ON A PROBLEM OF ERDOS AND TAYLOR

Let {S-n, n greater than or equal to 0} be a centered d-dimensional ra ndom walk (d greater than or equal to 3) and consider the so-called fu ture infima process J(n) =(df)inf(k greater than or equal to n) \\S-k\ \. This paper is concerned with obtaining precise integral criteria fo r a function to be in the Levy upper class of J. This solves an old pr oblem of Erdos and Taylor, who posed the problem for the simple symmet ric random walk on Z(d), d greater than or equal to 3. These results a re obtained by a careful analysis of the future infima of transient Be ssel processes and using strong approximations. Our results belong to a class of Ciesielski-Taylor theorems which relate d- and (d - 2)-dime nsional Bessel processes.