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We study the asymptotic behaviour of random walks on topological abelian groups $G$. Our main result is a sufficient condition for one random walk to overtake another in the stochastic order induced by any suitably large positive cone $G_+ subseteq G$, assuming that both walks have Radon distributions and compactly supported steps. We explain in which sense our sufficient condition is very close to a necessary one. Our result is a direct application of a recently proven theorem of real algebra, namely a Positivstellensatz for preordered semirings. It is due to Aubrun and Nechita in the one-dimensional case, but new already for $R^n$ with $n > 1$. We use our result to derive a formula for the rate at which the probabilities of a random walk decay relative to those of another, again for walks on $G$ with compactly supported Radon steps. In the case where one walk is a constant, this formula specializes to a version of Cramers large deviation theorem.
In this article we prove existence of the asymptotic entropy for isotropic random walks on regular Fuchsian buildings. Moreover, we give formulae for the asymptotic entropy, and prove that it is equal to the rate of escape of the random walk with res
In this article we consider transient random walks on HNN extensions of finitely generated groups. We prove that the rate of escape w.r.t. some generalised word length exists. Moreover, a central limit theorem with respect to the generalised word len
Random walks on a group $G$ model many natural phenomena. A random walk is defined by a probability measure $p$ on $G$. We are interested in asymptotic properties of the random walks and in particular in the linear drift and the asymptotic entropy. I
In this short note we give various near optimal characterizations of random walks over finite Abelian groups with large maximum discrepancy from the uniform measure. We also provide several interesting connections to existing results in the literature.
We study the frog model on Cayley graphs of groups with polynomial growth rate $D geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one