds on the left tail probability of $X$, this method requires first setting a continuous, positive, and strictly increasing function $g (x)$ such that $f (x)/g' (x)$ is an increasing and decreasing function, $\forall x<x_0$, which results in upper and lower bounds, respectively, given in the form $f (x) g (x)/g' (x)$, $\forall x<x_0$. We provide some examples of good candidates for the function $g (x)$. We also establish connections between the new bounds and Markov's inequality and Chernoff's bound. In addition, we provide an iterative method for obtaining ever tighter lower and upper bounds, under certain conditions. As an application, we use the proposed method to derive a novel closed-form asymptotic expression of the converse bound on the capacity of the additive white Gaussian noise (AWGN) channel in the finite-blocklength regime, which is tighter than the closed-form asymptotic expression by Polyanskiy-Poor-Verd\'u. Finally, we provide numerical examples where we show the tightness of the bounds obtained by the proposed method.