Bernoulli

• Bernoulli
• Volume 19, Number 5B (2013), 2294-2329.

Ergodicity and mixing bounds for the Fisher–Snedecor diffusion

Abstract

We consider the Fisher–Snedecor diffusion; that is, the Kolmogorov–Pearson diffusion with the Fisher–Snedecor invariant distribution. In the nonstationary setting, we give explicit quantitative rates for the convergence rate of respective finite-dimensional distributions to that of the stationary Fisher–Snedecor diffusion, and for the $\beta$-mixing coefficient of this diffusion. As an application, we prove the law of large numbers and the central limit theorem for additive functionals of the Fisher–Snedecor diffusion and construct $P$-consistent and asymptotically normal estimators for the parameters of this diffusion given its nonstationary observation.

Article information

Source
Bernoulli, Volume 19, Number 5B (2013), 2294-2329.

Dates
First available in Project Euclid: 3 December 2013

https://projecteuclid.org/euclid.bj/1386078604

Digital Object Identifier
doi:10.3150/12-BEJ453

Mathematical Reviews number (MathSciNet)
MR3160555

Zentralblatt MATH identifier
1296.60215

Citation

Kulik, A.M.; Leonenko, N.N. Ergodicity and mixing bounds for the Fisher–Snedecor diffusion. Bernoulli 19 (2013), no. 5B, 2294--2329. doi:10.3150/12-BEJ453. https://projecteuclid.org/euclid.bj/1386078604

References

• [1] Abourashchi, N. and Veretennikov, A.Y. (2009). On exponential mixing bounds and convergence rate for reciprocal gamma diffusion processes. Math. Commun. 14 331–339.
• [2] Aburashchi, N. and Veretennikov, O.Y. (2009). On exponential bounds for mixing and the rate of convergence for Student processes. Teor. Ĭmovīr. Mat. Stat. 81 1–12.
• [3] Anulova, S.V., Veretennikov, A.Y., Krylov, N.V., Liptser, R.Sh. and Shiryaev, A.N. (1989). Stochastic Calculus: Probability Theory—3. In Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. 45. Moscow: VINITI.
• [4] Avram, F., Leonenko, N.N. and Šuvak, N. (2011). On spectral properties and statistical analysis of Fisher–Snedecor diffusion. Preprint, available at arXiv:1007.4909.
• [5] Avram, F., Leonenko, N.N. and Šuvak, N. (2011). Parameter estimation for Fisher–Snedecor diffusion. Statistics 45 27–42.
• [6] Bhattacharya, R.N. (1982). On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrsch. Verw. Gebiete 60 185–201.
• [7] Bibby, B.M., Skovgaard, I.M. and Sørensen, M. (2005). Diffusion-type models with given marginal distribution and autocorrelation function. Bernoulli 11 191–220.
• [8] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
• [9] Bradley, R.C. (2005). Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 107–144. Update of, and a supplement to, the 1986 original.
• [10] Down, D., Meyn, S.P. and Tweedie, R.L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 1671–1691.
• [11] Forman, J.L. and Sørensen, M. (2008). The Pearson diffusions: A class of statistically tractable diffusion processes. Scand. J. Statist. 35 438–465.
• [12] Genon-Catalot, V., Jeantheau, T. and Larédo, C. (2000). Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6 1051–1079.
• [13] Gīhman, Ĭ.Ī. andSkorohod, A.V. (1972). Stochastic Differential Equations. Ergebnisse der Mathematik und ihrer Grenzgebiete 72. New York: Springer. Translated from the Russian by Kenneth Wickwire.
• [14] Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application. Probability and Mathematical Statistics. New York: Academic Press [Harcourt Brace Jovanovich Publishers].
• [15] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. Amsterdam: North-Holland.
• [16] Karlin, S. and Taylor, H.M. (1981). A Second Course in Stochastic Processes. New York: Academic Press [Harcourt Brace Jovanovich Publishers].
• [17] Kolmogoroff, A. (1931). Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 415–458.
• [18] Kulik, A.M. (2009). Exponential ergodicity of the solutions to SDE’s with a jump noise. Stochastic Process. Appl. 119 602–632.
• [19] Kulik, A.M. (2011). Asymptotic and spectral properties of exponentially $\phi$-ergodic Markov processes. Stochastic Process. Appl. 121 1044–1075.
• [20] Kulik, A.M., Leonenko, N.N. and Šuvak, N. (2012). Statistical inference for Fisher–Snedecor diffusion process. Preprint.
• [21] Ladyzhenskaja, O.A., Solonnikov, V.A. and Ural’ceva, N.N. (1968). Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23. Providence, RI: Amer. Math. Soc. Translated from the Russian by S. Smith.
• [22] Leonenko, N.N. and Šuvak, N. (2010). Statistical inference for reciprocal gamma diffusion process. J. Statist. Plann. Inference 140 30–51.
• [23] Leonenko, N.N. and Šuvak, N. (2010). Statistical inference for student diffusion process. Stoch. Anal. Appl. 28 972–1002.
• [24] Pearson, K. (1914). Tables for Statisticians and Biometricians. Cambridge: Cambridge Univ. Press.
• [25] Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
• [26] Shaw, T.W. and Munir, A. (2009). Dependency without copulas or ellipticity. European Journal of Finance 15 661–674.
• [27] Shiryayev, A.N. (1992). On analytic methods in probability theory. In Selected Works of A.N. Kolmogorov, Volume II 62–108. Dordrecht: Kluwer Academic Publisher.
• [28] Veretennikov, A.Y. (1987). Estimates of the mixing rate for stochastic equations. Theory Probab. Appl. 32 299–308.