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

Cherny, A. S. and Shiryaev, A. N. and Yor, M. (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

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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 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
Depositing User: Ms Lucy van Russelt
Date Deposited: 19 Nov 2007
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/913

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