Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits

Sandra Fortini; Lucia Ladelli; Eugenio Regazzini

Bollettino dell'Unione Matematica Italiana (2012)

  • Volume: 5, Issue: 3, page 515-542
  • ISSN: 0392-4041

Abstract

top
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.

How to cite

top

Fortini, Sandra, Ladelli, Lucia, and Regazzini, Eugenio. "Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits." Bollettino dell'Unione Matematica Italiana 5.3 (2012): 515-542. <http://eudml.org/doc/290847>.

@article{Fortini2012,
abstract = {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.},
author = {Fortini, Sandra, Ladelli, Lucia, Regazzini, Eugenio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {515-542},
publisher = {Unione Matematica Italiana},
title = {Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits},
url = {http://eudml.org/doc/290847},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Fortini, Sandra
AU - Ladelli, Lucia
AU - Regazzini, Eugenio
TI - Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/10//
PB - Unione Matematica Italiana
VL - 5
IS - 3
SP - 515
EP - 542
AB - 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.
LA - eng
UR - http://eudml.org/doc/290847
ER -

References

top
  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. Zbl1161.82337MR2456941DOI10.1007/s10955-008-9630-z
  2. BILLINGLEY, P., Convergence of Probability Measures, 2nd. ed. Wiley, New York (1999). MR1700749DOI10.1002/9780470316962
  3. BLUM, J. R. - CHERNOFF, H. - ROSENBLATT, M. - TEICHER, H., Central limit theorems for interchangeable processes. Canad. J. Math., 10 (1958), 222-229. Zbl0081.35203MR96298DOI10.4153/CJM-1958-026-0
  4. CHOW, Y. S. - TEICHER, H., Probability Theory. Independence, Interchangeability, Martingales, 3rd ed. Springer-Verlag, New York (1997). MR1476912DOI10.1007/978-1-4612-1950-7
  5. DALEY, D. J. - VERE-JONES, D., An Introduction to the Theory of Point Processes. Elementary Theory and Methods, 2nd ed. 1Springer-Verlag, New York (2003). Zbl1026.60061MR950166
  6. DIACONIS, P. - HOLMES, S., A Bayesian peek into Feller volume I. Sankhyā Ser. A, 64 (2002), 820-841. Zbl1192.60003MR1981513
  7. DOLERA, E. - REGAZZINI, E., Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions. arXiv: 1206.5147 v1. Zbl1319.60041MR3256816DOI10.1007/s00440-013-0530-z
  8. DUDLEY, R. M., Real Analysis and Probability. Cambridge Univ. Press, Cambridge (2002). Zbl1023.60001MR1932358DOI10.1017/CBO9780511755347
  9. FORTINI, S. - LADELLI, L. - REGAZZINI, E., A central limit problem for partially exchangeable random variables. Theory Probab. Appl., 41 (1996), 224-246. Zbl0881.60019MR1445757DOI10.1137/S0040585X97975459
  10. FRISTEDT, B. - GRAY, L., A Modern Approach to Probability Theory. Birkhäuser, Boston (1997). Zbl0869.60001MR1422917DOI10.1007/978-1-4899-2837-5
  11. GABETTA, E. - REGAZZINI, E., Central limit theorem for the solution of the Kac equation. Ann. Appl. Probab., 18 (2006), 2320-2336. Zbl1161.82018MR2474538DOI10.1214/08-AAP524
  12. GABETTA, E. - REGAZZINI, E., Complete characterization of convergence to equilibrium for an inelastic Kac model. J. Statist. Phys., 147 (2012), 1007-1019. Zbl1254.82034MR2946634DOI10.1007/s10955-012-0505-y
  13. GALAMBOS, J., Advanced Probability Theory. 2nd ed. Marcel Dekker Inc., New York (1995). Zbl0841.60001MR1350792
  14. IBRAGIMOV, I. A. - LINNIK, Y. V., Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen (1971). Zbl0219.60027MR322926
  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. Zbl1027.60019MR2009193DOI10.1023/A:1025612330587
  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. Zbl1237.60019MR2886390DOI10.1007/s10959-012-0405-8
  17. KALLENBERG, O., Probabilistic Symmetries and Invariance Principles. Springer-Verlag, New York (2005). Zbl1084.60003MR2161313
  18. LOÈVE, M., Probability Theory, 4th ed. 1Springer-Verlag, New York (1977). MR513230
  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. Zbl0912.60045MR1618750DOI10.1137/S0040585X97976520
  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. Zbl1177.82093MR2493650
  21. STOICA, G., An extension of the weak law of large numbers for exchangeable sequences. Acta Appl. Math., 109 (2010), 759-763. Zbl1193.60029MR2596173DOI10.1007/s10440-008-9344-x

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.