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

This article is cited in 13 scientific papers (total in 13 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

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

Bibliographic databases:

Received: 30.08.2001

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
\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.
\yr 2002
\vol 47
\issue 3
\pages 498--517
\jour Theory Probab. Appl.
\yr 2003
\vol 47
\issue 3
\pages 377--394

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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
    4. Ya. A. Lyulko, “On the distribution of time spent by Markov chain at different levels until achieving a fixed state”, Theory Probab. Appl., 56:1 (2012), 140–149  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
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    11. Csaki E., Csorgo M., Foldes A., Revesz P., “Limit Theorems For Local and Occupation Times of Random Walks and Brownian Motion on a Spider”, J. Theor. Probab., 32:1 (2019), 330–352  crossref  mathscinet  zmath  isi  scopus
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    13. Seol Y., “On Weak Limiting Distributions For Random Walks on a Spider”, Symmetry-Basel, 12:12 (2020), 2000  crossref  isi
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