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Random Walks Crossing High Level Curved Boundaries

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  • Harry Kesten
  • R. A. Maller

Abstract

Let }S n} be a random walk, generated by i.i.d. increments X i which drifts weakly to ∞ in the sense that $$S_n \xrightarrow{P}\infty$$ as n→ ∞. Suppose k≥0, k≠1, and E|X 1|1k = ∞ if k>1. Then we show that the probability that S. crosses the curve n↦an K before it crosses the curve n ↦ −an k tends to 1 as a → ∞. This intuitively plausible result is not true for k = 1, however, and for 1/2

Suggested Citation

  • Harry Kesten & R. A. Maller, 1998. "Random Walks Crossing High Level Curved Boundaries," Journal of Theoretical Probability, Springer, vol. 11(4), pages 1019-1074, October.
    • Handle: RePEc:spr:jotpro:v:11:y:1998:i:4:d:10.1023_a:1022621016708
    DOI: 10.1023/A:1022621016708
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    References listed on IDEAS

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    1. Harry Kesten & Ross A. Maller, 1997. "Divergence of a Random Walk Through Deterministic and Random Subsequences," Journal of Theoretical Probability, Springer, vol. 10(2), pages 395-427, April.
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    Cited by:

    1. R. A. Doney & R. A. Maller, 2007. "Almost Sure Relative Stability of the Overshoot of Power Law Boundaries," Journal of Theoretical Probability, Springer, vol. 20(1), pages 47-63, March.
    2. R. A. Doney & R. A. Maller, 2002. "Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity," Journal of Theoretical Probability, Springer, vol. 15(3), pages 751-792, July.

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