Abstract.
For families of processes with independent increments ηɛ(t), 0≤t≤T, with frequent small jumps, limit theorems for expectations of the functionals F(ηɛ[0, T]) are proved of the form where diD , are positive numbers, A di are linear integro-differential or differential operators acting on functionals, and some differentiability conditions are imposed on the functional F. The case of power ‘tails’ of the jump distribution is considered.
References
Banys, J.: Convergence in the mean for densities in the case of a stable limit law. Litovsk. Mat. Sb. 15, 71–78, 249 (1975)
Cramér, H.: On asymptotic expansions for sums of independent random variables with a limiting stable distribution. Sankhyā A25, 13–24 (1963)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. J. Wiley & Sons: New York et al., 1986
Godovan’chuk, V.V.: Asymptotic probabilities of large deviations due to large jumps of a Markov process. Teor. veroyatn. i primen. 26, 314–327 (1981)
Sviridenko, M.N.: Weak convergence to processes connected with stable distributions. Mat. Zametki 50, 764–765 (1991)
Wentzell, A.D.: Infinitesimal characteristics of Markov processes in a function space, which describe the past. Teor. veroyatn. i primen. 30, 625–639 (1985)
Wentzell, A.D.: Limit Theorems on Large Deviations for Markov Stochastic Processes. Kluwer: Dordrecht et al., 1990
Wentzell, A.D.: On differentiability of the expectations of functionals of a Markov process. Stochastics and Stochastic Reports. 39, 53–65 (1992)
Wentzell, A.D.: Asymptotic expansions in limit theorems for stochastic processes. -I. Probability Theory and Related Fields 106, 331–350 (1996)
Wentzell, A.D.: Asymptotic expansions in limit theorems for stochastic processes. -II. Probability Theory and Related Fields 113, 255–271 (1999)
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Mathematics Subject Classification (2000): 60F17
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Wentzell, A. Asymptotic expansions in limit theorems for stochastic processes. – III. Probab. Theory Relat. Fields 128, 63–81 (2004). https://doi.org/10.1007/s00440-003-0294-y
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DOI: https://doi.org/10.1007/s00440-003-0294-y
Keywords
- Stochastic Process
- Differential Operator
- Asymptotic Expansion
- Limit Theorem
- Independent Increment