On the Sub-Exponential Decay of Detection Error Probabilities in Long Tandems

We consider the problem of decentralized binary hypothesis testing in a network of sensors arranged in a tandem. We show that the rate of error probability decay is always sub-exponential, establishing the validity of a long-standing conjecture. Under the additional assumption of bounded Kullback-Leibler divergences, we show that for all d > 1/2, the error probability is Omega(e^-cnd), where c is a positive constant. Furthermore, the bound Omega(e^-c(logn)d), for all d > 1, holds under an additional mild condition on the distributions. This latter bound is shown to be tight.