On the necessary condition for Baum-Katz type theorem for non-identically distributed and negatively dependent random fields

Zbigniew Łagodowski

Abstract


Let  \(\{ X_{\bf n}, {\bf n}\in \mathbb{N}^d \}\) be a random field of negatively dependent  random variables.  The complete  convergence results for negatively dependent  random fields  are refined. To obtain the main theorem several lemmas  for convergence of families indexed by \(\mathbb{N}^d\)   have been proved. Auxiliary lemmas have wider application to study  the random walks on the lattice.

Keywords


Baum-Katz type theorems; complete convergence; negatively dependent random fields; convergence of families indexed by directed sets; metric space

Full Text:

PDF

References


Baum, L. E., Katz, M., Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), 108-123.

Bulinski, A., Shashkin, A., Limit Theorems for Associated Random Fields and Related Systems, World Scientific Publishing, Singapore, 2007.

Gut, A., Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann. Probability 6 (3) (1978), 469-482.

Gut, A., Stadtmuller, U., An asymmetric Marcinkiewicz-Zygmund LLN for random fields, Statist. Probab. Lett. 79 (8) (2009), 1016-1020.

Gut, A., Stadtmuller, U., On the Hsu-Robbins-Erdos-Spitzer-Baum-Katz theorem for random fields, J. Math. Anal. Appl. 387 (1) (2012), 447-463.

Hsu, P. L., Robbins, H., Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31.

Klesov, O. I., The strong law of large numbers for multiple sums of independent identically distributed random variables, Matem. Zametki 38 (1985), 915-930 (English transl. in Math. Notes 38 (1986), 1006-1014).

Klesov, O. I., Limit Theorems for Multi-Indexed Sums of Random Variables, Springer-Verlag, Berlin-Heidelberg, 2014.

Łagodowski, Z. A., An approach to complete convergence theorems for dependent random fields via application of Fuk–Nagaev inequality, J. Math. Anal. Appl. 437 (2016), 380-395.

Lehmann, E. L., Some concepts of dependence, Ann. Math. Statist. 37 (1966), 1137-1153.

Neveu, J., Discrete-Parameter Martingales, North-Holland, Amsterdam; American Elsevier, New York, 1975.




DOI: http://dx.doi.org/10.17951/a.2018.72.2.1
Date of publication: 2018-12-22 22:03:10
Date of submission: 2018-12-21 13:00:51


Statistics


Total abstract view - 854
Downloads (from 2020-06-17) - PDF - 473

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2018 Zbigniew Łagodowski