On exponential bounds for mixing and the rate of convergence for Student processes
Authors:
N. Abourashchi and A. Yu. Veretennikov
Translated by:
The authors
Journal:
Theor. Probability and Math. Statist. 81 (2010), 1-13
MSC (2010):
Primary 60H10, 60J60
DOI:
https://doi.org/10.1090/S0094-9000-2010-00805-2
Published electronically:
January 14, 2011
MathSciNet review:
2667305
Full-text PDF Free Access
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Additional Information
Abstract: Exponential bounds for the $\beta$-mixing coefficient are established for the Student diffusion process. The latter is a Markov diffusion process with Student distribution as a stationary measure. The method is based on a direct estimation of moments and on polynomial Lyapunov functions for evaluating exponential functionals of hitting times.
References
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- I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR 0322926
- N. V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 691–708 (Russian). MR 0339338
- N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239 (Russian). MR 563790
- N. N. Leonenko and N. Šuvak, Statistical inference for Student diffusion process, Stoch. Anal. Appl. 28 (2010), 972–1002.
- A. Yu. Veretennikov, Estimates of the mixing rate for stochastic equations, Teor. Veroyatnost. i Primenen. 32 (1987), no. 2, 299–308 (Russian). MR 902757
- A. Yu. Veretennikov, On polynomial mixing and the rate of convergence for stochastic differential and difference equations, Teor. Veroyatnost. i Primenen. 44 (1999), no. 2, 312–327 (Russian, with Russian summary); English transl., Theory Probab. Appl. 44 (2000), no. 2, 361–374. MR 1751475, DOI https://doi.org/10.1137/S0040585X97977550
- A. Yu. Veretennikov, On Approximations of Diffusions with Equilibrium, Institute of Mathematics Reports C17, Helsinki University of Technology, 2004; on-line version at http://math.tkk.fi/visitors0405/AVslides.pdf.
- A. Yu. Veretennikov and S. A. Klokov, On the subexponential rate of mixing for Markov processes, Teor. Veroyatn. Primen. 49 (2004), no. 1, 21–35 (Russian, with Russian summary); English transl., Theory Probab. Appl. 49 (2005), no. 1, 110–122. MR 2141328, DOI https://doi.org/10.1137/S0040585X97980841
References
- C. C. Heyde and N. N. Leonenko, Student processes, Adv. Appl. Probab. 37(2) (2005), 342–365. MR 2144557 (2005m:62161)
- I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publ., Groningen, 1971. MR 0322926 (48:1287)
- N. V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 691–708; English transl., Math. USSR Izv. 7 (1973), 691–709. MR 0339338 (49:4097)
- N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44(1) (1980), 161–175, 239. (Russian) MR 563790 (83c:35059)
- N. N. Leonenko and N. Šuvak, Statistical inference for Student diffusion process, Stoch. Anal. Appl. 28 (2010), 972–1002.
- A. Yu. Veretennikov, Estimates of the mixing rate for stochastic equations, Teor. Veroyatnost. Primenen. 32 (1987), no. 2, 299–308; English transl. in Theory Probab. Appl. 32 (1987), no. 2, 273–281. MR 902757 (89b:60144)
- A. Yu. Veretennikov, On polynomial mixing and the rate of convergence for stochastic differential and difference equations, Teor. Veroyatn. Primenen. 44 (1999), no. 2, 312–327; English transl. in Theory Probab. Appl. 44 (2000), no. 2, 361–374. MR 1751475 (2001k:60083)
- A. Yu. Veretennikov, On Approximations of Diffusions with Equilibrium, Institute of Mathematics Reports C17, Helsinki University of Technology, 2004; on-line version at http://math.tkk.fi/visitors0405/AVslides.pdf.
- A. Yu. Veretennikov and S. A. Klokov, On the subexponential rate of mixing for Markov processes, Teor. Veroyatn. Primenen. 49 (2004), no. 1, 21–35; English transl. in Theory Probab. Appl. 49 (2005), no. 1, 110–122. MR 2141328 (2006b:60169)
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Additional Information
N. Abourashchi
Affiliation:
School of Mathematics, University of Leeds, LS2 9JT Leeds, United Kingdom
Email:
niloufar@leeds.ac.uk
A. Yu. Veretennikov
Affiliation:
Institute for Information Transmission Problems, Moscow, Russia
Address at time of publication:
School of Mathematics, University of Leeds, LS2 9JT Leeds, United Kingdom
Email:
A.Veretennikov@leeds.ac.uk
Keywords:
Student diffusion,
exponential mixing,
heavy tails
Received by editor(s):
May 28, 2009
Published electronically:
January 14, 2011
Article copyright:
© Copyright 2010
American Mathematical Society
