Contribution to the study of random walks with memory.

Summary In this work, we study several types of walks with memory. We first study *-correlated random walks, i.e. additive functionals of a markov process on a finite state space, for which we establish an invariance theorem. We also explain a method for computing the limit covariance matrix, which we apply to the case of p-correlated markets on zd. Using the techniques of *-correlated markets on zd, we solve the recurrence/transience problem for the canonical random walk on the alternate Manhattan graph td, and we establish an invariance theorem for this walk. In the particular case of 1-correlated walks on z, we obtain a time reversal theorem as well as a result similar to the classical pitman theorem for simple walks. Then, we study the process of increasing magnitude on z by determining more particularly the law of the limit frequencies of visit of each site. Finally, we study the vertex-reinforced random walk on z, partially solving a conjecture of r. Pemantle and s. Volkov concerning the asymptotic behavior of the weights associated with the sites visited by the walk.