Random ergodic sequences on LCA groups

Abstract:

Let ${\{ X(t,\omega )\} _{t \in {{\mathbf {R}}^ + }}}$ be a stochastic process on a locally compact abelian group $G$, which has independent stationary increments. We show that under mild restrictions on $G$ and $\{ X(t,\omega )\}$ the random families of probability measures \[ {\mu _T}( \cdot ,\omega ) = B_T^{ - 1}\int \limits _0^T {f(t){x_{( \cdot )}}} (X(t,\omega ))dt\quad {\text {for}}\;T > 0{\text {,}}\] where $f(t)$ is a function from ${{\mathbf {R}}^ + }$ to ${{\mathbf {R}}^ + }$ of polynomial growth and ${B_T} = \int _0^T {f(t)} \;dt$, converge weakly to Haar measure of the Bohr compactification of $G$. As a consequence we obtain mean and individual ergodic theorems and asymptotic occupancy times for these processes.