The Annals of Probability

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

Yuji Hamana

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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

Keywords
Multiple point range random walk intersection local time

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


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References

  • [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR381718
  • [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.
    Mathematical Reviews (MathSciNet): MR13,852b
    Zentralblatt MATH: 0044.14001
  • [3] Erd ¨os, P. and Taylor, S. J. (1960). Some problems concerning the structure of random walk paths. Acta Math. Hungar. 11 137-162.
    Mathematical Reviews (MathSciNet): MR2212599
  • [4] Flatto, L. (1976). The multiple range of two-dimensional recurrent walk. Ann. Probab. 4 229-248.
    Mathematical Reviews (MathSciNet): MR554388
    Zentralblatt MATH: 0349.60067
    Digital Object Identifier: doi:10.1112/blms/11.3.241
  • [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.
    Mathematical Reviews (MathSciNet): MR93h:60112
    Zentralblatt MATH: 0762.60061
  • [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.
    Mathematical Reviews (MathSciNet): MR93k:60173
    Zentralblatt MATH: 0768.60060
    Digital Object Identifier: doi:10.3792/pjaa.68.195
    Project Euclid: euclid.pja/1195511706
  • [7] Hamana, Y. (1994). The law of the iterated logarithm for the single point range of random walk. Tokyo J. Math. 17 171-180.
    Mathematical Reviews (MathSciNet): MR95h:60106
    Zentralblatt MATH: 0804.60059
    Digital Object Identifier: doi:10.3836/tjm/1270128195
    Project Euclid: euclid.tjm/1270128195
  • [8] Hamana, Y. (1995). The fluctuation results for the single point range of random walks in low dimensions. Japan. J. Math. 17 287-333.
    Mathematical Reviews (MathSciNet): MR96j:60117
    Zentralblatt MATH: 0851.60073
  • [9] Hamana, Y. (1995). The limit theorems for the single point range of strongly transient random walks. Osaka J. Math. 32 869-886.
    Mathematical Reviews (MathSciNet): MR97c:60176
    Zentralblatt MATH: 0851.60074
    Project Euclid: euclid.ojm/1200786474
  • [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.
    Mathematical Reviews (MathSciNet): MR436984
    Zentralblatt MATH: 0194.49205
    Digital Object Identifier: doi:10.1002/nav.3800220102
  • [11] Jain, N. C. and Pruitt, W. E. (1971). The range of transient random walk. J. Anal. Math. 24 369-393.
    Mathematical Reviews (MathSciNet): MR441120
    Zentralblatt MATH: 0249.60038
  • [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.
    Mathematical Reviews (MathSciNet): MR499955
    Zentralblatt MATH: 0247.60042
  • [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.
    Mathematical Reviews (MathSciNet): MR410936
  • [14] Jain, N. C. and Pruitt, W. E. (1974). Further limit theorems for the range of random walk. J. Anal. Math. 27 94-117.
    Mathematical Reviews (MathSciNet): MR5717844
    Zentralblatt MATH: 0293.60063
  • [15] Kesten, H. and Spitzer, F. (1963). Ratio theorems for random walks I. J. Anal. Math. 11 323-379.
    Mathematical Reviews (MathSciNet): MR285478
    Zentralblatt MATH: 0121.35201
  • [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.
    Mathematical Reviews (MathSciNet): MR92j:60083
    Zentralblatt MATH: 0729.60066
    Digital Object Identifier: doi:10.1214/aop/1176990446
    Project Euclid: euclid.aop/1176990446
  • [19] Pitt, J. H. (1974). Multiple points of transient random walk. Proc. Amer. Math. Soc. 43 195-199.
    Mathematical Reviews (MathSciNet): MR526880
    Digital Object Identifier: doi:10.1007/BF02925748
  • [20] Spitzer, F. (1976). Principles of Random Walk. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR529383

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