On edgeworth expansions for dependency-neighborhoods chain structures with strong mixing characteristics

V. I. Rotar'

Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 1, pp. 21-40

Résumé

Let $W$ be the sum of dependent random variables, and let $h(x)$ be a function. This paper modifies results of [Y. Rinott and V. I. Rotar, Probab. Theory Related Fields, 126 (2003), pp. 528–570] concerning Edgeworth expansions for $\mathbf E\{h(W)\}$ in the case of the so-called dependency-neighborhoods structures. The most typical example of such structures is mixing on graphs. The main improvement consists in the replacement of $\phi$-mixing dependency characteristics by those of strong (or $\alpha$-) mixing, which essentially widens the possibilities of applications. Another improvement concerns the smoothness conditions. We show how conditions on the smoothness of $h$ may be replaced by conditions on the smoothness of the distribution of $W$.

Bibliographie

[1] Barbour A. D., “Asymptotic expansions based on smooth functions in the central limit theorem”, Probab. Theory Related Fields, 72:2 (1986), 289–303 | DOI | MR | Zbl

[2] Barbour A. D., Karoński M., Ruciński A., “A central limit theorem for decomposable random variables with applications to random graphs”, J. Combin. Theory, Ser. B, 47:2 (1989), 125–145 | DOI | MR | Zbl

[3] Bening V. E., “On the rate of convergence and asymptotic expansions for $U$-statistics under alternatives”, J. Math. Sci. (New York), 99:4 (2000), 1403–1407 | DOI | MR | Zbl

[4] Bentkus V., Götze F., van Zwet W. R., “An Edgeworth expansion for symmetric statistics”, Ann. Statist., 25:2 (1997), 851–896 | DOI | MR | Zbl

[5] Bhattacharya R., Ranga Rao R., Normal Approximation and Asymptotic Expansions, Krieger, Melbourne, 1986, 291 pp. | MR | Zbl

[6] Bickel P. J., Götze F., van Zwet W. R., “The Edgeworth expansion for $U$-statistics of degree two”, Ann. Statist., 14:4 (1986), 1463–1484 | DOI | MR | Zbl

[7] Bloznelis M., Götze F., “One-term Edgeworth expansion for finite population $U$-statistics of degree two”, Acta Appl. Math., 58:1–3 (1999), 75–90 | DOI | MR | Zbl

[8] Bradley R. C., “Basic properties of strong mixing conditions”, Dependence in Probability and Statistics: A Survey of Recent Results (Oberwolfach, 1985), Progr. Probab. Statist., 11, Birkhäuser, Boston, 1985, 165–192 | MR

[9] Bradley R., Introduction to Strong Mixing Conditions., V. 1. Technical Report, Department of Mathematics, Custom Publishing of IU, Indiana University, Bloomington, 2002 | MR

[10] Davydov Yu. A., “Printsip invariantnosti dlya statsionarnykh protsessov”, Teoriya veroyatn. i ee primen., 15:3 (1970), 498–509 | MR | Zbl

[11] Götze F., Hipp C., “Asymptotic expansions for sums of weakly dependent random vectors”, Z. Wahrscheinlichkeitstheor. Verw. Geb., 64:2 (1983), 211–239 | DOI | MR | Zbl

[12] Heinrich L., “Some remarks on asymptotic expansions in the central limit theorem for $m$-dependent random variables”, Math. Nachr., 122 (1985), 151–155 | DOI | MR | Zbl

[13] Heinrich L., “Asymptotic expansions in the central limit theorem for a special class of $m$-dependent random fields. I”, Math. Nachr., 134 (1987), 83–106 | DOI | MR | Zbl

[14] Heinrich L., “Asymptotic expansions in the central limit theorem for a special class of $m$-dependent random fields. II. Lattice case”, Math. Nachr., 145 (1990), 309–327 | DOI | MR | Zbl

[15] Hipp C., “Asymptotic expansions in the central limit theorem for compound and Markov processes”, Z. Wahrscheinlichkeitstheor. Verw. Geb., 69:3 (1985), 361–385 | DOI | MR | Zbl

[16] Ibragimov I. A., “Nekotorye predelnye teoremy dlya statsionarnykh v uzkom smysle veroyatnostnykh protsessov”, Dokl. AN SSSR, 125:4 (1959), 711–714 | Zbl

[17] Iosifescu M., Theodorescu R., Random Processes and Learning, Grundlehren Math. Wiss., 150, Springer-Verlag, Berlin, 1969, 304 pp. | MR

[18] Jensen J. G., “Asymptotic expansions for strongly mixing Harris recurrent Markov chains”, Scand. J. Statist., 16:1 (1989), 47–63 | MR | Zbl

[19] Jensen J. G., “A note on asymptotic expansions for sums over a weakly dependent random field with application to the Poisson and Strauss processes”, Ann. Inst. Statist. Math., 45:2 (1993), 353–360 | DOI | MR | Zbl

[20] Korolyuk V. S., Borovskikh Yu. V., “Razlozheniya dlya $U$-statistik i funktsionalov Mizesa”, Ukr. matem. zhurn., 37:4 (1985), 450–456 | MR | Zbl

[21] Lahiri S. N., “Refinements in asymptotic expansions for sums of weakly dependent random vectors”, Ann. Probab., 21:2 (1993), 791–799 | DOI | MR | Zbl

[22] Loh W.-L., “An Edgeworth expansion for $U$-statistics with weakly dependent observations”, Statist. Sinica, 6:1 (1996), 171–186 | MR | Zbl

[23] Maesono Y., “Asymptotic mean square errors of variance estimators for $U$-statistics and their Edgeworth expansions”, J. Japan Statist. Soc., 28:1 (1998), 1–19 | MR | Zbl

[24] Malinovskii V. K., “O predelnykh teoremakh dlya kharrisovskikh tsepei Markova, I”, Teoriya veroyatn. i ee primen., 31:2 (1986), 315–332 | MR

[25] Mykland P. A., “Asymptotic expansions for martingales”, Ann. Probab., 21:2 (1993), 800–818 | DOI | MR | Zbl

[26] Petrov V. V., Limit Theorems of Probability Theory. Sequences of Independent Random Variables, Oxford Stud. Probab., 4, Clarendon Press, New York, 1995, 292 pp. | MR | Zbl

[27] Rhee W., “An Edgeworth expansion for a sum of $m$-dependent random variables”, Internat. J. Math. Math. Sci., 8:3 (1985), 563–569 | DOI | MR | Zbl

[28] Rinott Y., Rotar V., “On Edgeworth expansions for dependency-neighborhoods chain structures and Stein's method”, Probab. Theory Related Fields, 126:4 (2003), 528–570 | DOI | MR | Zbl

[29] Rinott Y., Rotar V., “A multivariate CLT for local dependence with $n^{-1/2}\log n$ rate, and applications to multivariate graph related statistics”, J. Multivariate Anal., 56:2 (1996), 333–350 | DOI | MR | Zbl

[30] Rotar V. I., “Stein's method, Edgeworth's expansions, and a formula of Barbour”, Stein's Method and Applications, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5, eds. A. D. Barbour and L. H. Y. Chen, Singapore Univ. Press, Singapore, 2005, 59–84 | MR

[31] Rotar V. I., Teoriya veroyatnostei, Vysshaya shkola, M., 1992, 368 pp. | MR | Zbl

[32] Rosenblatt M., “A central limit theorem and a strong mixing condition”, Proc. Natl. Acad. Sci. USA, 42 (1956), 43–47 | DOI | MR | Zbl

[33] Statulyavichus V. A., “Predelnye teoremy dlya summ sluchainykh velichin, svyazannykh v tsep Markova. I, II, III”, Litov. matem. sb., 9:2 (1969), 345–362; 9:3 (1969), 635–672; 10:1 (1970), 161–169

[34] Stein C., “A bound for the error in the normal approximation to the distribution of a sum of dependent random variables”, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2 (Berkeley, 1970/1971), Univ. California Press, Berkeley, 1972, 583–602 | MR | Zbl

[35] Stein C., Approximate Computation of Expectations, Inst. Math. Statist., Hayward, 1986, 164 pp. | MR