32 research outputs found
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Rough invariance principle for delayed regenerative processes
We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the p-variation settings, where a rough Donsker Theorem is available under the second moment condition. The key renewal theorem is applied to obtain an optimal moment condition
Zero-one law for directional transience of one-dimensional random walks in dynamic random environments
We prove the trichotomy between transience to the right, transience to the
left and recurrence of one-dimensional nearest-neighbour random walks in
dynamic random environments under fairly general assumptions, namely:
stationarity under space-time translations, ergodicity under spatial
translations, and a mild ellipticity condition. In particular, the result
applies to general uniformly elliptic models and also to a large class of
non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An
immediate consequence is the recurrence of models that are symmetric with
respect to reflection through the origin.Comment: 14 pages, 1 figure. Added Corollary 2.
Linear -additivity and some applications
We show that countable increasing unions preserve a large family of
well-studied covering properties, which are not necessarily sigma-additive.
Using this, together with infinite-combinatorial methods and simple forcing
theoretic methods, we explain several phenomena, settle problems of Just,
Miller, Scheepers and Szeptycki [COC2], Gruenhage and Szeptycki [FUfin], Tsaban
and Zdomskyy [SFT], and Tsaban [o-bdd, OPiT], and construct topological groups
with very strong combinatorial properties