## The Annals of Probability

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

#### 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

https://projecteuclid.org/euclid.aop/1029867131

Digital Object Identifier
doi:10.1214/aop/1029867131

Mathematical Reviews number (MathSciNet)
MR1920111

Zentralblatt MATH identifier
1031.60031

#### 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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• 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