Uniform integrability of sequences of random elements with respect to weak topologies and weak integrals

Year 2023, , 91 - 102, 15.02.2023

Havva Uluçay Mehmet Ünver

https://doi.org/10.15672/hujms.1065594

Cited By: 1

Abstract

In probability theory, uniform integrability of families of random variables or random elements plays an important role in the mean convergence. In this paper, we introduce a new version of uniform integrability for sequences in normed spaces in the weak sense. We study the relationship of this new concept with summability theory by considering statistical convergence. We also define a new type of uniform integrability of random elements taking values in topological vector spaces by considering weak integrals. Moreover, we study the connection of summability theory with this new concept as well.

Keywords

weak uniform integrability, statistical convergence, strong convergence

References

• [1] R. G. Antonini, M. Unver and A. Volodin, On the concept of A-statistical uniform integrability and the law of large numbers, Lobachevskii J. Math. 40 (12), 2034-2042, 2019.
• [2] P. Billingsley, Convergence of Probability Measures, John Wiley and Sons, 1999.
• [3] S. Bochner, Monotone funktionen, stieltjessche integrale und harmonische analyse, Math. Ann. 108 (1), 378-410, 1933.
• [4] K. L. Chung, A Course in Probability Theory, Academic Press, 2000.
• [5] J. Connor, M. Ganichev and V. Kadets A characterization of Banach spaces with separable duals via weak statistical convergence, J. Math. Anal. Appl. 244 (1), 251- 261, 2000.
• [6] J. A. Cuesta and C. Matràn, Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions, J. Multivar. Anal. 25, 311-322, 1988.
• [7] H. Çakalli and M. K. Khan, Summability in topological spaces, Appl. Math. Lett. 24 (3), 348-352, 2011.
• [8] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.
• [9] J. A. Fridy, On statistical convergence, Analysis 5 (4), 301-313, 1985.
• [10] J. A. Fridy and H. I. Miller, A matrix characterization of statistical convergence, Analysis 11 (1), 59-66, 1991.
• [11] C. Godet-Thobie and B. Satco, Decomposability and uniform integrability in Pettis integration, Quaest. Math. 29 (1), 39-58, 2006.
• [12] M. K. Khan and C. Orhan, Characterizations of strong and statistical convergences, Publ. Math. Debr. 76 (1-2), 77-88, 2010.
• [13] E. Kolk, Statistically convergent sequences in normed spaces, Methods of algebra and analysis. Tartu. 63-66, 1988.
• [14] G. Di Maio and L. D. R. Kočinac, Statistical convergence in topology, Topology Appl. 156 (1), 28-45, 2008.
• [15] M. Ordóñez Cabrera, Convergence of weighted sums of random variables and uniform integrability concerning the weights, Collect. Math. 45 (2), 121-132, 1994.
• [16] M. Ordóñez Cabrera, A. Rosalsky, M. Ünver, A. Volodin, A new type of compact uniform integrability with application to degenerate mean convergence of weighted sums of Banach space valued random elements, J. Math. Anal. Appl. 487 (1), 123975, 2020.
• [17] M. Ordóñez Cabrera, A. Rosalsky, M. Ünver, A. Volodin, On the concept of Bstatistical uniform integrability of weighted sums of random variables and the law of large numbers with mean convergence in the statistical sense, TEST 30 (1), 83-102, 2021.
• [18] M. Ordóñez Cabrera, Convergence in mean of weighted sums of $\{a_{n,k}\}$-compactly uniformly integrable random elements in Banach spaces, Int. J. Math. Sci. 20 (3), 443-450, 1997.
• [19] V. S. Pugachev and I. N. Sinitsyn, Lectures on Functional Analysis and Applications, World Scientific, 1999.
• [20] M. Ünver and H. Uluçay, Compactly uniform Bochner integrability of random elements, Positivity 21 (4), 1261-1272, 2017.
• [21] X. C. Wang and M. B. Rao, Some results on the convergence of weighted sums of random elements in separable Banach spaces, Studia Math. 86 (2), 131-153, 1987.