On the random coding exponent of nonlinear gaussian channels

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

“…(2.4.8) and (2.4.9) in [16]). We show in the following that Proposition 4 suggests an improvement over the bound in (68) (that is introduced in [16, Eq. (2.4.10)]).…”

“…The improvement of the exponent (70) over the exponent in [16, Eq. (2.4.10)]) (see (68)) holds since Theorem 5: Let {S n , F n } ∞ n=0 be a discrete-parameter realvalued martingale such that S 0 = 0, and Y k S k − S k−1 ≤ 1 a.s. for every k ∈ N. Let us define the random variables…”

“…from an unknown distribution, but the alphabet size and the source distribution both scale with the block length (so the empirical distribution does not converge to the true distribution as the block length tends to infinity). In another recent work [68], Azuma's inequality was used to derive achievable rates and random coding error exponents for non-linear additive white Gaussian noise channels. This was followed by another work of the same authors [69] who used some other concentration inequalities, for discrete-parameter martingales with bounded jumps, to derive achievable rates and random coding error exponents for non-linear Volterra channels (where their bounding technique can be also applied to intersymbol-interference (ISI) channels, as was noted in [69]).…”