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

Kôhei Uchiyama

Stochastic Processes and their Applications, 2011, vol. 121, issue 9, 1938-1961

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.

Keywords: Ladder; height; Ladder; epoch; Potential; function; Spitzer's; condition (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (2)

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