Blackwell's renewal theorem for certain linear submartingales and coupling

Abstract

The present work surveys some extensions of Blackwell's renewal theorem for a certain class of linear submartingalesS ℕ which have been recently obtained by the author. The basic assumption onS ℕ is that their conditional increment distribution functions with respect to some filtration $$\mathcal{F}_\mathbb{N} $$ are bounded from above and below by integrable distribution functions. Under a further mean stability condition these random walks turn out to be natural candidates for satisfying Blackwell-type renewal theorems. The latter are derived by employing a coupling argument similar to that which has been used in the i.i.d. case by Lindvallet al. A number of applications are also presented.