Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Method For Bounding Tail Probabilities

Nikola Zlatanov

TL;DR

This work introduces a unified tail-bounding framework for continuous random variables that uses a positive, monotone function $g(x)$ to produce right-tail bounds $1-F(x) \leq -f(x)\frac{g(x)}{g'(x)}$ and left-tail bounds $F(x) \leq f(x)\frac{g(x)}{g'(x)}$, with concrete practical cases and an iterative refinement strategy. It establishes connections to Markov's and Chernoff's bounds via specific choices of $g$ and auxiliary constructions, and provides a systematic method to tighten bounds through iterations. The framework is then applied to the AWGN finite-blocklength problem, deriving new closed-form asymptotic expressions for the converse bound and demonstrating tight numerical performance on Gaussian, beta-prime, and non-central chi-squared tails. The results enable accurate, computationally efficient evaluation of rare-event probabilities and capacity-type bounds without numerical integration, with clear implications for communications and reliability analysis.

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-Verdú. Finally, we provide numerical examples where we show the tightness of the bounds obtained by the proposed method.

A Method For Bounding Tail Probabilities

TL;DR

This work introduces a unified tail-bounding framework for continuous random variables that uses a positive, monotone function to produce right-tail bounds and left-tail bounds , with concrete practical cases and an iterative refinement strategy. It establishes connections to Markov's and Chernoff's bounds via specific choices of and auxiliary constructions, and provides a systematic method to tighten bounds through iterations. The framework is then applied to the AWGN finite-blocklength problem, deriving new closed-form asymptotic expressions for the converse bound and demonstrating tight numerical performance on Gaussian, beta-prime, and non-central chi-squared tails. The results enable accurate, computationally efficient evaluation of rare-event probabilities and capacity-type bounds without numerical integration, with clear implications for communications and reliability analysis.

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 with probability density function , this method requires first setting a continuous, positive, and strictly decreasing function such that is a decreasing and increasing function, , which results in upper and lower bounds, respectively, given in the form , , where is some point. Similarly, for the upper and lower bounds on the left tail probability of , this method requires first setting a continuous, positive, and strictly increasing function such that is an increasing and decreasing function, , which results in upper and lower bounds, respectively, given in the form , . We provide some examples of good candidates for the function . 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-Verdú. Finally, we provide numerical examples where we show the tightness of the bounds obtained by the proposed method.
Paper Structure (35 sections, 18 theorems, 279 equations, 5 figures, 2 algorithms)

This paper contains 35 sections, 18 theorems, 279 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounds On The Right Tail
  4. The General Bounds
  5. Two Special Cases For $g (x)$
  6. Third Special Case For $g (x)$ and Connections to Markov's Inequality and Chernoff's Bound
  7. The Iterative Method For The Right Tail
  8. Bounds On The Left Tail
  9. The General Bounds
  10. A Special Case For $g (x)$
  11. Another Special Case For $g (x)$ and Constructing Left Tail Bounds Similar to Markov's Inequality and Chernoff's Bound
  12. The Iterative Method For The Left Tail
  13. Rate of Convergence
  14. Bounding and Approximating The Converse Bound Of The AWGN Channel Capacity
  15. Problem Formulation
  16. ...and 20 more sections

Key Result

Lemma 1

Let $g (x)$ be any continuous, positive, and strictly decreasing function on a given interval $I$, i.e., $g (x):\;$$g (x)>0$ and $g' (x)< 0$, $\forall x\in I$. For such a $g (x)$, if is a deceasing function on the interval $I$, i.e., if the following holds then the following upper holds Otherwise, if is an increasing function on the interval $I$, i.e., if the following holds then the followin

Figures (5)

  • Figure 1: Rate of convergence for the upper and lower bounds on the right tail of the Gaussian RV with $\sigma=1.9$, $\mu=-1.7$.
  • Figure 2: Rate of convergence for the upper and lower bounds on the right tail of the beta prime RV with $\alpha=2.1$, $\beta=1.3$.
  • Figure 3: Rate of convergence for the upper and lower bounds on the left tail for the non-central chi-squared RV with $k=10$, $\lambda=2$.
  • Figure 4: Comparison of converse bounds with with bounds and approximatons for $\Omega=1$ and $\epsilon -10^{-3}$
  • Figure 5: Comparison of converse bounds with with bounds and approximatons for $\Omega=5$ and $\epsilon =10^{-5}$

Theorems & Definitions (21)

  • Lemma 1
  • Theorem 1
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 11 more