Ph.D Student | Alexander Roitershtein |
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Subject | Random Walks in Random Environments |

Department | Department of Applied Mathematics |

Supervisors | Professor Mayer-Wolf Eddy |

Full Professor Zeitouni Ofer |

his thesis deals with limit theorems for one-dimensional random walks in random environment (RWRE). We obtain non-Gaussian limit laws for transient random walks in semi-Markov environments. The main result is applicable also to some other environments that have a regenerative property, e.g. induced by certain Gibbs states. This line of research has its roots in the seminal work of Kesten, M. Kozlov and Spitzer who treated random walks in i.i.d. environments.

The limit laws for the random walks are derived from stable limit laws for its hitting times, and we direct our efforts to establish the latter. The proof is by an approximation of the distribution of the hitting times by that of partial sums of one-dependent random variables to which a general stable limit theorem is applied. The approximation is based on a duality introduced by Kesten, M. Kozlov and Spitzer between the RWRE and certain branching processes with immigration in a random environment (BPRE).

The required tail
estimates for the branching process follow from a regular variation of the
distribution tail of the random variable_{} with Markov-dependent real
coefficients_{}.
An important part of the work is an extension of a theorem of Kesten that
establishes the regular variation of the distribution tail of *R*.

In addition, using a method based on a study of the generating function of the associated branching process and a large deviation analysis of the tail, we prove a ``log-scale" limit laws for strongly mixing environments and transient random walks with zero asymptotic speed.