Maximum likelihood estimator consistency for recurrent random walk in a parametric random environment with finite support

We consider a one-dimensional recurrent random walk in random environment (RWRE) when the environment is i.i.d. with a parametric, finitely supported distribution. Based on a single observation of the path, we provide a maximum likelihood estimation procedure of the parameters of the environment. Unlike most of the classical maximum likelihood approach, the limit of the criterion function is in general a nondegenerate random variable and convergence does not hold in probability. Not only the leading term but also the second order asymptotics is needed to fully identify the unknown parameter. We present different frameworks to illustrate these facts. We also explore the numerical performance of our estimation procedure

Spectral gaps and exponential integrability of hitting times for linear diffusions

Let X be a regular continuous positively recurrent Markov process with state space R, scale function S and speed measure m. For a ∈ R denote B+a = sup x≥a m(]x,+∞[)(S(x) − S(a)) B−a = sup x≤a m(]−∞;x[)(S(a) − S(x)) It is well known that the finiteness of B±a is equivalent to the existence of spectral gaps of generators associated with X. We show how these quantities appear independently in the study of the exponential moments of hitting times of X. Then we establish a very direct relation between exponential moments and spectral gaps, all by improving their classical bounds

Maximum likelihood estimator consistency for ballistic random walk in a parametric random environment

International audienceWe consider a one dimensional ballistic random walk evolving in an i.i.d. parametric random environment. We provide a maximum likelihood estimation procedure of the environment parameters based on a single observation of the path till the time it reaches a distant site, and prove that this estimator is consistent as the distant site tends to infinity. We also explore the numerical performances of our estimation procedure

Limit of the environment viewed from Sinaï's walk

For Sinai's walk we show that the empirical measure of the environment seen from the particle converges in law to some random measure. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov (1984). As a consequence an "in law" ergodic theorem holds. When the limit is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov (1984)