The Annals of Probability

Law of the iterated logarithm for the range of random walks in two and three dimensions

Richard F. Bass and Takashi Kumagai

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Abstract

Let $S_n$ be a random walk in $\bz^d$ and let $R_n$ be the range of $S_n$. We prove an almost sure invariance principle for $R_n$ when $d=3$ and a law of the iterated logarithm for $R_n$ when $d=2$.

Article information

Source
Ann. Probab., Volume 30, Number 3 (2002), 1369-1396.

Dates
First available in Project Euclid: 20 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1029867131

Digital Object Identifier
doi:10.1214/aop/1029867131

Mathematical Reviews number (MathSciNet)
MR1920111

Zentralblatt MATH identifier
1031.60031

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F15: Strong theorems 60G17: Sample path properties

Keywords
Range of random walk law of the iterated logarithm law of the iterated logarithm intersection local time

Citation

Bass, Richard F.; Kumagai, Takashi. Law of the iterated logarithm for the range of random walks in two and three dimensions. Ann. Probab. 30 (2002), no. 3, 1369--1396. doi:10.1214/aop/1029867131. https://projecteuclid.org/euclid.aop/1029867131


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References

  • [1] BASS, R. F. and KHOSHNEVISAN, D. (1992). Local times on curves and uniform invariance principles. Probab. Theory Related Fields 92 465-492.
  • [2] BASS, R. F. and KUMAGAI, T. (2000). Laws of the iterated logarithm for some sy mmetric diffusion processes. Osaka J. Math. 37 625-650.
  • [3] BENNETT, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33-45.
  • [4] DONSKER, M. D. and VARADHAN, S. R. S. (1979). On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32 721-747.
  • [5] DVORETZKY, A. and ERD OS, P. (1951). Some problems on random walk in space. Proc. Second Berkeley Sy mp. Math. Statist. Probab. 353-367. Univ. California Press, Berkeley.
  • [6] HAMANA, Y. (1995). On the multiple point range of three-dimensional random walks. Kobe J. Math. 12 95-122.
  • [7] HAMANA, Y. (1997). The fluctuation result for the multiple point range of two-dimensional recurrent random walks. Ann. Probab. 25 598-639.
  • [8] HAMANA, Y. (1998). An almost sure invariance principle for the range of random walks. Stochastic Process. Appl. 78 131-143.
  • [9] HAMANA, Y. (2000). Personal communication.
  • [10] HAMANA, Y. and KESTEN, H. (2001). A large-deviation result for the range of random walk and for the Wiener sausage. Probab. Theory Related Fields 120 183-208.
  • [11] HAMANA, Y. and KESTEN, H. (2002). Large deviations for the range of an integer-valued random walk. Ann. Inst. H. Poincaré Probab. Statist. 38 17-58.
  • [12] JAIN, N. C. and PRUITT, W. E. (1970). The range of recurrent random walk in the plane. Z. Wahrsch. Verw. Gebiete 16 279-292.
  • [13] JAIN, N. C. and PRUITT, W. E. (1971). The range of transient random walk. J. Anal. Math. 24 369-393.
  • [14] 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.
  • [15] JAIN, N. C. and PRUITT, W. E. (1972). The range of random walk. Proc. Sixth Berkeley Sy mp. Math. Statist. Probab. 3 31-50. Univ. California Press, Berkeley.
  • [16] JAIN, N. C. and PRUITT, W. E. (1974). Further limit theorems for the range of random walk. J. Anal. Math. 27 94-117.
  • [17] KALLENBERG, O. (1997). Foundations of Modern Probability. Springer, New York.
  • [18] LE GALL, J.-F. (1986). Propriétés d'intersection des marches aléatoires. I. Convergence vers le temps local d'intersection. Comm. Math. Phy s. 104 471-507.
  • [19] LE GALL, J.-F. (1988). Fluctuation results for the Wiener sausage. Ann. Probab. 16 991-1018.
  • [20] LE GALL, J.-F. (1994). Exponential moments for the renormalized self-intersection local time of planar Brownian motion. Séminaire de Probabilités XXVIII. Lecture Notes in Math. 1583 172-180. Springer, Berlin.
  • [21] LE GALL, J.-F. and ROSEN, J. (1991). The range of stable random walks. Ann. Probab. 19 650-705.
  • [22] SKOROHOD, A. V. (1965). Studies in the Theory of Random Processes. Addison-Wesley, Reading, MA.
  • [23] SPITZER, F. (1976). Principles of Random Walk. Springer, Berlin.
  • STORRS, CONNECTICUT 06269 E-MAIL: bass@math.uconn.edu RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES Ky OTO UNIVERSITY Ky OTO 606-8502 JAPAN E-MAIL: kumagai@kurims.ky oto-u.ac.jp