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Contents of PMS, Vol. 18, Fasc. 1,
pages 39 - 81
 

LEVEL CROSSINGS AND LOCAL TIME FOR REGULARIZED GAUSSIAN PROCESSES

Corinne Berzin
José R. León
Joaquín Ortega

Abstract: Let (X ,t (- [0,1]) n be a centred stationary Gaussian process defined on (_O_,A,P ) with covariance function satisfying

r(t) ~ 1- C |t| 2a, 0 < a < 1, ast-- > 0.
Define the regularized process

Xe = fe * X and Ye = Xe/se, wheres2e = varXet,
fe is a kernel which approaches the Dirac delta function as e --> 0 and * denotes the convolution. We study the convergence of

integral oo [ Ye ] Ze(f) = e- a(a) N---(x)- LX(x) f(x)dx as e-- > 0, - oo c(e)
where N V(x) and LV(x) denote, respectively, the number of crossings and the local time at level x for the process V in [0,1] and

c(e) = (2var(Xet)/pvar(Xet))1/2.
The limit depends on the value of a.

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

Key words and phrases: -

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