Limit behavior of additive functionals of semistable processes and processes attracted to semistable processes

Abstract

The limit behavior of distributions of functionals of the form $$v_T = \frac{1}{T}\int\limits_o^T {h\left( {x_t } \right)dt,} $$ is studied in the paper; here xt is a process with values in Rm which is semistable or is attracted to a semistable process, and h, h:Rm→-R1, is bounded and measurable. Sufficient conditions are obtained for the existence of a limit distribution. It is shown that in the one-dimensional case these conditions are actually also necessary. Results are also presented on the joint behavior of several functionals of similar type.