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A note on summability of ladder heights and the distributions of ladder epochs for random walks

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  • Uchiyama, Kôhei

Abstract

This paper concerns a recurrent random walk on the real line and obtains a purely analytic result concerning the characteristic function, which is useful for dealing with some problems of probabilistic interest for the walk of infinite variance: it reduces them to the case when the increment variable X takes only values from {...,-2,-1,0,1}. Under the finite expectation of ascending ladder height of the walk, it is shown that given a constant 1 [infinity]) if and only if , where is a de Bruijn [alpha]-conjugate of L and T denotes the first epoch when the walk hits (-[infinity],0]. Analogous results are obtained in the cases [alpha]=1 or 2. The method also provides another derivation of Chow's integrability criterion for the expectation of the ladder height to be finite.

Suggested Citation

  • Uchiyama, Kôhei, 2011. "A note on summability of ladder heights and the distributions of ladder epochs for random walks," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 1938-1961, September.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:9:p:1938-1961
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    References listed on IDEAS

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    1. Doney, R. A., 1982. "On the exact asymptotic behaviour of the distribution of ladder epochs," Stochastic Processes and their Applications, Elsevier, vol. 12(2), pages 203-214, March.
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    Cited by:

    1. Uchiyama, Kôhei, 2019. "Asymptotically stable random walks of index 1<α<2 killed on a finite set," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5151-5199.
    2. Gerold Alsmeyer & Alexander Iksanov & Matthias Meiners, 2015. "Power and Exponential Moments of the Number of Visits and Related Quantities for Perturbed Random Walks," Journal of Theoretical Probability, Springer, vol. 28(1), pages 1-40, March.

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