Upper Functions for Symmetric Processes with Stationary, Independent Increments

Characterization of unitary processes with independent and stationary increments

This is a continuation of the earlier work [13] to characterize stationary unitary increment Gaussian processes. The earlier assumption of uniform continuity is replaced by weak continuity and with a technical assumption on the domain of the generator, unitary equivalence of the processes to the solution of Hudson-Parthasarathy equation is proved.

Representations of Gaussian processes with stationary increments

Maxima of Long Memory Stationary Symmetric Α-stable Processes, and Self-similar Processes with Stationary Max-increments

We derive a functional limit theorem for the partial maxima process based on a long memory stationary α-stable process. The length of memory in the stable process is parameterized by a certain ergodic theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fréchet process. It is a self-similar process with α-Fréchet marginals, a...

On the Measure of Hilbert Neighborhoods for Processes with Stationary, Independent Increments

where G(u) is a bounded, nondecreasing function with G(— °o)=0 and where 7 is a real-valued constant. Below it is shown that for certain processes of this type the measure of the Hilbert neighborhood of the origin is related to the solution of a certain differential system. In fact, (A) if {x(t), 0fkt< °° } is a separable stochastic process with symmetric, stationary, and independent increments...

A Convolution Equation and Hitting Probabilities of Single Points for Processes with Stationary Independent Increments

Our purpose is to determine when h(r) is strictly positive, respectively zero. An old and obvious result is that h(r) >0 for all r if Xt is Brownian motion. Somewhat more difficult is the behavior of h(r) for symmetric stable processes of index a in dimension one. For such processes it was shown by Levy, Erdös and Kac (see [6]; also [12] for simplified proofs) that h(r) >0 for all rER il Ka^2 a...