UNIVERSITY
OF WROCŁAW
 
Main Page
Contents of previous volumes
Forthcoming papers
General Information
Instructions for authors


VOLUMES
38.2 38.1 37.2 37.1 36.2 36.1 35.2
35.1 34.2 34.1 33.2 33.1 32.2 32.1
31.2 31.1 30.2 30.1 29.2 29.1 28.2
28.1 27.2 27.1 26.2 26.1 25.2 25.1
24.2 24.1 23.2 23.1 22.2 22.1 21.2
21.1 20.2 20.1 19.2 19.1 18.2 18.1
17.2 17.1 16.2 16.1 15 14.2 14.1
13.2 13.1 12.2 12.1 11.2 11.1 10.2
10.1 9.2 9.1 8 7.2 7.1 6.2
6.1 5.2 5.1 4.2 4.1 3.2 3.1
2.2 2.1 1.2 1.1
 
 
WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 11, Fasc. 1,
pages 19 - 36
 

CONDITIONS FOR CONVERGENCE OF NUMBER OF CROSSINGS TO THE LOCAL TIME

APPLICATION TO STABLE PROCESSES WITH INDEPENDENT INCREMENTS AND TO GAUSSIAN PROCESSES

J. M. Azaďs

Abstract: Let X(t), t  (-  R, be a real valued stochastic process admitting a local time and let X (t),e  (-  R+,
 e be a family of smooth processes which converge in some sense to X(t). We exhibit sufficient conditions for L2  -convergence of the number of crossings of X (t)
 e to the local time of X(t), after normalization.

Two main cases are considered for X(t), stable processes and Gaussian processes.

Two main cases are considered for X (t) : X (t)
 e      e being the convolution of X(t) with a size e approximate identity and X (t)
 e being the size e polygonal approximation of X(t).

Such a convergence is shown to hold for both approximations when X(t) is a stable process with independent increments with index a > 1.

Convergence of crossings of the polygonal approximation is shown to hold for a Gaussian process under technical conditions.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

Download:    Abstract    Full text   Abstract + References