RANDOM-WALKS WITH STOCHASTICALLY BOUNDED INCREMENTS - RENEWAL THEORY

This paper develops renewal theory for a rather general class of rando m walks S-N including linear submartingales with positive drift. The b asic assumption on S-N is that their conditional increment distributio n functions with respect to some filtration F-N are bounded from above and below by integrable distribution functions. Under a further mean stability condition these random walks turn out to be natural candidat es for satisfying Blackwell-type renewal theorems. In a companion pape r [2], certain uniform lower and upper drift bounds for S-N, describin g its average growth on finite remote time intervals, have been introd uced and shown to be equal in case the afore-mentioned mean stability condition holds true. With the help of these bounds we give lower and upper estimates for H U(B), where U denotes the renewal measure of S -N, H a suitable delay distribution and B a Borel subset of R. This is then further utilized in combination with a coupling ar ument to prov e the principal result, namely an extension of Blackwell's renewal the orem to random walks of the previous type whose conditional increment distribution additionally contain a subsequence with a common componen t in a certain sense. A number of examples are also presented.