Large deviations of probability rank

Abstract

Consider a pair of random variables (X,Y) with distribution P. The probability rank function is defined so that G(x|y) = 1 for the most probable outcome x conditional on Y = y, G(x|y) = 2 for the second most probable outcome, and so on, resolving ties between elements with equal probabilities arbitrarily. The function G was considered in [1] in the context of finding the unknown outcome of a random experience by asking question of the form 'Is the outcome equal to x?' sequentially until the actual outcome is determined. The primary focus in [1], and the subsequent works [2], [3], was to find tight bounds on the moments E[G(X|Y)θ]. The present work is closely related to these works but focuses more directly on the large deviations properties of the probability rank function.