On Stability of Volterra Difference Equations of Convolution Type

S. Elaydi obtained a characterization of the stability of the null solution of the Volterra difference equation $$x_n=\sum _{i=0}^{n-1}{a}_{n-i}{x}_i\text{,}n\ge 1\text{,}$$ by localizing the roots of its characteristic equation $$1-\sum _{n=1}^{\mathrm{\infty }}{a}_n{z}^n=0\text{.}$$ The assumption that $(a_n)\in {\mathrm{\ell }}^1$ was the single hypothesis considered for the validity of that characterization, which is an insufficient condition if the ratio $R$ of convergence of the power series of the previous equation equals one. In fact, when $R=1$, this characterization conflicts with a result obtained by Erd"os, Feller and Pollard. Here, we analyze the $R=1$ case and show that some parts of that characterization still hold. Furthermore, studies on stability for the $R<1$ case are presented. Finally, we state some new results related to stability via finite approximation.