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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 $$[0,1]$$.

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