Some Joint Laws in Fluctuation Theory
1. Summary. Let SO = 0, Sn =Ej Xi, n = 1, 2, *. be the sequence of partial sums of independent, identically distributed real valued random variables X1, X2, The sum Sk is a (strict) ladder sum if Si < Sk for 0 < j < k. Contrary to usual practice, we always count SO as a ladder sum. A run of ladder sums of length i starts at Sk if Sk, Sk+1, *--, Sk+i-I are ladder sums but Sk-1 and Sk+i are not. Thus SO is always the beginning of a run, of length one if S, ? 0. Except at the beginning of Section 3, we assume that X1 has continuous law, symmetric with respect to 0. Thus ties S, = Sj can be disregarded. Let Ln = "index of max {S, 0 < i < n}," (1) Gn = "number of ladder sums among 0, SI, S," Rn = "number of runs they form."