Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits
Fortini, Sandra ; Ladelli, Lucia ; Regazzini, Eugenio
Bollettino dell'Unione Matematica Italiana, Tome 5 (2012), p. 515-542 / Harvested from Biblioteca Digitale Italiana di Matematica

The problem of convergence in law of normed sums of exchangeable random variables is examined. First, the problem is studied w.r.t. arrays of exchangeable random variables, and the special role played by mixtures of products of stable laws - as limits in law of normed sums in different rows of the array - is emphasized. Necessary and sufficient conditions for convergence to a specific form in the above class of measures are then given. Moreover, sufficient conditions for convergence of sums in a single row are proved. Finally, a potentially useful variant of the formulation of the results just summarized is briefly sketched, a more complete study of it being deferred to a future work.

Publié le : 2012-10-01
@article{BUMI_2012_9_5_3_515_0,
     author = {Sandra Fortini and Lucia Ladelli and Eugenio Regazzini},
     title = {Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5},
     year = {2012},
     pages = {515-542},
     zbl = {1286.60025},
     mrnumber = {3051735},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2012_9_5_3_515_0}
}
Fortini, Sandra; Ladelli, Lucia; Regazzini, Eugenio. Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits. Bollettino dell'Unione Matematica Italiana, Tome 5 (2012) pp. 515-542. http://gdmltest.u-ga.fr/item/BUMI_2012_9_5_3_515_0/

[1] Bassetti, F. - Ladelli, L. - Regazzini, E., Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model. J. Statist. Phys., 133 (2008), 683-710. | MR 2456941 | Zbl 1161.82337

[2] Billingley, P., Convergence of Probability Measures, 2nd. ed. Wiley, New York (1999). | MR 1700749

[3] Blum, J. R. - Chernoff, H. - Rosenblatt, M. - Teicher, H., Central limit theorems for interchangeable processes. Canad. J. Math., 10 (1958), 222-229. | MR 96298 | Zbl 0081.35203

[4] Chow, Y. S. - Teicher, H., Probability Theory. Independence, Interchangeability, Martingales, 3rd ed. Springer-Verlag, New York (1997). | MR 1476912

[5] Daley, D. J. - Vere-Jones, D., An Introduction to the Theory of Point Processes. Elementary Theory and Methods, 2nd ed. 1 Springer-Verlag, New York (2003). | MR 950166 | Zbl 1026.60061

[6] Diaconis, P. - Holmes, S., A Bayesian peek into Feller volume I. Sankhyā Ser. A, 64 (2002), 820-841. | MR 1981513 | Zbl 1192.60003

[7] Dolera, E. - Regazzini, E., Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions. arXiv: 1206.5147 v1. | MR 3256816 | Zbl 1319.60041

[8] Dudley, R. M., Real Analysis and Probability. Cambridge Univ. Press, Cambridge (2002). | MR 1932358 | Zbl 1023.60001

[9] Fortini, S. - Ladelli, L. - Regazzini, E., A central limit problem for partially exchangeable random variables. Theory Probab. Appl., 41 (1996), 224-246. | MR 1445757 | Zbl 0881.60019

[10] Fristedt, B. - Gray, L., A Modern Approach to Probability Theory. Birkhäuser, Boston (1997). | MR 1422917 | Zbl 0869.60001

[11] Gabetta, E. - Regazzini, E., Central limit theorem for the solution of the Kac equation. Ann. Appl. Probab., 18 (2006), 2320-2336. | MR 2474538 | Zbl 1161.82018

[12] Gabetta, E. - Regazzini, E., Complete characterization of convergence to equilibrium for an inelastic Kac model. J. Statist. Phys., 147 (2012), 1007-1019. | MR 2946634 | Zbl 1254.82034

[13] Galambos, J., Advanced Probability Theory. 2nd ed. Marcel Dekker Inc., New York (1995). | MR 1350792 | Zbl 0841.60001

[14] Ibragimov, I. A. - Linnik, Y. V., Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen (1971). | MR 322926 | Zbl 0219.60027

[15] Jiang, X. - Hahn, M. G., Central limit theorems for exchangeable random variables when limits are scale mixtures of Normals. J. Theoret. Probab., 16 (2003), 543-570. | MR 2009193 | Zbl 1027.60019

[16] Jiang, X. - Hahn, M. G., Erratum to: Central limit theorems for exchangeable random variables when limits are scale mixtures of normals. J. Theoret. Probab., 25 (2012), 310-311. | MR 2886390 | Zbl 1237.60019

[17] Kallenberg, O., Probabilistic Symmetries and Invariance Principles. Springer-Verlag, New York (2005). | MR 2161313 | Zbl 1084.60003

[18] Loève, M., Probability Theory, 4th ed. 1Springer-Verlag, New York (1977). | MR 513230

[19] Regazzini, E. - Sazonov, V. V., On the central limit problem for partially exchangeable random variables with values in a Hilbert space. Theory Probab. Appl., 42 (1997), 796-812. | MR 1618750 | Zbl 0912.60045

[20] Regazzini, E., Convergence to equilibrium of the solution of Kac's kinetic equation. A probabilistic view. Boll. Unione Mat. Ital. (9), 2 (2009), 175-198. | MR 2493650 | Zbl 1177.82093

[21] Stoica, G., An extension of the weak law of large numbers for exchangeable sequences. Acta Appl. Math., 109 (2010), 759-763. | MR 2596173 | Zbl 1193.60029