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 Detailed view Article in peer-reviewed journal
 Theory of Probability & Its Applications 42, 4 (1997) 757-771
 Weak convergence of the integrated number of level crossings to the local time for Wiener processes
 Let $\{X_{t}, t \in[0,1]\}$ be a standard Wiener process defined on $(\Omega,A,\bf{P})$. We define the regularized process $X^{\varepsilon}_{t}= \varphi_{\varepsilon}*X_{t}$, with $\varphi_{\varepsilon}(t)=\ve^{-1}\varphi(t/\ve)$, a kernel that approaches Dirac's delta function as $\ve \rightarrow 0$. We study the convergence of $Z_{\varepsilon}(f) = \varepsilon^{-1/2} \int_{-\infty}^{+\infty} \bigg [ \frac{N^{X^{\varepsilon}}(x)}{c(\ve)} - L_{X}(x)\bigg]\, f(x)\, dx,$ 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 $[0,1]$. As a by-product of our method we also obtain a weak convergence result for the increments of the process $X$.
 Keyword(s) : Wiener processes – local time – crossings – increments
 hal-00319153, version 1 http://hal.archives-ouvertes.fr/hal-00319153 oai:hal.archives-ouvertes.fr:hal-00319153 From: Corinne Berzin <> Submitted on: Friday, 5 September 2008 18:07:38 Updated on: Tuesday, 9 September 2008 13:27:49