Asymptotic Behavior of Sequence Models

In this paper we study the limiting dynamics of a sequential process that generalizes Polya's urn.  This process has been studied also in the context of language generation, discrete choice, repeat consumption, and models for the web graph.  The process we study generates future items by copying from past items.  It is parameterized by a sequence of weights describing how much to prefer copying from recent versus more distant locations.  We show that, if the weight sequence follows a power law with exponent $\alpha \in [0,1)$, then the sequences generated by the model tend toward a limiting behavior in which the eventual frequency of each token in the alphabet attains a limit. Moreover, in the case $\alpha > 2$, we show that the sequence converges to a token being chosen infinitely often, and each other token being chosen only constantly many times.