Abstract
The asymptotic behavior of local renewal probabilities is studied in the case when the contraction of the distribution F(x) of a step in a random walk to the nonnegative semiaxis is a mixture of geometric distributions and, in addition, has an infinite expectation. In contrast to preceding works, F(x), generally speaking, does not get attracted to a stable law. The method applied is based in the proof on extending the characteristic function of $F(x)$ in the right half-plane with the subsequent changing of the contour of integration in the inversion formula is applied. The final conclusion is that the local renewal probability behaves as some completely monotone sequence for which the explicit expression is given.
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