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Open AccessArticle

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

by Dadja Anade 1,‡, Jean-Marie Gorce 1,‡, Philippe Mary 2,‡ and Samir M. Perlaza 3,4,*,‡
1
Laboratoire CITI, a Joint Laboratory between INRIA, the Université de Lyon and the Institut National de Sciences Appliquées (INSA) de Lyon. 6 Av. des Arts, 69621 Villeurbanne, France
2
IETR and the Institut National de Sciences Appliquées (INSA) de Rennes, 20 Avenue des Buttes de Coësmes, CS 70839, 35708 Rennes, France
3
INRIA, Centre de Recherche de Sophia Antipolis—Méditerranée, 2004 Route des Lucioles, 06902 Sophia Antipolis, France
4
Princeton University, Electrical Engineering Department, Princeton, NJ 08544, USA
*
Author to whom correspondence should be addressed.
Parts of this paper appear in the proceedings of the IEEE international Conference on Communications (ICC), Dublin, Ireland, June 2020, and in the INRIA technical report RR-9329.
Authors are listed in alphabetical order.
Entropy 2020, 22(6), 690; https://doi.org/10.3390/e22060690
Received: 12 May 2020 / Revised: 10 June 2020 / Accepted: 11 June 2020 / Published: 20 June 2020
(This article belongs to the Special Issue Wireless Networks: Information Theoretic Perspectives)
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 α -stable noise channel. View Full-Text
Keywords: sums of independent and identically random variables; saddlepoint approximation; memoryless channels sums of independent and identically random variables; saddlepoint approximation; memoryless channels
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Anade, D.; Gorce, J.-M.; Mary, P.; Perlaza, S.M. An Upper Bound on the Error Induced by Saddlepoint Approximations—Applications to Information Theory. Entropy 2020, 22, 690.

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