Abstract
As the concept of uniform integrability has an essential role in the convergence of moments and martingales in the probability theory it is important to study this concept. In the present paper using Bochner integral we define a new type of uniform integrability for sequences of random elements. We give some independent necessary and sufficient conditions for this concept. We also generalize the concepts of strong and statistical convergences to sequences of random elements and we obtain the relationship between these two important concepts of summability theory via this new type of uniform integrability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chung, K.L.: A Course in Probability Theory. Academic Press, San Diego (2000)
Cabrera, M.O.: Convergence of weighted sums of random variables and uniform integrability concerning the weights. Collect. Math. 45(2), 121–132 (1994)
Wang, X.C., Rao, M.B.: Some results on the convergence of weighted sums of random elements in separable Banach spaces. Studia Math. 86(2), 131–153 (1987)
Cuesta, J.A., Matran, C.: Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions. J. Multivar. Anal. 25, 311–322 (1988)
Cabrera, M.O.: Convergence in mean of weighted sums of \(\{a_{n, k}\}\)-compactly uniformly integrable random elements in Banach spaces. Internat J. Math. Math. Sci. 20(3), 443–450 (1997)
Khan, M.K., Orhan, C.: Characterizations of strong and statistical convergences. Publ. Math. Debr. 76, 77–88 (2010)
Fridy, J.A.: On statistical convergence. Analysis 5, 301–313 (1985)
Connor, J.S.: On strong matrix summability with respect to a modulus and statistical convergence. Can. Math. Bull. 32, 194–198 (1989)
Connor, J.S.: The statistical and strong p-Cesàro convergence of sequences. Analysis 8, 47–63 (1988)
Khan, M.K., Orhan, C.: Matrix characterization of A-statistical convergence. J. Math. Anal. Appl. 335, 406–417 (2007)
Fridy, J.A., Miller, H.I.: A matrix characterization of statistical convergence. Analysis 11, 59–66 (1991)
Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)
Di Maio, G., Kočinac, L.D.R.: Statistical convergence in topology. Topol. Appl. 156(1), 28–45 (2008)
Cakalli, H., Khan, M.K.: Summability in topological spaces. Appl. Math. Lett. 24(3), 348–352 (2011)
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer Science and Business Media, Berlin (2007)
Pugachev, V.S., Sinitsyn, I.N.: Lectures on Functional Analysis and Applications. World Scientific, Singapore (1999)
Wilansky, A.: Summability Through Functional Analysis. Elsevier, Amsterdam (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
This study was supported by TÜBiTAK during the project 3001-115F292.
Rights and permissions
About this article
Cite this article
Ünver, M., Uluçay, H. Compactly uniform Bochner integrability of random elements. Positivity 21, 1261–1272 (2017). https://doi.org/10.1007/s11117-017-0465-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-017-0465-1