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A Stochastic Difference Equation with Stationary Noise on Groups

  Published:2011-12-23
 Printed: Oct 2012
  • Chandiraraj Robinson Edward Raja,
    Stat-Math Unit, Indian statistical institure, 8th Mile Mysore Road, Karnataka 56059, INDIA
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Abstract

We consider the stochastic difference equation $$\eta _k = \xi _k \phi (\eta _{k-1}), \quad k \in \mathbb Z $$ 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 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-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu$ is supported on $Kz= z\phi (K)$ 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: dissipating, distal automorphisms, probability measures, pointwise distal groups, shifted convolution powers dissipating, distal automorphisms, probability measures, pointwise distal groups, shifted convolution powers
MSC Classifications: 60B15, 60G20 show english descriptions Probability measures on groups or semigroups, Fourier transforms, factorization
Generalized stochastic processes
60B15 - Probability measures on groups or semigroups, Fourier transforms, factorization
60G20 - Generalized stochastic processes
 

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