Extended convergence to continuous in probability processes with independent increments
Criteria for Aldous-extended convergence [\textit{D. Aldous}, A concept of weak convergence for stochastic processes viewed in the Strasbourg manner. Preprint, Statist. Laboratory, Univ. Cambridge (1979)] of semimartingales to processes of the kind mentioned in the title are obtained in terms of the known assumptions in functional limit theorems. The extended in Aldous sense convergence of real stochastic processes \(X^ n\) on probabilistic basis with filtrations \(F^ n=(F^ n_ t)_{R^+}\), \(n\geq 1\), is by definition the convergence of random elements \((X^ n,Z^ n)\) in D(R\(\times {\mathcal M}_{w}(D(R)))\) by distribution. Here D(S) is the Skorokhod space of functions in a Polish space S, and \(Z^ n=Z^ n_ t\) is the regular \(F^ n_ t\)- conditional distribution of D(R)-valued random element \(X^ n.\) Assume that the known Jacod conditions of the usual functional convergence of semimartingales \((X^ n,F^ n)\) to a quasi-continuous process (X,F) with independent increments are fulfilled [\textit{J. Jacod}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 63, 109-136 (1983; Zbl 0526.60065), theorem 1.21]. The main theorem: By this assumptions \((X^ n,F^ n)\) converges extendedly to (X,F). Corollaries: The assumptions of CLT's for semimartingales due to \textit{R. Liptser} and \textit{A. Shiryaev} [ibid. 59, 311-318 (1982; Zbl 0482.60030)] are sufficient and necessary for extended convergence of \((X^ n,F^ n)\) to a Gaussian local martingale. Theorem: \((X^ n,F^ n)\) converges extendedly to (X,F) iff the family \((X^ n,F^ n)\) satisfies the Aldous condition of C-tightness, and there exists a sequence of small-scaling on compacts discretisations \(\rho\) (n) of the time line such that \((X^ n_{\rho (n)},F^ n_{\rho (n)})\) fulfills the discrete analogue of the Jacod conditions.
functional convergence of semimartingales
scientific article; zbMATH DE number 3960670
scientific article