Csiszar´s forward cutoff rate for testing between two arbitrary sources

The Csisza forward β-cutoff rate (β<0) for hypothesis testing is defined as the largest rate R₀≥0 such that for all rates 00, 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 Csiszar (see IEEE Transactions on Information Theory, vol.41, p.26-34, January 1995) that the forward β-cutoff rate for testing between a hypothesis X against an alternative hypothesis X~ based on independent and identically distributed samples, is given by Renyi´s α-divergence D_α(X||X~), 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).