×
Giap, Duong Xuan; van Huan, Nguyen; Ngoc, Bui Nguyen Tram; van Quang, Nguyen

Multivalued strong laws of large numbers for triangular arrays with gap topology. (English) Zbl 1456.60068

J. Convex Anal. 27, No. 4, 1157-1176 (2020).
Summary: We state some strong laws of large numbers for triangular arrays of random sets in separable Banach spaces with the gap topology and with or without compactly uniformly integrable condition.

MSC:

60F15 Strong limit theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections

Keywords:

random set; gap topology; triangular array; compactly uniformly integrable
Cite Review PDF
Full Text: Link

References:

[1] Z. Artstein, S. Hart:Law of large numbers for random sets and allocation processes, Math. Oper. Res. 6 (1981) 485-492. · Zbl 0524.28015
[2] Z. Artstein, R. A. Vitale:A strong law of large numbers for random compact sets, Ann. Probability 3 (1975) 879-882. · Zbl 0313.60012
[3] G. Beer:Topologies on closed and closed convex sets and the Effros measurability of set valued functions, Sém. Anal. Convexe 21 (1991) 44 pp. · Zbl 0824.46089
[4] G. Beer:Wijsman convergence of convex sets under renorming, Nonlinear Analysis 22 (1994) 207-216. · Zbl 0815.46010
[5] A. Bozorgnia, R. F. Patterson, R. L. Taylor:Chung type strong laws for arrays of random elements and bootstrapping, Stochastic Anal. Appl. 15 (1997) 651-669. · Zbl 0899.60028
[6] C. Castaing, N. V. Quang, D. X. Giap:Mosco convergence of strong law of large numbers for double array of closed valued random variables in Banach space, J. Nonlinear Convex Analysis 13 (2012) 615-636. · Zbl 1279.60038
[7] C. Castaing, N. V. Quang, D. X. Giap:Various convergence results in strong law of large numbers for double array of random sets in Banach spaces, J. Nonlinear Convex Analysis 13 (2012) 1-30. · Zbl 1243.60028
[8] D. X. Giap, N. V. Quang:Multidimensional and multivalued ergodic theorems for measure-preserving transformations, Set-Valued Var. Analysis 24 (2016) 637-658. · Zbl 1356.28008
[9] C. Hess:Loi forte des grands nombres pour des ensembles aléatoires non bornés à valeurs dans un espace de Banach séparable (French), C. R. Acad. Sci. Paris Sér. I Math. 300 (1985) 177-180. · Zbl 0582.60009
[10] C. Hess:The distribution of unbounded random sets and the multivalued strong law of large numbers in nonreflexive Banach spaces, J. Convex Analysis 6 (1999) 163-182. · Zbl 0945.60024
[11] F. Hiai:Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Amer. Math. Soc. 291 (1985) 613-627. · Zbl 0583.60007
[12] J. Hoffmann-Jørgensen, G. Pisier:The law of large numbers and the central limit theorem in Banach spaces, Ann. Probability 4 (1976) 587-599. · Zbl 0368.60022
[13] T. C. Hu, F. Móricz, R. L. Taylor:Strong laws of large numbers for arrays of rowwise independent random variables, Acta Math. Hungar. 54 (1989) 153-162. · Zbl 0685.60032
[14] T. C. Hu, R. L. Taylor:On the strong law for arrays and for the bootstrap mean and variance, Internat. J. Math. Math. Sci. 20 (1997) 375-382. · Zbl 0883.60024
[15] N. V. Quang, D. X. Giap:Mosco convergence of SLLN for triangular arrays of rowwise independent random sets, Statist. Probab. Lett. 83 (2013) 1117-1126. · Zbl 1267.60033
[16] N. V. Quang, D. X. Giap, B. N. T. Ngoc, T. C. Hu:Some strong laws of large numbers for double arrays of random sets with gap topology, J. Convex Analysis 26 (2019) 719-738. · Zbl 1480.60070
[17] P.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.