You are here: MIMS > EPrints
MIMS EPrints

2007.179: Limit Behavior of the "Horizontal-Vertical" Random Walk and Some Extensions of the Donsker-Prokhorov Invariance Principle

2007.179: A. S. Cherny, A. N. Shiryaev and M. Yor (2003) Limit Behavior of the "Horizontal-Vertical" Random Walk and Some Extensions of the Donsker-Prokhorov Invariance Principle. Theory of Probability and its Applications, 47 (3). pp. 377-394. ISSN 1095-7219

Full text available as:

PDF - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
267 Kb

DOI: 10.1137/S0040585X97979834

Abstract

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 ≤ 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.

Item Type:Article
Uncontrolled Keywords:limit theorems for degenerate processes; Donsker-Prokhorov invariance principle; local time of Brownian motion; skew Brownian motions; Skorokhod embedding problem
Subjects:MSC 2000 > 60 Probability theory and stochastic processes
MIMS number:2007.179
Deposited By:Mrs Louise Healey
Deposited On:19 November 2007

Download Statistics: last 4 weeks
Repository Staff Only: edit this item