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Teor. Veroyatnost. i Primenen., 2002, Volume 47, Issue 3, Pages 498–517 (Mi tvp3689)  

This article is cited in 11 scientific papers (total in 11 papers)

Limit behavior of the “horizontal-vertical” random walk and some extensions of the Donsker–Prokhorov invariance principle

A. S. Chernya, A. N. Shiryaevb, M. Yorc

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Steklov Mathematical Institute, Russian Academy of Sciences
c Université Pierre & Marie Curie, Paris VI

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 $\{y\le x\}$. 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.

DOI: https://doi.org/10.4213/tvp3689

Full text: PDF file (1566 kB)

English version:
Theory of Probability and its Applications, 2003, 47:3, 377–394

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Received: 30.08.2001
Language:

Citation: 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”, Teor. Veroyatnost. i Primenen., 47:3 (2002), 498–517; Theory Probab. Appl., 47:3 (2003), 377–394

Citation in format AMSBIB
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\by A.~S.~Cherny, A.~N.~Shiryaev, M.~Yor
\paper Limit behavior of the ``horizontal-vertical'' random walk and some extensions of the Donsker--Prokhorov invariance principle
\jour Teor. Veroyatnost. i Primenen.
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\issue 3
\pages 498--517
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\zmath{https://zbmath.org/?q=an:1034.60076}
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\jour Theory Probab. Appl.
\yr 2003
\vol 47
\issue 3
\pages 377--394
\crossref{https://doi.org/10.1137/S0040585X97979834}
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    2. A. S. Mishchenko, “On a probability distribution of some random walk functionals”, Theory Probab. Appl., 50:4 (2006), 710–717  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Bertacchi D., “Asymptotic behaviour of the simple random walk on the 2–dimensional comb”, Electronic Journal of Probability, 11 (2006), 1184–1203  crossref  mathscinet  zmath  isi  scopus
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    6. A. T. Abakirova, “On some functional inequalities for skew Brownian motion”, Proc. Steklov Inst. Math., 287:1 (2014), 3–13  mathnet  crossref  crossref  isi  elib
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    8. Csaki E., Csorgo M., Foldes A., Revesz P., “Some Limit Theorems for Heights of Random Walks on a Spider”, J. Theor. Probab., 29:4 (2016), 1685–1709  crossref  mathscinet  zmath  isi  elib  scopus
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  • Теория вероятностей и ее применения Theory of Probability and its Applications
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