Geodesic
Weak convergence of the integrated number of~level crossings to~the~local time for~Wiener processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 4, pp. 757-771

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\{X_{t},\ t\in[0,1]\}$ be a standard Wiener process defined on $(\Omega,A,\mathbb P)$. We define the regularized process $X^{\varepsilon}_{t}= \varphi_{\varepsilon}*X_{t}$, with $\varphi_{\varepsilon}(t)=\varepsilon^{-1}\varphi(t/\varepsilon)$, a kernel that approaches Dirac's delta function as $\varepsilon \rightarrow 0$. We study the convergence of $$ Z_{\varepsilon}(f) = \varepsilon^{-1/2} \int_{-\infty}^{+\infty} \biggl [ \frac{N^{X^{\varepsilon}}(x)}{c(\varepsilon)} - L_{X}(x)\biggr]f(x)\, dx, $$ 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 $X$.
Keywords: Wiener processes, local time, crossings
Mots-clés : increments. 
@article{TVP_1997_42_4_a6,
     author = {C. Berzin-Joseph and J. R. Le\'on},
     title = {Weak convergence of the integrated number of~level crossings to~the~local time {for~Wiener} processes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {757--771},
     publisher = {mathdoc},
     volume = {42},
     number = {4},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_1997_42_4_a6/}
}
TY  - JOUR
AU  - C. Berzin-Joseph
AU  - J. R. León
TI  - Weak convergence of the integrated number of~level crossings to~the~local time for~Wiener processes
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 1997
SP  - 757
EP  - 771
VL  - 42
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TVP_1997_42_4_a6/
LA  - en
ID  - TVP_1997_42_4_a6
ER  - 
%0 Journal Article
%A C. Berzin-Joseph
%A J. R. León
%T Weak convergence of the integrated number of~level crossings to~the~local time for~Wiener processes
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1997
%P 757-771
%V 42
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TVP_1997_42_4_a6/
%G en
%F TVP_1997_42_4_a6
C. Berzin-Joseph; J. R. León. Weak convergence of the integrated number of~level crossings to~the~local time for~Wiener processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 4, pp. 757-771. http://geodesic.mathdoc.fr/item/TVP_1997_42_4_a6/