Limit behavior of the ``horizontal-vertical'' random walk and some extensions of the Donsker--Prokhorov invariance principle
Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 3, pp. 498-517
Voir la notice de l'article provenant de la source Math-Net.Ru
We consider a two-dimensional random walk that moves in the horizontal direction on the half-plane $\{y>x\}$ and in the vertical direction on the half-plane $\{y\le x\}$. The limit behavior (as the time interval between two steps and the size of each step tend to zero) of this “horizontal-vertical” random walk is investigated. In order to solve this problem, we prove an extension of the Donsker–Prokhorov invariance principle. The extension states that the discrete-time stochastic integrals with respect to the appropriately renormalized one-dimensional random walk converge in distribution to the corresponding stochastic integral with respect to a Brownian motion. This extension enables us to construct a discrete-time approximation of the local time of a Brownian motion. We also provide discrete-time approximations of skew Brownian motions.
Keywords:
limit theorems for degenerate processes, Donsker–Prokhorov invariance principle, local time of Brownian motion, skew Brownian motions, Skorokhod embedding problem.
@article{TVP_2002_47_3_a3,
author = {A. S. Cherny and A. N. Shiryaev and M. Yor},
title = {Limit behavior of the ``horizontal-vertical'' random walk and some extensions of the {Donsker--Prokhorov} invariance principle},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {498--517},
publisher = {mathdoc},
volume = {47},
number = {3},
year = {2002},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_2002_47_3_a3/}
}
TY - JOUR AU - A. S. Cherny AU - A. N. Shiryaev AU - M. Yor TI - Limit behavior of the ``horizontal-vertical'' random walk and some extensions of the Donsker--Prokhorov invariance principle JO - Teoriâ veroâtnostej i ee primeneniâ PY - 2002 SP - 498 EP - 517 VL - 47 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_2002_47_3_a3/ LA - en ID - TVP_2002_47_3_a3 ER -
%0 Journal Article %A A. S. Cherny %A A. N. Shiryaev %A M. Yor %T Limit behavior of the ``horizontal-vertical'' random walk and some extensions of the Donsker--Prokhorov invariance principle %J Teoriâ veroâtnostej i ee primeneniâ %D 2002 %P 498-517 %V 47 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/TVP_2002_47_3_a3/ %G en %F TVP_2002_47_3_a3
A. S. Cherny; A. N. Shiryaev; M. Yor. Limit behavior of the ``horizontal-vertical'' random walk and some extensions of the Donsker--Prokhorov invariance principle. Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 3, pp. 498-517. http://geodesic.mathdoc.fr/item/TVP_2002_47_3_a3/