DICHOTOMIES FOR CERTAIN PRODUCT MEASURES AND STABLE
PROCESSES
Mauro S. de F. Marques
Stamatis Cambanis
Abstract: Necessary and sufficient conditions for equivalence or 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 independent stable
random variables are considered and the singularity of sequences with different indexes of
stability is proved. By using these results the dichotomy, ”two processes are either equivalent
or singular”, is established for certain classes of stable processes, such as independently
scattered measures and harmonizable processes. Also sufficient conditions for singularity and
necessary conditions for absolute continuity are given for order processes.
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -