Csiszár's forward cutoff rate for testing between two arbitrary sources

The Csiszár forward β-cutoff rate (β < 0) for hypothesis testing is defined as the largest rate R₀ ≥ 0 such that for all rates 0 < E < R₀, the smallest probability of type 1 error of sample size-n tests with probability of type 2 error ≤ e^-nE is asymptotically vanishing as e^-nβ(E-R0). It was shown by Csiszár that the forward β-cutoff rate for testing between a null hypothesis X̄ against an alternative hypothesis X based on independent and identically distributed samples, is given by Rényi's α-divergence D_α(X∥X̄), where α = 1/(1 - β). In this work, we show that the forward β-cutoff rate for the general hypothesis testing problem is given by the lim inf α-divergence rate. The result holds for an arbitrary abstract alphabet (not necessarily countable).