Weak quenched limiting distributions for transient one-dimensional random walk in a random environment [article]

Jonathon Peterson, Gennady Samorodnitsky
2016 arXiv   pre-print
We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter κ>0 that determines the fluctuations of the process. When 0<κ<2, the averaged distributions of the hitting times of the random walk converge to a κ-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true
more » ... t for almost every fixed environment, the distributions of the hitting times (centered and scaled in any manner) converge to a non-degenerate distribution. We show, however, that the quenched distributions do have a limit in the weak sense. That is, the quenched distributions of the hitting times -- viewed as a random probability measure -- converge in distribution to a random probability measure, which has interesting stability properties. Our results generalize both the averaged limiting distribution and the non-existence of quenched limiting distributions.
arXiv:1011.6366v4 fatcat:6uhvuc7375a2zp3x6tdyucm3xi