## Abstract

Let X_{n} for n≤ 1 be independent random variables with EX_{n} = 0 and EX_{n} ^{2}. Set S_{k,n} = Σ _{i1} < ··· < _{ik} ≤ _{n} X _{i1} ··· X_{ik}. 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 language | English (US) |
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Pages (from-to) | 503-512 |

Number of pages | 10 |

Journal | Journal of the London Mathematical Society |

Volume | 57 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1998 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)