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|| = (i2+j2) 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/(log2n)3/4+\epsilon) a.s. for all finitely many n where logk 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

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