A Study on Lebesgue Decomposition of Measures Induced by Stable Processes.

Abstract

Abstract : The Lebesgue decomposition of measures induced by symmetric stable process is considered. An upper bound for the set of admissible translates of a general pth order process is presented, which is a partial analog of the reproducing kernel Hilbert space of a second order process. For invertible processes a dichotomy is established: each translate is either admissible or singular, and the admissible translates are characterized. As a consequence, most continuous time moving averages and all harmonizable processes with nonatomic spectral measure have no admissible translate. Necessary and sufficient conditions for equivalence and singularity of certain product measures are given and applied to the problem of distinguishing a sequence of random vectors from affine transformations of itself; in particular sequences of stable random variables are considered and the singularity of sequences with different indexes of stability is proved. Sufficient conditions for singularity and necessary conditions for absolute continuity are given for the pth order processes. Finally the dichotomy 'two processes are either equivalent or singular', is shown to hold for certain stable processes such as independently scattered random measures and harmonizable processes.