Abstract
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points.
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This research was supported by grant 436RUS113/467/81-2 from the Deutsche Forschungsgemeinschaft and by grants 05-01-04004 and 04-01-00700 from the Russian Foundation of Fundamental Researches. The first author worked on the paper during a visit at the Laboratory of Probability Theory and Random Models of the University Paris VI in 2006. He is grateful for the hospitality during his stay. We would like to thank Stephane Menozzi, two referees and the associate editor for helpful comments
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Konakov, V., Mammen, E. Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains. Probab. Theory Relat. Fields 143, 137–176 (2009). https://doi.org/10.1007/s00440-007-0123-9
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DOI: https://doi.org/10.1007/s00440-007-0123-9
Keywords
- Markov chains
- Diffusion processes
- Transition densities
- Edgeworth expansions
Mathematics Subject Classification (2000)
- Primary: 62G07
- Secondary: 60G60