CONDITIONS FOR CONVERGENCE OF NUMBER OF CROSSINGS TOTHE LOCAL TIME
APPLICATION TO STABLE PROCESSES WITH INDEPENDENTINCREMENTS AND TO GAUSSIAN PROCESSES
J. M. Azaďs
Abstract: Let be a real valued stochastic process admitting a local time and
let be a family of smooth processes which converge in some sense to
We exhibit sufficient conditions for -convergence of the number of crossings of to
the local time of after normalization.
Two main cases are considered for stable processes and Gaussian processes.
Two main cases are considered for being the convolution of with a
size approximate identity and being the size polygonal approximation of
Such a convergence is shown to hold for both approximations when is a stable
process with independent increments with index
Convergence of crossings of the polygonal approximation is shown to hold for a Gaussian
process under technical conditions.