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
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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 |
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