Recurrent random walks on homogeneous spaces

1986 ◽  
pp. 146-152
Author(s):  
R. Schott
Keyword(s):  
Random Walks ◽  
Homogeneous Spaces

scholarly journals Limit profiles for reversible Markov chains

2021 ◽  
Author(s):  
Evita Nestoridi ◽  
Sam Olesker-Taylor
Keyword(s):  
Random Walk ◽  
Markov Chains ◽  
Random Walks ◽  
Gibbs Sampler ◽  
Statistical Physics ◽  
Homogeneous Spaces ◽  
New Method ◽  
Reversible Markov Chains ◽  
Similar Approximation ◽  
Random Transpositions

AbstractIn a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the k-cycle shuffle, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli  (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008).

Sharp ergodic theorems for group actions and strong ergodicity

1999 ◽  
Vol 19 (4) ◽  
pp. 1037-1061 ◽  
Author(s):  
ALEX FURMAN ◽  
YEHUDA SHALOM
Keyword(s):  
Probability Measure ◽  
Random Walks ◽  
Homogeneous Spaces ◽  
Locally Compact Group ◽  
Invariant Mean ◽  
Spectral Condition ◽  
Ergodic Theorems ◽  
Averaging Operator ◽  
The Mean ◽  
Probability Measure Space

Let $\mu$ be a probability measure on a locally compact group $G$, and suppose $G$ acts measurably on a probability measure space $(X,m)$, preserving the measure $m$. We study ergodic theoretic properties of the action along $\mu$-i.i.d. random walks on $G$. It is shown that under a (necessary) spectral assumption on the $\mu$-averaging operator on $L^2(X,m)$, almost surely the mean and the pointwise (Kakutani's) random ergodic theorems have roughly $n^{-1/2}$ rate of convergence. We also prove a central limit theorem for the pointwise convergence. Under a similar spectral condition on the diagonal $G$-action on $(X\times X,m\times m)$, an almost surely exponential rate of mixing along random walks is obtained.The imposed spectral condition is shown to be connected to a strengthening of the ergodicity property, namely, the uniqueness of $m$-integration as a $G$-invariant mean on $L^\infty(X,m)$. These related conditions, as well as the presented sharp ergodic theorems, never occur for amenable $G$. Nevertheless, we provide many natural examples, among them automorphism actions on tori and actions on Lie groups' homogeneous spaces, for which our results can be applied.

Random walks on finite volume homogeneous spaces

2011 ◽  
Vol 187 (1) ◽  
pp. 37-59 ◽  
Author(s):  
Yves Benoist ◽  
Jean-Francois Quint
Keyword(s):  
Random Walks ◽  
Finite Volume ◽  
Homogeneous Spaces
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