An Upper Bound on the Error Induced by Saddlepoint Approximations -- Applications to Information Theory

TL;DR

This paper develops a precise upper bound on the error of saddlepoint approximations for sums of i.i.d. variables, and applies it to improve bounds on decoding error probabilities in information theory.

Contribution

It introduces a new upper bound on saddlepoint approximation errors and uses it to derive tighter bounds on error probabilities in communication channels.

Findings

New upper and lower bounds on the DT and MC bounds.

Numerical validation on BSC, AWGN, and symmetric alpha-stable channels.

Enhanced accuracy of error probability estimates in large deviation regimes.

Abstract

This paper introduces an upper bound on the absolute difference between: (a) the cumulative distribution function (CDF) of the sum of a finite number of independent and identically distributed random variables with finite absolute third moment; and (b) a saddlepoint approximation of such CDF. This upper bound, which is particularly precise in the regime of large deviations, is used to study the dependence testing (DT) bound and the meta converse (MC) bound on the decoding error probability (DEP) in point-to-point memoryless channels. Often, these bounds cannot be analytically calculated and thus lower and upper bounds become particularly useful. Within this context, the main results include, respectively, new upper and lower bounds on the DT and MC bounds. A numerical experimentation of these bounds is presented in the case of the binary symmetric channel, the additive white Gaussian noise channel, and the additive symmetric $\alpha$-stable noise channel.