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Functional convergence of stochastic integrals with application to statistical inference

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  • Davis, Richard A.
  • Song, Li

Abstract

Assuming that {(Un,Vn)} is a sequence of càdlàg processes converging in distribution to (U,V) in the Skorohod topology, conditions are given under which {∬fn(β,u,v)dUndVn} converges weakly to ∬f(β,x,y)dUdV in the space C(R), where fn(β,u,v) is a sequence of “smooth” functions converging to f(β,u,v). Integrals of this form arise as the objective function for inference about a parameter β in a stochastic model. Convergence of these integrals play a key role in describing the asymptotics of the estimator of β which optimizes the objective function. We illustrate this with a moving average process.

Suggested Citation

  • Davis, Richard A. & Song, Li, 2012. "Functional convergence of stochastic integrals with application to statistical inference," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 725-757.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:3:p:725-757
    DOI: 10.1016/j.spa.2011.10.007
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    References listed on IDEAS

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    1. Davis, Richard A. & Knight, Keith & Liu, Jian, 1992. "M-estimation for autoregressions with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 145-180, February.
    2. Davis, Richard A. & Dunsmuir, William T.M., 1996. "Maximum Likelihood Estimation for MA(1) Processes with a Root on or near the Unit Circle," Econometric Theory, Cambridge University Press, vol. 12(1), pages 1-29, March.
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    Cited by:

    1. Andreas S{o}jmark & Fabrice Wunderlich, 2023. "Functional CLTs for subordinated L\'evy models in physics, finance, and econometrics," Papers 2312.15119, arXiv.org, revised Jan 2024.

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