The discrete renewal equation with nonsummable inhomogeneous term (Q2077452)

The discrete renewal equation is given as \[ z_{j}=\sum_{k=-\infty}^{\infty} z_{j-k} f_{k}+g_{j}, \quad j \in \mathbb{Z}, \] where \(F=\left\{f_{j}\right\}_{j=-\infty}^{\infty}\) is an arithmetic probability distribution with span \(1\), \(\left\{z_{j}\right\}_{j=-\infty}^{\infty}\) is the sequence sought, and the inhomogeneous term \(\left\{g_{j}\right\}_{j=-\infty}^{\infty}\) is a given sequence of complex numbers. In the paper, when the inhomogeneous term behaves like a nondecreasing submultiplicative sequence, some asymptotic properties of the solution are established.