##
**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 655.60055

**Autor: ** Erdös, Paul; Révész, P.

**Title: ** On the area of the circles covered by a random walk. (In English)

**Source: ** J. Multivariate Anal. 27, No.1, 169-180 (1988).

**Review: ** A simply symmetric random walk on the plane is considered. Let Q(N) = **{**x = (i,j): ||x|| = (i^{2}+j^{2})^{ ½} \leq N**}**. The circle Q(N) is covered by the random walk in time n if \xi(x,n) > 0 for every x in Q(N) where \xi(x,n) means the number of passings through the point x during time n. Let R(n) be the largest integer for which Q(R(n)) is covered in n. For R(n) the following lower estimate is proved:

for any \epsilon > 0 R(n) \geq \exp ((log n)^{ ½}/(log_{2}n)^{3/4+\epsilon}) a.s. for all finitely many n where log_{k} is the k times iterated logarithm. An estimate is obtained for the density K(N,n) of the points of Q(N) covered by the random walk. Some further related problems are formulated.

**Reviewer: ** L.Lakatos

**Classif.: ** * 60J15 Random walk

60F15 Strong limit theorems

60G17 Sample path properties

**Keywords: ** random walk on the plane; iterated logarithm

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag