Large deviations of probability rank

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 Arikan (1996) in the context of finding the unknown outcome of a random experiment by asking questions of the form `is the outcome equal to x?´ sequentially until the actual outcome is determined. The primary focus in Arakan (1996) and Arakan and Merhav (1998) 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