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Convergence in distribution and Skorokhod Convergence for the general theory of processes
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  • Published: September 1991

Convergence in distribution and Skorokhod Convergence for the general theory of processes

  • D. N. Hoover1 

Probability Theory and Related Fields volume 89, pages 239–259 (1991)Cite this article

Summary

This paper proves some Skorokhod Convergence Theorems for processes with filtration. Roughly, these are theorems which say that if a family of processes with filtration (X n,ℱ n),n∈ℕ, converges in distribution in a suitable sense, then there exists a family of equivalent processes (Y n,ℊ n),n∈ℕ, which converges almost surely. The notion of equivalence used is that of adapted distribution, which guarantees that each (X n,ℊ n) has the same stochastic properties as (X n,ℱ n), with respect to its filtration, such as the martingale property or the Markov property. The appropriate notion of convergence in distribution is convergence in adapted distribution, which is developed in the paper. Fortunately, any tight sequence of processes has a subsequence which converges in adapted distribution. For discrete time processes, (Y n,ℊ n),n∈ℕ, and their limit (Y, ℊ) may be taken as all having the same fixed filtrationℊ n=ℊ. In the continuous time case, theY n,ℊ n may require different filtrationsℊ n, which converge toℊ. To handle this, convergence of filtrations is defined and its theory developed.

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Authors and Affiliations

  1. Odyssey Research Associates, 301A Harris B. Dates Dr, 14850-1313, Ithaca, NY, USA

    D. N. Hoover

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  1. D. N. Hoover
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Additional information

During part of the time this work was in progress, it was supported by an NSERC operating grant, and the author was an NSERC University Research Fellow. The author wishes to thank the Steklov Mathematical Institute of the Soviet Academy of Sciences for its hospitality while the principle research in this paper was being begun, A.N. Shiryaev and P.C. Greenwood, who made the author's visit there possible, and Ján Mináč for his hospitality while that research was being finished. We thank the referee who suggested the results in Sect. 12

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Hoover, D.N. Convergence in distribution and Skorokhod Convergence for the general theory of processes. Probab. Th. Rel. Fields 89, 239–259 (1991). https://doi.org/10.1007/BF01198786

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  • Received: 24 November 1989

  • Revised: 13 December 1990

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01198786

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Keywords

  • Filtration
  • Time Process
  • Stochastic Process
  • General Theory
  • Probability Theory
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