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

Abstract

This paper concerns a recurrent random walk on the real line R 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 < α < 2 and a slowly varying function L ( x ) at infinity, P [ X < − x ] ∼ − x − α / Γ ( 1 − α ) L ( x ) ( x → ∞ ) if and only if P [ T > n ] ∼ n − 1 + 1 / α / Γ ( α ) L α ∗ ( n ) , where L α ∗ is a de Bruijn α -conjugate of L and T denotes the first epoch when the walk hits ( − ∞ , 0 ] . Analogous results are obtained in the cases α = 1 or 2. The method also provides another derivation of Chow’s integrability criterion for the expectation of the ladder height to be finite.