On some limit properties for the power series distribution
-
Yu Miao
,
Yanyan Tang
Abstract
In the present paper, we consider random variables with the power series distribution which is often used in the study of the generalized allocation scheme. We establish some asymptotic properties which include law of large numbers, moderate deviation principle, almost sure central limit theorem and the rate of convergence in the local limit theorem. These results supplements results obtained by A. V. Kolchin.
Note: Originally published in Diskretnaya Matematika (2022) 34,№1, 88–102 (in Russian).
Acknowledgment
The authors are very grateful to the referee for his/her careful reading and helpful comments which improved the presentation of this paper.
-
Funding: This work is supported by NSFC (11971154) and the Foundation of Young Scholar of the Educational Department of Henan province grant 2019GGJS012.
References
[1] Billingsley P., Convergence of probability measures, J. Wiley & Sons, Inc., New York, 1968.Search in Google Scholar
[2] Dembo A., Zeitouni O., Large Deviations Techniques and Applications, Second edition, Springer-Verlag, New York, 1998.10.1007/978-1-4612-5320-4Search in Google Scholar
[3] Dudley R. M., Real Analysis and Probability, Cambridge University Press, Cambridge, 2002.10.1017/CBO9780511755347Search in Google Scholar
[4] Kolchin A. V., “Limit theorems for a generalized allocation scheme”, Discrete Math. Appl., 13:6 (2003), 627–636.10.1515/156939203322733336Search in Google Scholar
[5] Kolchin V. F., Sevast’yanov B. A., Chistyakov V. P., Random Allocations, V. H. Winston & Sons, Washington, DC, 1978.Search in Google Scholar
[6] Kolchin V. F., Random Graphs, Cambridge Univ. Press, Cambridge, 1999.10.1017/CBO9780511721342Search in Google Scholar
[7] Park C. J., “A note on the classical occupancy problem”, Ann. Math. Statist., 43:5 (1972), 1698–1701.10.1214/aoms/1177692405Search in Google Scholar
[8] Peligrad M., Shao Q. M., “A note on the almost sure central limit theorem for weakly dependent random variables”, Statist. Probab. Lett., 22:2 (1995), 131–136.10.1016/0167-7152(94)00059-HSearch in Google Scholar
[9] Petrov V. V., Sums of Independent Random Variables, Springer-Verlag, New York-Heidelberg, 1975.10.1515/9783112573006Search in Google Scholar
[10] Weiss I., “Limiting distributions in some occupancy problems”, Ann. Math. Statist., 29:3 (1958), 878–884.10.1214/aoms/1177706544Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Diagnostic tests for discrete functions defined on rings
- New lower bound for the minimal number of edges of simple uniform hypergraph without the property Bk
- On some invariants under the action of an extension of GA(n, 2) on the set of Boolean functions
- On some limit properties for the power series distribution
- Estimates of lengths of shortest nonzero vectors in some lattices. I
Articles in the same Issue
- Frontmatter
- Diagnostic tests for discrete functions defined on rings
- New lower bound for the minimal number of edges of simple uniform hypergraph without the property Bk
- On some invariants under the action of an extension of GA(n, 2) on the set of Boolean functions
- On some limit properties for the power series distribution
- Estimates of lengths of shortest nonzero vectors in some lattices. I
