A Method For Bounding Tail Probabilities

Résumé : 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 $f_X(x)$, this method requires first setting a continuous, positive, and strictly decreasing function $g_X(x)$ such that $-f_X(x)/g'_X(x)$ is a decreasing and increasing function, $\forall x>x_0$, which results in upper and lower bounds, respectively, given in the form $-f_X(x) g_X(x)/g'_X(x)$, $\forall x>x_0$, 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 $g_X(x)$ such that $f_X(x)/g'_X(x)$ is an increasing and decreasing function, $\forall x<x_0$, which results in upper and lower bounds, respectively, given in the form $f_X(x) g_X(x)/g'_X(x)$, $\forall x<x_0$. We provide some examples of good candidates for the function $g_X(x)$. 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. Finally, we provide numerical examples, where we show the tightness of these bounds, for some chosen $g_X(x)$.