Variational sums of infinitesimal systems

Abstract

In this paper we consider the limits of variational sums of an infinitesimal system S={{X nj }}. For a function g we let $$V_n (g,S) = \mathop \Sigma \limits_j g(X_{nj} )$$ . When S is formed from the increments of a stochastic process with stationary independent increments, there has been much work done on conditions to insure convergence of V n (g, S). The methods used in these studies however fail in the case of a general system. We define an index a for the system S and show it is the natural index for studying convergence of V n (g, S). For systems formed by the domain of attraction of a stable distribution with index γ we have α=γ. For systems from increments of processes with (not necessarily stationary) independent increments a generalizes the Blumenthal-Getoor index. Convergence in distribution and convergence of expectations of V n (g, S) are studied.