On the rate of convergence of the range of cumulative sums

Abstract

Let [X~, n>=l} be a sequence of independent random variables, and S write ~ = ~ X~. The range of cumulative sums, max (0, S~, 9 -., S~)min (0, $ 1 , . . . , S~) has been the subject of considerable research in the li terature (see, for example, [3], [5] where fur ther details and references may also be found). In what follows the random variables X1, X 2 , . . . are assumed to be independent and identically distributed with common law ~(X). We write M~=maxS~, m ~ = m i n S~ and call R ~ = M ~ m ~ , l ~ k N u 1NkN_n the range of cumulative sums S~. Take 0 ntJ~), for the sequence {P(R~>nlJ~s), n>=l} to converge to zero for arbitrary s>0 at specified ra tes (Theorem 2).