Blackwell-type Theorems for Weighted Renewal Functions

For a numerical sequence ${a_n}$ satisfying broad assumptions on its "behaviour on average" and a random walk $S_n=\xi_1 +...+\xi_n$ with i.i.d. jumps $\xi_j$ with positive mean $\mu$, we establish the asymptotic behaviour of the sums [\sum_{n\ge 1} a_n \pr (S_n\in[x, x+\D)) \quad as \quad x\to \infty,] where $\D>0$ is fixed. The novelty of our results is not only in much broader conditions on the weights ${a_n}$, but also in that neither the jumps $\xi_j$ nor the weights $a_j$ need to be positive. The key tools in the proofs are integro-local limit theorems and large deviation bounds. For the jump distribution $F$, we consider conditions of four types: (a) the second moment of $\xi_j$ is finite, (b) $F$ belongs to the domain of attraction of a stable law, (c) the tails of $F$ belong to the class of the so-called locally regularly varying functions, (d) $F$ satisfies the moment Cram\'er condition. Regarding the weights, in cases (a)--(c) we assume that ${a_n}$ is a so-called $\psi$-locally constant on average sequence, $\psi(n)$ being the scaling factor ensuring convergence of the distributions of $(S_n - \mu n)/\psi (n)$ to the respective stable law. In case (d) we consider sequences of weights of the form $a_n=b_n e^{qn},$ where ${b_n}$ has the properties assumed about the sequence ${a_n}$ in cases (a)--(c) for $\psi(n)=\sqrt{n}.$