Ratio Limit Theorems for Random Walks in Homogeneous Spaces II

1990 ◽  
Vol 34 (3) ◽  
pp. 464-473 ◽  
Author(s):  
M. G. Shur
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Homogeneous Spaces ◽  
Ratio Limit Theorems ◽  
Ratio Limit

scholarly journals Spread out random walks on homogeneous spaces

2020 ◽  
pp. 1-35
Author(s):  
ROLAND PROHASKA
Keyword(s):  
Random Walks ◽  
Haar Measure ◽  
Limit Theorems ◽  
Homogeneous Spaces ◽  
Locally Compact Group ◽  
Quadratic Growth ◽  
Infinite Volume ◽  
Starting Position ◽  
Markov Chain Theory ◽  
Ratio Limit

Abstract A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.

Ratio Limit Theorems for Random Walks and L�vy Processes

2005 ◽  
Vol 23 (4) ◽  
pp. 357-380
Author(s):  
Minzhi Zhao ◽  
Jiangang Ying
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Ratio Limit Theorems ◽  
Ratio Limit

Ratio Limit Theorems

1976 ◽  
Vol 28 (2) ◽  
pp. 403-407
Author(s):  
A. G. Mucci
Keyword(s):  
Limit Theorems ◽  
Probability Space ◽  
Random Variables ◽  
Ratio Limit Theorems ◽  
Ratio Limit ◽  
Image Position

Let be an adapted sequence of integrable random variables on the probability space . Let us set .The following result can be immediately derived from Brown [2]:

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