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Weak convergence of the integrated number of level crossings to the local time for Wiener processes. (English. Abridged French version) Zbl 0812.60069
Summary: Let \(\{X_ t,\;t\in[0,1]\}\) be a Wiener process defined on \((\Omega, A, P)\) with covariance function \(r(t,s) = E(X_ tX_ s) = \inf \{t,s\}\). We define the regularized process \(X^ \varepsilon_ t = \varphi_ \varepsilon* X_ t\), with \(\varphi_ \varepsilon\) a kernel that approaches Dirac’s delta function. We study the convergence of \[ Z_ \varepsilon (f) = \varepsilon^{-1/2} \int^{+ \infty}_{- \infty} \bigl[ N^{X^ \varepsilon} (x)/c (\varepsilon)-L_ X (x) \bigr] f(x) dx \] when \(\varepsilon\) goes to zero, with \(N^{X^ \varepsilon} (x)\) the number of crossings for \(X^ \varepsilon\) at level \(x\) in \([0,1]\) and \(L_ X(x)\) the local time of \(X\) in \(x\) on \([0,1]\).

60J65 Brownian motion
60F05 Central limit and other weak theorems
60J55 Local time and additive functionals