Paper detail

A Method For Bounding Tail Probabilities

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 $-f (x)/g&#39; (x)$ is a decreasing and increasing function, $\forall x>x_0$, which results in upper and lower bounds, respectively, given in the form $-f (x) g (x)/g&#39; (x)$, $\forall x>x_0$, 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 $f (x)/g&#39; (x)$ is an increasing and decreasing function, $\forall x<x_0$, which results in upper and lower bounds, respectively, given in the form $f (x) g (x)/g&#39; (x)$, $\forall x<x_0$. We provide some examples of good candidates for the function $g (x)$. We also establish connections between the new bounds and Markov&#39;s inequality and Chernoff&#39;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-Verdú. Finally, we provide numerical examples where we show the tightness of the bounds obtained by the proposed method.