On Refined Versions of the Azuma-Hoeffding Inequality with Applications in Information Theory

Abstract

This paper derives some refined versions of the Azuma-Hoeffding inequality for discrete-parameter martingales with uniformly bounded jumps, and it considers some of their potential applications in information theory and related topics. The first part of this paper derives these refined inequalities, followed by a discussion on their relations to some classical results in probability theory. It also considers a geometric interpretation of some of these inequalities, providing an insight on the interconnections between them. The second part exemplifies the use of these refined inequalities in the context of hypothesis testing and information theory, communication, and coding theory. The paper is concluded with a discussion on some directions for further research. This work is meant to stimulate the use of some refined versions of the Azuma-Hoeffding inequality in information-theoretic aspects.

Moderate Deviations in Channel Coding

“…160]) R cr < C and hence for all sufficiently large n, C − δ n > R cr . This observation, coupled with (36), ensures that for all sufficiently large n, we have…”

“…Polyanskiy and Verdú [34] improved the result in [29] by relaxing the positivity assumption and extending it to Gaussian channels, among other contributions. More recently, moderate deviations in lossy source coding and hypothesis testing problems have been investigated by Tan [35] and Sason [36], respectively.…”

“…P n and ρ n achieve the maxima in(36) at rate C − δ n , i.e.,E SP (C − δ n , W ) = −ρ n (C − δ n ) + E o (ρ n , P n ). Now E SP (C − δ n , W ) > 0 for all n, which is evident from (4).…”

Tight finite-key security for twin-field quantum key distribution

“…The main tool we use to relate each of the above average terms to their actual occurrences, N ph and M nm , is a refined version of Azuma's inequality [44,45]. Azuma's inequality [46] is widely used in security analyses of QKD to bound sums of observables over a set of rounds of the protocol (in our case, the set of successful rounds after sifting), when the independence between the observables corresponding to different rounds cannot be guaranteed.…”

“…Here, ∆ nm is a statistical fluctuation term that depends on the specific method used. For (n, m) = (0, 0), we use the refined Azuma's inequality [44,45], for which ∆ nm = ∆ = 5 4 √ 2…”