Csiszár's cutoff rates for the general hypothesis testing problem

In [6], Csiszár established the concept of forward β-cutoff rate for the error exponent hypothesis testing problem based on independent and identically distributed (i.i.d.) observations. Given β < 0, he defined the forward β-cutoff rate as the number R₀ ≥ 0 that provides the best possible lower bound in the form β(E - R_o) to the type 1 error exponent function for hypothesis testing where 0 < E < R₀ is the rate of exponential convergence to 0 of the type 2 error probability. He then demonstrated that the forward β-cutoff rate is given by D_1/(1-β) (X∥X̄), where D_α(X∥X̄) denotes the Rényi α-divergence [19], α > 0, α ≠ 1. Similarly, for 0 < β < 1, Csiszár also established the concept of reverse β-cutoff rate for the correct exponent hypothesis testing problem. In this work, we extend Csiszár's results by investigating the forward and reverse β-cutoff rates for the hypothesis testing between two arbitrary sources with memory. We demonstrate that the lim inf Rényi α-divergence rate provides the expression for the forward β-cutoff rate. We also show that if the log-likelihood large deviation spectrum admits a limit, then the reverse β-cutoff rate equals the liminf α-divergence rate, where α = 1/1-β and 0 < β < β_max, where β_max is the largest β < 1 for which the lim inf 1/1-β-divergence rate is finite. For β_max ≤ β ≤ 1, we show that the reverse cutoff rate is in general only upper-bounded by the lim inf Rényi divergence rate. Unlike in [4], where the alphabet for the source coding cutoff rate problem was assumed to be finite, we assume arbitrary (countable or continuous) source alphabet. We also provide several examples to illustrate our forward and reverse β-cutoff rates results and the techniques employed to establish them.