Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains. (English) Zbl 1158.60034
Authors’ 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.
Reviewer: Josef Steinebach (Köln)
MSC:
| 60J05 | Discrete-time Markov processes on general state spaces |
| 62G07 | Density estimation |
| 60G60 | Random fields |
| 60J60 | Diffusion processes |
Keywords:
Markov chain; diffusion process; transition density; Edgeworth expansion; parametrix methodReferences:
| [1] | Bally, V.; Talay, D., The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function, Probab. Theory Relat. Fields, 104, 43-60 (1996) · Zbl 0838.60051 · doi:10.1007/BF01303802 |
| [2] | Bally, V.; Talay, D., The law of the Euler scheme for stochastic differential equations: II. Convergence rate of the density, Monte Carlo Methods Appl., 2, 93-128 (1996) · Zbl 0866.60049 |
| [3] | Bertail, P.; Clémenčon, S., Edgeworth expansions of suitably normalized sample mean statistics for atomic Markov chains, Probab. Theory Relat. Fields, 130, 388-414 (2004) · Zbl 1075.62075 · doi:10.1007/s00440-004-0360-0 |
| [4] | Bertail, P.; Clémenčon, S., Regenerative block bootstrap for Markov chains, Bernoulli, 12, 689-712 (2006) · Zbl 1125.62037 · doi:10.3150/bj/1155735932 |
| [5] | Bhattacharya, R.; Rao, R., Normal Approximations and Asymptotic Expansions (1976), New York: Wiley, New York · Zbl 0331.41023 |
| [6] | Bolthausen, E., The Berry-Esseen theorem for functionals of discrete Markov chains, Z. Wahrsch. verw. Geb., 54, 59-73 (1980) · Zbl 0431.60019 · doi:10.1007/BF00535354 |
| [7] | Bolthausen, E., The Berry-Esseen theorem for strongly mixing Harris recurrent Markov chains, Z. Wahrsch. verw. Geb., 60, 283-289 (1982) · Zbl 0476.60022 · doi:10.1007/BF00535716 |
| [8] | Friedman, A., Partial differential equations of parabolic type (1964), Englewood Cliffs: Prentice-Hall, Englewood Cliffs · Zbl 0144.34903 |
| [9] | Fukasawa, M.: Edgeworth expansion for ergodic diffusions. Probab. Theory Relat. Fields (in print) (2007a) · Zbl 1154.60018 |
| [10] | Fukasawa, M.: Regenerative block bootstrap for ergodic diffusions (preprint) (2007b) |
| [11] | Götze, F., Edgeworth expansions in functional limit theorems, Ann. Probab., 17, 4, 1602-1634 (1989) · Zbl 0689.60038 · doi:10.1214/aop/1176991176 |
| [12] | Götze, F.; Hipp, C., Asymptotic expansions for sums of weakly dependent random vectors, Z. Wahrsch. verw. Geb., 64, 211-239 (1983) · Zbl 0497.60022 · doi:10.1007/BF01844607 |
| [13] | Guyon, J., Euler scheme and tempered distributions, Stoch. Proc. Appl., 116, 877-904 (2006) · Zbl 1106.60053 · doi:10.1016/j.spa.2005.11.011 |
| [14] | Jacod, J., The Euler scheme for Levy driven stochastic differential equations: limit theorems, Ann. Probab., 32, 1830-1872 (2004) · Zbl 1054.65008 · doi:10.1214/009117904000000667 |
| [15] | Jacod, J.; Protter, P., Asymptotic error distributions for the Euler method for stochastic differential equations, Ann. Probab., 26, 267-307 (1998) · Zbl 0937.60060 · doi:10.1214/aop/1022855419 |
| [16] | Jacod, J.; Kurtz, T.; Meleard, S.; Protter, P., The approximate Euler method for Levy driven stochastic differential equations, Ann. de l’I.H.P., 41, 523-558 (2005) · Zbl 1071.60046 |
| [17] | Jensen, J. L., Asymptotic expansions for strongly mixing Harris recurrent Markov chains, Scand. J. Statist., 16, 47-63 (1989) · Zbl 0674.60067 |
| [18] | Konakov, V., Molchanov, S.: On the convergence of Markov chains to diffusion processes. Teoria veroyatnostei i matematiceskaya statistika, 31, 51-64 (1984) (in russian) [English translatiion in Theory Probab. Math. Stat. 31, 59-73 (1985)] · Zbl 0594.60077 |
| [19] | Konakov, V.; Mammen, E., Local limit theorems for transition densities of Markov chains converging to diffusions, Probab. Theory Relat. Fields., 117, 551-587 (2000) · Zbl 0996.60083 · doi:10.1007/PL00008735 |
| [20] | Konakov, V.; Mammen, E., Edgeworth type expansions for Euler schemes for stochastic differential equations, Monte Carlo Methods Appl., 8, 271-286 (2002) · Zbl 1028.65005 · doi:10.1515/mcma.2002.8.3.271 |
| [21] | Konakov, V.; Mammen, E., Edgeworth-type expansions for transition densities of Markov chains converging to diffusions, Bernoulli, 11, 4, 591-641 (2005) · Zbl 1098.60069 · doi:10.3150/bj/1126126762 |
| [22] | Konakov, V., Mammen, E.: Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains. Extended version (preprint). ArXiv: 0705.3139 (available under http://fr.arxiv.org/abs/0705.3139) (2007) · Zbl 1158.60034 |
| [23] | Kusuoka, S.; Yoshida, N., Malliavin calculus, geometric mixing, and expansion of diffusion functionals, Probab. Theory Relat. Fields, 116, 457-484 (2000) · Zbl 0970.60061 · doi:10.1007/s004400070001 |
| [24] | Ladyzenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.: Linear and quasi-linear equations of parabolic type. Amer. Math. Soc., Providence, Rhode Island (1968) · Zbl 0174.15403 |
| [25] | Malinovskii, V. K., Limit theorems for Harris Markov chains, 1, Theory Probab. Appl., 31, 269-285 (1987) · Zbl 0657.60087 · doi:10.1137/1131033 |
| [26] | Mykland, P. A., Aymptotic expansions and bootstrapping distributions for dependent variables: A martingale approach, Ann. Statist., 20, 623-654 (1992) · Zbl 0759.62011 · doi:10.1214/aos/1176348649 |
| [27] | McKean, H. P.; Singer, I. M., Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., 1, 43-69 (1967) · Zbl 0198.44301 |
| [28] | Protter, P.; Talay, D., The Euler scheme for Levy driven stochastic differential equations, Ann. Probab., 25, 393-323 (1997) · Zbl 0876.60030 · doi:10.1214/aop/1024404293 |
| [29] | Skorohod, A.V.: Studies in the Theory of Random Processes. Addison-Wesley, Reading [English translation of Skorohod A. V. (1961). Issledovaniya po teorii sluchainykh processov. Kiev University Press] (1965) |
| [30] | Stroock, D. W.; Varadhan, S. R., Multidimensional Diffusion Processes (1979), Berlin: Springer, Berlin · Zbl 0426.60069 |
| [31] | Yoshida, N., Partial mixing and Edgeworth expansion, Probab. Theory Relat. Fields, 129, 559-624 (2004) · Zbl 1057.62017 · doi:10.1007/s00440-003-0325-8 |
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