PERTURBATION AND REARRANGEMENT ASPECTS OF RENEWAL AND HARMONIC RENEWAL SEQUENCES RUDOLF GRUBEL AND SUSAN M. PITTS ABSTRACT Two convolution series arising naturally in various stochastic models are the renewal and the harmonic renewal measure, 00 00 1 u = £ p* n , v = £ -p*\ n-0 n=i n These can be regarded as sequences if the underlying probability distribution p is concentrated on the integers. We investigate the behaviour of these series under infinitesimal perturbations of p and also obtain alternative series representations with better convergence properties. 1. Introduction Let X v X 2 ,... be independent and identically distributed random variables which only take integer values and put p n = P(X 1 = n) for all n e Z. The sequence of partial sums, S o = 0, S fc = £ A ; forallfceN, 1-1 is called a random walk on Z with start at 0 and step distribution p = (p n ) ne2 . For each n e Z let be the expected number of visits of the random walk to the state n. The sequence u = (wJnez obviously depends on the random walk {S n } only via the step distribution p and is called the renewal sequence associated with p. This generalizes the classical (discrete time) renewal model where p is assumed to be concentrated on the non- negative integers. Renewal sequences arise in various contexts in probability theory and its applications; see [3, Chapter XIII; 4, Chapter XI] for the background and a general treatment. From a more analytic point of view we may regard p as an element of nel the space of all absolutely summable sequences, and u as being given by a convolution series in p, namely, u = XXoP**- H ere the convolution a*b of a, be I 1 is defined by {a *b) n = £ a m b n _ m for all n e Z. meZ Received 29 October 1988; revised 14 December 1988. 1980 Mathematics Subject Classification (1985 Revision) 60K05. J. London Math. Soc. (2) 40 (1989) 563-576