The Annals of Probability

The fluctuation result for the multiple point range of two-dimensional recurrent random walks

Yuji Hamana

Abstract

We study the fluctuation problem for the multiple point range of random walks in the two dimensional integer lattice with mean 0 and finite variance. The $p$-multiple point range means the number of distinct sites with multiplicity $p$ of random walk paths before time $n$. The suitably normalized multiple point range is proved to converge to a constant, which is independent of the multiplicity, multiple of the renormalized self-intersection local time of a planar Brownian motion.

Article information

Source
Ann. Probab., Volume 25, Number 2 (1997), 598-639.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1024404413

Digital Object Identifier
doi:10.1214/aop/1024404413

Mathematical Reviews number (MathSciNet)
MR1434120

Zentralblatt MATH identifier
0890.60066

Subjects
Primary: 60J15
Secondary: 60F05: Central limit and other weak theorems

Citation

Hamana, Yuji. The fluctuation result for the multiple point range of two-dimensional recurrent random walks. Ann. Probab. 25 (1997), no. 2, 598--639. doi:10.1214/aop/1024404413. https://projecteuclid.org/euclid.aop/1024404413

References

• [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• [2] Dvoretzky, A. and Erd ¨os, P. (1951). Some problems on random walk in space. Proc. Second Berkeley Symp. Math. Statist. Probab. 353-367. Univ. California Press, Berkeley.
• [3] Erd ¨os, P. and Taylor, S. J. (1960). Some problems concerning the structure of random walk paths. Acta Math. Hungar. 11 137-162.
• [4] Flatto, L. (1976). The multiple range of two-dimensional recurrent walk. Ann. Probab. 4 229-248.
• [5] Hamana, Y. (1992). On the central limit theorem for the multiple point range of random walk. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39 339-363.
• [6] Hamana, Y. (1992). The variance of the single point range of two dimensional recurrent random walk. Proc. Japan Acad. Ser. A Math. Sci. 68 195-197.
• [7] Hamana, Y. (1994). The law of the iterated logarithm for the single point range of random walk. Tokyo J. Math. 17 171-180.
• [8] Hamana, Y. (1995). The fluctuation results for the single point range of random walks in low dimensions. Japan. J. Math. 17 287-333.
• [9] Hamana, Y. (1995). The limit theorems for the single point range of strongly transient random walks. Osaka J. Math. 32 869-886.
• [10] Jain, N. C. and Pruitt, W. E. (1970). The range of recurrent random walk in the plane. Z. Wahrsch. Verw. Gebiete 16 279-292.
• [11] Jain, N. C. and Pruitt, W. E. (1971). The range of transient random walk. J. Anal. Math. 24 369-393.
• [12] Jain, N. C. and Pruitt, W. E. (1972). The law of the iterated logarithm for the range of random walk. Ann. Math. Statist. 43 1692-1697.
• [13] Jain, N. C. and Pruitt, W. E. (1973). The range of random walk. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 31-50. Univ. California Press, Berkeley.
• [14] Jain, N. C. and Pruitt, W. E. (1974). Further limit theorems for the range of random walk. J. Anal. Math. 27 94-117.
• [15] Kesten, H. and Spitzer, F. (1963). Ratio theorems for random walks I. J. Anal. Math. 11 323-379.
• [16] Le Gall, J.-F. (1986). Propri´et´es d'intersection des marches al´eatoires I. Comm. Math. Phys. 104 471-507.
• [17] Le Gall, J.-F. (1985). Sur le temps local d'intersection du mouvement brownien plan et la m´ethode de renormalisation de Varadhan. S´eminaire de Probabiliti´es XIX. Lecture Notes in Math. 1123 314-331. Springer, Berlin.
• [18] Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks. Ann. Probab. 19 650-705.
• [19] Pitt, J. H. (1974). Multiple points of transient random walk. Proc. Amer. Math. Soc. 43 195-199.
• [20] Spitzer, F. (1976). Principles of Random Walk. Springer, Berlin.