scholarly journals Rates of convergence in invariance principles for random walks on linear groups via martingale methods

2020 ◽  
Vol 374 (1) ◽  
pp. 137-174
Author(s):  
C. Cuny ◽  
J. Dedecker ◽  
F. Merlevède
Keyword(s):  
Random Walks ◽  
Rates Of Convergence ◽  
Linear Groups ◽  
Invariance Principles ◽  
Martingale Methods

scholarly journals Strong approximation of continuous local martingales by simple random walks

2004 ◽  
Vol 41 (1) ◽  
pp. 101-126
Author(s):  
B. Székely ◽  
T. Szabados
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Rates Of Convergence ◽  
Quadratic Variation ◽  
Local Martingale ◽  
Motion Time ◽  
Time Changes ◽  
Necessary And Sufficient ◽  
Nested Sequence ◽  
The Given

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].

scholarly journals A random walk on the rook placements on a Ferrer's board

10.37236/1284 ◽  
1996 ◽  
Vol 3 (2) ◽  
Author(s):  
Phil Hanlon
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Rate Of Convergence ◽  
Spectral Methods ◽  
Transition Matrix ◽  
Rates Of Convergence ◽  
Sharp Estimates ◽  
Rook Placements ◽  
Ferrers Board

Let $B$ be a Ferrers board, i.e., the board obtained by removing the Ferrers diagram of a partition from the top right corner of an $n\times n$ chessboard. We consider a Markov chain on the set $R$ of rook placements on $B$ in which you can move from one placement to any other legal placement obtained by switching the columns in which two rooks sit. We give sharp estimates for the rate of convergence of this Markov chain using spectral methods. As part of this analysis we give a complete combinatorial description of the eigenvalues of the transition matrix for this chain. We show that two extremes cases of this Markov chain correspond to random walks on groups which are analyzed in the literature. Our estimates for rates of convergence interpolate between those two results.

scholarly journals Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces

2011 ◽  
Vol 32 (4) ◽  
pp. 1313-1349 ◽  
Author(s):  
Y. GUIVARC’H ◽  
C. R. E. RAJA
Keyword(s):  
Random Walks ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Local Fields ◽  
Quadratic Growth ◽  
Linear Groups ◽  
Compact Groups ◽  
Large Classes ◽  
Skew Products ◽  
Recurrent Random Walk

AbstractWe discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular, we show that a closed subgroup of a product of finitely many linear groups over local fields supports an adapted recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces and for associated homeomorphisms with infinite invariant measure. The structural properties of closed subgroups of linear groups over local fields and the properties of group actions with respect to certain Radon measures associated with random walks play an important role in the proofs.

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