Limits and rates of convergence for the distribution of search cost under the move-to-front rule

Abstract

We derive upper and lower bounds on total variation distance to stationarity for the distribution of search cost under the move-to-front (MTF) rule for self-organizing lists with i.i.d. record requests. These enable us to obtain sharp rates of convergence for several standard examples of weights, including Zipf's law and geometric weights, as the length of the list becomes large. The upper bound also shows that a number of moves of the order of the length of the list is uniformly sufficient for near-stationarity over all choices of weights. Concerning the stationary search cost distribution itself, we use a representation obtained by considering the continuized MTF Markov chain to derive, for each of the standard examples, the asymptotic distribution for long lists.