Zeros of a two-parameter random walk

D. Khoshnevisan and P. Révész

Abstract. We prove that the number γN of a two-parameter simple random walk in its first N×N time steps is almost surely equal to N1+o(1) as N→∞. This is in contrast with our earlier joint effort with Z. Shi [4]; that work shows that the number of zero crossings in the first N×N times steps is N(3/2)+o(1) as N→∞. We prove also that the number of zeros on the diagonal in the first N time steps is ((2π)-1/2 +o(1))log N almost surely.

Keywords. Random walks, local time, random fields.

AMS Classification (2000) Primary. 60G50; Secondary. 60G60, 60F15.

Support. Research supported in part by a grant from the NSF (DMS-0704024). Pre/E-Prints. This paper is available in

Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
Pál Révész
Institut für Statistik
    und Wahrscheinlichkeitstheorie
Technische Universität Wien
Wiedner Haupstrasse 8-10/1071
A-1040, Wien, Austria

Last Update: July 2, 2009
© 2009 - Davar Khoshnevisan and Pál Révész