Blackwell's renewal theorem for certain linear submartingales and coupling

The present work surveys some extensions of Blackwell's renewal theorem for a certain class of linear submartingalesS which have been recently obtained by the author. The basic assumption onS is that their conditional increment distribution functions with respect to some filtration 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.