An upper bound on the error induced by saddlepoint approximations: Applications to information theory

In this talk we discuss a new upper-bound on the absolute difference between: (a) the cumulative distribution function (c.d.f.) of the sum of a finite number of independent and identically distributed (i.i.d) random variables; and (b) a saddlepoint approximation of such c.d.f. This upperbound is general and particularly precise in the regime of large deviations. This result is of the same kind of the Berry-Esseen theorem (BET) that provides an upper bound on the absolute difference between the c.d.f. of the sum of a finite number of i.i.d random variables and the c.d.f. of a Gaussian random variable whose mean and variance are the same as those of the sum. The BET has played a central role in recent developments in information theory, in particular in source and channel coding. Nonetheless, it often leads to loose bounds on the absolute difference mentioned above. In the light of this observation, the new result is used to study the dependence testing (DT) bound on the minimum decoding error probability (DEP) in memoryless channels. Within this context, an upper bound on the absolute difference between the exact value of the DT bound and its saddlepoint approximation is obtained. Numerical analysis of these bounds are presented for the case of the binary symmetric channel and the additive white Gaussian noise channel, in which the new bounds are observed to be several orders of magnitude better than those obtained with the BET.

Joint work with Dadja Anade, Jean-Marie Gorce and Philippe Mary.