Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


We consider stable type p Banach spaces. We extend results known for independent random variables to the mixing random variables.

In particular we prove moment in equalities, low of large numbers and almost sure convergence of the series in the case of mixing random variables.

First Page


Last Page



1. Denker M. and Jakubowski A. Stable limit distributions for strongly mixing sequences. Statist. Probab. Lett. 8, pp. 477–483 (1989).

2. Hoffmann-Jørgensen J. Sums of independent Banach space valued random variables, Studia Math. 52, pp. 159–186 (1974).

3. Hoffmann-Jørgensen J., Pisier G. The law of large numbers and the central limit theorem in Banach spaces. Ann. Probab. 4, pp. 587–599 (1976).

4. Iosifescu M., Theodorescu R. Random Processes and Learning. Springer Verlag. Berlin, (1969).

5. Jajte R. On stable distributions in Hilbert space. Studia Math. XXX, pp. 63-71 (1968).

6. Jakubowski A. and Kobus M. p−stable limit theorems for sums of dependent random vectors. J. Multivariate Anal. 29, pp. 219–251 (1989).

7. Jakubowski A. On multidimensional domains of attraction for stationary sequences. Statist. Probab. Lett. 19, pp. 321–326 (1994).

8. Jakubowski A. Minimal conditions in p−stable limit theorems. Stoch. Proc. Appl. 44, pp. 291–327 (1993).

9. Jakubowski A. Minimal conditions in p−stable limit theorems - II. Stoch. Proc. Appl. 68, pp. 1–20 (1997).

10. Johnson O. and Samworth R. Central limit theorem and convergence to stable laws in Mallows distance. Bernoulli 11(5), pp. 829–845 (2005).

11. Jurek Z. and Urbanik K. Remarks on stable measures on Banach spaces. Colloquium mathematicum, Vol. 38, Fasc. 2, (1978).

12. Korzeniowski A. On Marcinkiewicz SLLN in Banach spaces. Ann. Probab. 12, pp. 279–280 (1984).

13. Klosowska M. The domain of attraction of a non-gaussian stable distribution in a Hilbert space. Colloq. Math. 32, pp. 127–136 (1972).

14. Kuelbs J. and Mandrekar V. Domains of attraction of stable measures on a Hilbert space. Studia Math. 59, pp. 149–162 (1974).

15. Ledoux M., Talagrand M. Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin (1991).

16. Marcus M.B. and Woyczynski W.A. Stable measures and central limit theorems in spaces of stable type. Transactions of the American Mathematical Society, Volume 251, pp. 71–102 (July 1979).

17. Maurey B., Pisier G. Series de variables aleatoires vectorielles independantes et proprietes geometriques des espaces de Banach. Stud. Math. 58, pp. 45–90 (1976).

18. Nadarajah S., Chan S. The exact distribution of the sum of stable random variables. Journal of Computational and Applied Mathematics. 349, pp. 187–196 (2019).

19. Pisier G. Probabilistic methods in the geometry of Banach spaces. In: Probability and Analysis, Lectures Given at the 1st 1985 Session of the Centro Internazionale Matematico Estivo (C.I.M.E). Lecture Notes in Mathematics, Springer, Berlin, vol. 1206, pp. 167–241 (1986).

20. Rosinski J. Remarks on Banach spaces of stable type. Probab. Math. Statist. 1, pp. 67–71 (1980).

21. Sharipov O.Sh. On probabilistic characterization of Banach spaces by weakly dependent random elements. Uzb. Math. J. № 2, pp. 86–91 (1998). (in Russian)

22. Woyczynski W.A. Geometry and martingales in Banach spaces, Part II: Independent increments. In: Kuelbs, J., Ney, P. (eds.) Probability on Banach Spaces. Advances in Probability and Related Topics, Dekker, New York, vol. 4, pp. 267–517 (1978).

23. Woyczynski W.A. Geometry and martingales in Banach spaces. CRC Press; 1st edition, (2020).

24. Woyczynski W.A. Random series and laws of large numbers in some Banach spaces. Teorya Veroyatnostej i Prim. 18, pp. 371–377 (1973).

25. Zuparov T.M. Investigations on limit theorems for sums of weakly dependent Banach valued random variables. Doctor of Sciences thesis. Tashkent, 284 pp (1991). (in Russian)



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.