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