Abstract
Let \(\mathcal{M}^{(n)} \), n = 1, 2, ..., be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to \(\mathcal{M}^{(n)} \) converges almost surely and in the mean to a random variable W. For a large subclass of nonnegative and concave functions ƒ, we provide a criterion for the finiteness of \(\mathbb{E}\) Wf(W). The main assertions of the present paper generalize some results obtained recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton-Watson processes. In the process of the proof, we study the existence of the ƒ-moments of perpetuities.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 451–471, April, 2006.
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Iksanov, A.M., Rösler, U. Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities. Ukr Math J 58, 505–528 (2006). https://doi.org/10.1007/s11253-006-0082-9
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DOI: https://doi.org/10.1007/s11253-006-0082-9