Abstract view
A Stochastic Difference Equation with Stationary Noise on Groups


Published:20111223
Printed: Oct 2012
Chandiraraj Robinson Edward Raja,
StatMath Unit, Indian statistical institure, 8th Mile Mysore Road, Karnataka 56059, INDIA
Abstract
We consider the stochastic difference equation $$\eta _k = \xi _k
\phi (\eta _{k1}), \quad k \in \mathbb Z $$ 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
onedimensional 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 oneone
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.