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

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.

### MSC:

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

### Citations:

Zbl 0551.60046; Zbl 0501.60029
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