Moments of some stopping rules

Henry Teicher, Cun Hui Zhang

Research output: Contribution to journalArticlepeer-review


Let Xn for n≤ 1 be independent random variables with EXn = 0 and EXn 2. Set Sk,n = Σ i1 < ··· < ikn X i1 ··· Xik. Define T k,c,m= inf {n ≤ m: \k !S k,n \ > cn k/2}. We study critical values c k,p for k ≥ 2 and p > 0, such that ET k,c,m p < ∞ for c < c k,p and all m, and ET k,c,m p = ∞ for c > c k,p and all sufficiently large m. In particular, c 1,1 = c 2,1 = 1, c 3,1 = 2 and C 4,1 = 3 under certain moment conditions on X 1, when X n are identically distributed. We also investigate perturbed stopping rules of the torm T n,m = inf {n ≥ m : h(S 1,n/ n 1/2) < ξ n or > ζ n} for continuous function h and random variables ξ n ∼ a and ζ n ∼ b with a < b. Related stopping rules of the Wiener process are also considered via the Uhlenbeck process.

Original languageEnglish (US)
Pages (from-to)503-512
Number of pages10
JournalJournal of the London Mathematical Society
Issue number2
StatePublished - Apr 1998

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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