Random filters which perserve the stability of random inputs

A stationary stable random process goes through an independently distributed random linear filter. It is shown that when the input is Gaussian or harmonizable stable, then the output is also stable provided the filter’s transfer function has non-random gain. In contrast, when the input is a non-Gaussian stable moving average, then the output is stable provided the filter’s randomness is due only to a random global sign and time shift.