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Infinite sums of i.i.d. random variables discounted by a multiplicativerandom walk are called perpetuities and have been studied by many authors. Thepresent paper provides a log-type moment result for such random variables underminimal conditions which is then utilized for the study of related moments ofa.s. limits of certain martingales associated with the supercritical branchingrandom walk. The connection, first observed by the second author in [Iksanov,A.M. (2004). Elementary fixed points of the BRW smoothing transforms withinfinite number of summands. Stoch. Proc. Appl. 114, 27-50.], arises uponconsideration of a size-biased version of the branching random walk originallyintroduced by Lyons in [Lyons, R.(1997). A simple path to Biggins' martingaleconvergence for branching random walk. In Athreya, K.B., Jagers, P. (eds.).Classical and Modern Branching Processes, IMA Volumes in Mathematics and itsApplications, vol. 84, Springer, Berlin, 217-221.]. We also provide a necessaryand sufficient condition for uniform integrability of these martingales in themost general situation which particularly means that the classical(LlogL)-condition is not always needed.

A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks