Summary
As an application of general convergence results for semimartingales, exposed in their book ‘Limit Theorems for Stochastic Processes’, Jacod and Shiryaev obtained a fundamental result on the convergence of likelihood ratio processes to a Gaussian limit. We strengthen this result in a quantitative sense and show that versions of the likelihood ratio processes can be defined on the space of the limiting experiment such that we get pathwise almost sure approximations with respect to the uniform metric. The approximations are considered under both sequences of measures, the hypothesisP n and the alternative\(\overline {P^n } \). A consequence is e.g. an estimate for the speed of convergence in the Prohorov metric. New approximation techniques for stochastic processes are developed.
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Eberlein, E., Römersperger, M. Strong approximation of semimartingales and statistical processes. Probab. Th. Rel. Fields 104, 539–567 (1996). https://doi.org/10.1007/BF01198166
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DOI: https://doi.org/10.1007/BF01198166
Mathematics Subject Classification (1991)
- 62E20
- 60F17
- 60G07
- 60G15
- 60G17