Operator Decomposable Measures and Stochastic Difference Equations

摘要

We consider the following convolution equation (or equivalently stochastic difference equation) for a given bi-sequence of probability measures on and a linear map on . We study the solutions of Eq. (1) by realizing the process as a measure on and rewriting the stochastic difference equation as -any such measure on is known as -decomposable measure with co-factor where is a suitable weighted shift operator on . This enables one to study the solutions of (1) in the settings of -decomposable measures. A solution of (1) will be called a fundamental solution if any solution of (1) can be written as for some probability measure on . Motivated by the splitting/factorization theorems for operator decomposable measures, we address the question of existence of fundamental solutions when a solution exists and answer affirmatively via a one-one correspondence between fundamental solutions of (1) and strongly -decomposable measures on with co-factor . We also prove that fundamental solutions are extremal solutions and vice versa. We provide a necessary and sufficient condition in terms of a logarithmic moment condition for the existence of a (fundamental) solution when the noise process is stationary and when the noise process has independent -paths.