Random time changes and convergence in distribution under the Meyer-Zheng conditions. (English) Zbl 0742.60036

Kurtz, Thomas G.

Ann. Probab. 19, No. 3, 1010-1034 (1991).

Let \(\{X_ n\}\) be a sequence of stochastic processes with values in a separable metric space. In the paper the results of P. A. Meyer and W. A. Zheng [Ann. Inst. Henri Poincaré, Probab. Statist. 20, 353- 372 (1984; Zbl 0551.60046)] are extended in a number of ways. First suppose \(X_ n\) are cadlag processes. Under a generalized Meyer-Zheng condition it is proved the relative compactness under the Skorokhod topology of \((Y_ n,\gamma_ n)\) for suitably defined \(Y_ n,\gamma_ n\) satisfying \(X_ n(t)=Y_ n(\gamma_ n^{-1}(t))\), where \(\gamma_ n\) are continuous random time transformations. Moreover, conditions are found ensuring weak convergence of \(\{X_ n\}\) in the Skorokhod topology, thus extending results of J. Jacod, J. Mémin and M. Métivier [Stochastic Processes Appl. 14, 109-146 (1983; Zbl 0501.60029)]. Finally, suppose only that \(X_ n\) are measurable. Then conditions, weaker than that of Meyer-Zheng, for relative compactness of \(\{X_ n\}\) under the topology of convergence in measure are obtained.

Reviewer: T.Inglot (Wrocław)

MSC:

60F17	Functional limit theorems; invariance principles
60G44	Martingales with continuous parameter
60G99	Stochastic processes

Keywords:

weak convergence; relative compactness; convergence in measure; random time change; Skorokhod topology

Citations:

Zbl 0551.60046; Zbl 0501.60029