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Diffusion Approximation for Random Walks on Anisotropic Lattices

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Lajos Horváth
Lajos Horváth*
Affiliation:
University of Utah
*
∗Postal address: Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
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Abstract

We show that the horizontal position of a random walk on a two-dimensional anisotropic lattice converges weakly to a diffusion process.

Keywords

Wiener processweak convergencelocal time

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Type
Short Communications
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 206 - 212
Copyright
Copyright © Applied Probability Trust 1998 

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References

Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press, Cambridge.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.Google Scholar
Chung, K. L., and Williams, R. J. (1983). Introduction to Stochastic Integration. Birkhäuser, Boston.Google Scholar
den Hollander, F. (1994). On three conjectures by K. Shuler. J. Statist. Phys. 75, 891–918.CrossRefGoogle Scholar
Hall, P., and Heyde, C. C. (1980). Martingale Limit Theory and its Applications. Academic Press, New York.Google Scholar
Heyde, C. C. (1982). On the asymptotic behavior of random walks on an anisotropic lattice. J. Statist. Phys. 27, 721–730.Google Scholar
Heyde, C. C. (1993). Asymptotics for two-dimensional anisotropic random walks. In Stochastic Processes. Springer, New York. pp. 125–130.Google Scholar
Heyde, C. C., Westcott, M., and Williams, E. R. (1982). The asymptotic behavior of a random walk on a dual-medium lattice. J. Statist. Phys. 28, 375–380.CrossRefGoogle Scholar
Horváth, L. (1991). Weak convergence of discrete scattering processes. Adv. Appl. Prob. 23, 733–750.CrossRefGoogle Scholar
Kesten, H. (1965). An iterated logarithm for local time. Duke Math. J. 32, 447–456.Google Scholar
Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent R.V.'s and the sample DF. I. Z. Wahrscheinlichkeitsth. 32, 111–131.Google Scholar
Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent R.V.'s and the sample DF. II. Z. Wahrscheinlichkeitsth. 34, 33–58.Google Scholar
Kurtz, T. G., and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Prob. 19, 1035–1070.CrossRefGoogle Scholar
Révész, P. (1990). Random Walks in Random and Non-Random Environments. World Scientific, Singapore.CrossRefGoogle Scholar
Skorohod, A. V., and Slobodenjuk, N. P. (1965). Limit distributions for additive functionals of a sequence of sums of independent identically distributed lattice random variables. Ukraine Mat. Ž. 17, 97–105.Google Scholar
Westcott, M. (1982). Random walks on a lattice. J. Statist. Phys. 27, 75–82.Google Scholar