26544 articles – 20365 references  [version française]
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
Corinne Berzin1, José R. León

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 $X$.
1:  LJK - Laboratoire Jean Kuntzmann
Wiener processes – local time – crossings – increments