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A Stochastic Difference Equation with Stationary Noise on Groups

Published online by Cambridge University Press:  20 November 2018

Robinson Edward Raja Chandiraraj
Robinson Edward Raja Chandiraraj*
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560 059, INDIA email: [email protected]
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Abstract

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We consider the stochastic difference equation ${{\eta }_{k}}\,=\,{{\xi }_{k}}\phi \left( {{\eta }_{k-1}} \right),\,\,k\,\in \,\mathbb{Z}$ on a locally compact group $G$, where $\phi $ is an automorphism of $G$, ${{\xi }_{K}}$ are given $G$-valued random variables, and ${{\eta }_{k}}$ are unknown $G$-valued random variables. This equation was considered by Tsirelson and Yor on a one-dimensional torus. We consider the case when ${{\xi }_{K}}$ have a common law $\mu $ and prove that if $G$ is a distal group and $\phi $ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-to-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu $ is supported on $Kz\,=\,z\phi \left( K \right)$ for any $z$ in the support of $\mu $. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

Keywords

60B1560G20dissipatingdistal automorphismsprobability measurespointwise distal groupsshifted convolution powers
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 5 , 01 October 2012 , pp. 1075 - 1089
Copyright
Copyright © Canadian Mathematical Society 2012

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