Move-to-front rule is a heuristic updating a list of n items according to requests. Items are required with unknown probabilities (or ppopularities). The induced Markov chain is known to be ergodic . One main problem is the study of the distribution of the search cost defined as the position of the required item. Here we first establish the link between two recent papers [3, 8] that both extend results proved by Kingman  on the expected stationary search cost. Combining results contained in these papers, we obtain the limiting behavior for any moments of the stationary seach cost as n tends to infinity.