It seems clear to me that there should be some analogue to the notion of a region of convergence for a Volterra series.

However, it seems as though there are now many different subtle ways for things not to converge, and possibly even more so if we're talking about distributions rather than functions.

How does this work? Can Volterra series be "analytic" over some "radius" of input? Are there convergence theorems that are analogous to those for Taylor series that hold for Volterra series?