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Central limit theorem for the local time of a Gaussian process. (English) Zbl 0939.60006

Dalang, Robert C. (ed.) et al., Seminar on Stochastic analysis, random fields and applications. Centro Stefano Franscini, Ascona, Italy, September 1996. Basel: Birkhäuser. Prog. Probab. 45, 25-37 (1999).
Let \(\{X_t,\;t\in R\}\) be a real-valued, Gaussian, stationary process with \(E(X_s)=0\), \(E(X^2_s)=1\) and covariance function \(E(X_s,X_t)=r(|s-t|)\). As is known, under the first of the conditions \[ 1.\;\int^\infty_0 {ds \over \bigl(1- r^2(s)\bigr)^p}< \infty, \quad 2.\;\exists m>0 : \int\bigl |r(s)\bigr|^m ds< \infty, \] with \(p={1\over 2}\) the so-called local time \(l_t(x)\) exists and admits the following expansion in \(L^2(\Omega)\): \[ l_t (x)=p(x)\sum^\infty_{k=0}{H_k(x)\over k!}\int^t_0H_k(X_s)ds, \] where \(p(x)={1 \over \sqrt{2\pi}}e^{-x^2/2}\), \(H_k(x)=(-1)^k{p^{(k)}(x)\over p(x)}\) denote the normal density and the \(k\)th order Hermite polynomial, respectively.
It is proved that under conditions (1) the remainder of this expansion, \[ R_t (x)=p(x) \sum^\infty_{k=m} {H_k(x)\over k!}\int^t_0H_k(X_s)ds, \] suitably normalized, is asymptotically Gaussian for some \(m\), chosen as the smallest number, satisfying (1.2). Moreover, it is shown that under these conditions: 1) with \(p={1\over 2}\) the finite-dimensional distributions of the random process \(\{{1\over\sqrt t}R_t (x),\;x\in R\}\) converge to those of a Gaussian and centered process \(R(x)\) with covariance function \[ \Gamma(x,y)=2 \sum^\infty_{k=m} {H_k(x)p(x) H_k(y)p(y) \over k!}\int^\infty_0r^k(s)ds, \] 2) with \(p>1/2\) the limiting process \(R(\cdot) \) admits a modification in law with almost surely continuous sample paths, and 3) for \(p>1\) the previous convergence is functional (in the space \(C(R,R))\).
For the entire collection see [Zbl 0914.00071].

MSC:

60F05 Central limit and other weak theorems
60G15 Gaussian processes
60J55 Local time and additive functionals
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