Abstract：Let R(n)be the largest integ er for which the disc of radius R(n)around the origin is covered by the first n steps of a random walk.The main objective o f the present paper is to obtain be tter estimates fo r the upper tail of the distributionof R(n).Therefore , the co njecture of P.Ré vé sz (1990)remains if we change1/4 <λ<120 to1/4 <λ<8 .
