Details

Autor(en) / Beteiligte

• Berzin-Joseph, C.
• León, J. R.

Titel

Weak Convergence of the Integrated Number of Level Crossings to the Local Time for Wiener Processes

Ist Teil von

• Theory of probability and its applications, 1998-01, Vol.42 (4), p.568-579

Ort / Verlag

Philadelphia: Society for Industrial and Applied Mathematics

Erscheinungsjahr

1998

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• Let $\{X_{t},\ t\in[0,1]\}$ be a standard Wiener process defined on $(\Omega,A,\bP)$. 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 $X$.