Exact strong laws of large numbers for independent random fields

• Autorzy: Kurasiński P. , Matuła P. , Adler A.

Identyfikatory

DOI

10.4064/ba8153-9-2018

• Języki publikacji: EN

Abstrakty

EN

Let {X_ṉ, ṉ ϵ N^d} be a family of independent random variables with multidimensional indices (a random field) with the same distribution as the r.v. X. A necessary and sufficient condition for the strong law of large numbers in this setting is E|X| log^d-1₊|X| < ∞. Our goal is to study the almost sure convergence of normalized or weighted sums in the case when this moment condition is not satisfied.

Słowa kluczowe

EN

strong law of large numbers; slowly varying functions; random fields

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2018
• Tom: Vol. 66, no. 2
• Strony: 179--188
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Kurasiński P.

• Institute of Mathematics, Curie-Skłodowska University, Pl. M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland

autor

Matuła P.

• Institute of Mathematics, Curie-Skłodowska University, Pl. M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland

autor

Adler A.

• Department of Mathematics, Illinois Institute of Technology, Chicago, IL 60616, U.S.A.

Bibliografia

• [1] A. Adler, Exact strong laws, Bull. Inst. Math. Acad. Sinica 28 (2000), 141-166.
• [2] A. Adler, Exact strong laws for multidimensionally indexed random variables, J. Multivariate Anal. 77 (2001), 73-83.
• [3] A. Adler and P. Matuła, On exact strong laws of large numbers under general dependence conditions, Probab. Math. Statist. 38 (2018), 103-121.
• [4] A. Adler and Y. Qi, On exact strong laws for sums of multidimensionally indexed random variables, Probab. Math. Statist. 23 (2003), 305-313.
• [5] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge Univ. Press, Cambridge, 1987.
• [6] Y. S. Chow and H. Robbins, On sums of independent random variables with infinite moments and “fair” games, Proc. Nat. Acad. Sci. USA 47 (1961), 330-335.
• [7] A. Gut, Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann. Probab. 6 (1978), 469-482.
• [8] D. H. Hong and J. M. Park, Exact sequences for sums of pairwise i.i.d. random variables, Bull. Korean Math. Soc. 30 (1993), 167-170.
• [9] O. Klesov, Limit Theorems for Multi-Indexed Sums of Random Variables, Springer, 2014.
• [10] R. T. Smythe, Strong laws of large numbers for r-dimensional arrays of random variables, Ann. Probab. 1 (1973), 164-170.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).