The Annals of Probability

Temps de Sejour et Oscillation du Mouvement Brownien au Voisinage de la Sphere Euclidienne

A. Goldman

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Abstract

For a standard Brownian motion $\omega(t)$ in $R^p, p \geq 3$, let $t_a(\omega)$ be the last exit time from the ball $B(0, a)$ of radius $a$ centered at the origin and let $F(a, t, \omega)$ be the oscillation in the neighbourhood of sphere $S(0, a)$. The distribution of the functional $B_f(a, \omega) = \int^{+\infty}_0 1_{B(0,a)}(\omega(t))f\big(\frac{F(a, t, \omega)}{F(a, +\infty, \omega)}\big) dt,$ where $f: (0, 1) \rightarrow R^+$ is an arbitrary bounded measurable function, coincides with the limiting distribution, when $n \rightarrow +\infty$, of the weighted sojourn time $\frac{T_f(a\sqrt n, \omega)}{n} = \sum^{+\infty}_{k=0} 1_{B(0,a\sqrt n)}(S_k(\omega))f\big(\frac{n(a\sqrt n, k, \omega)}{n(a\sqrt n, +\infty, \omega)}\big)\big/n$ for a standard random walk in $Z^p$ where $n(b, k, \omega)$ denote the number of crossing $S(0, b)$ during the first $k$ steps. We give explicit formulas, in terms of Laplace transform, for the joint distribution of $B_f(a, \omega), F(a, +\infty, \omega)$ and $t_a(\omega)$.

Article information

Source
Ann. Probab., Volume 12, Number 3 (1984), 829-842.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993231

Digital Object Identifier
doi:10.1214/aop/1176993231

Mathematical Reviews number (MathSciNet)
MR744237

Zentralblatt MATH identifier
0541.60028

JSTOR
links.jstor.org

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60J15 60J65: Brownian motion [See also 58J65] 33A40 60J55: Local time and additive functionals 60G17: Sample path properties 60G60: Random fields

Keywords
Brownian motion random walk sojourn time and oscillation Bessel functions Hausforff measure

Citation

Goldman, A. Temps de Sejour et Oscillation du Mouvement Brownien au Voisinage de la Sphere Euclidienne. Ann. Probab. 12 (1984), no. 3, 829--842. doi:10.1214/aop/1176993231. https://projecteuclid.org/euclid.aop/1176993231


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