Abstract
In this paper, we introduce the concept of A-statistical uniform integrability of sequences of random variables which is not only more general than the concept of uniform integrability, but is also weaker than the concept of uniform integrability. We also give some characterizations of A-statistical uniform integrability and prove a law of large numbers.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. K. Chandra, “Uniform integrability in the Cesàro sense and the weak law of large numbers,” Sankhyā Ser. A 51, 309–317 (1989).
M. Ordóñez Cabrera, “Convergence in mean of weighted sums of {an,k}-compactly uniformly integrable random elements in Banach spaces,” Int. J. Math. Math. Sci. 20, 443–450 (1997).
M. Ordóñez Cabrera and A. Volodin, “Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability,” J. Math. Anal. Appl. 305, 644–658 (2005).
T. C. Hu, M. Ordóñez Cabrera, and A. Volodin, “Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights,” Statist. Probab. Lett. 51(2), 155–164 (2001).
R. Giuliano Antonini, T. C. Hu, Y. Kozachenko, and A. Volodin, “An application of φ-subgaussian technique to Fourier analysis,” J. Math. Anal. Appl. 408, 114–124 (2013).
A. D. Gadjiev and C. Orhan, “Some approximation theorems via statistical convergence,” Rocky Mountain J. Math. 32, 129–138 (2002).
D. Söylemez and M. Ünver, “Korovkin type theorems for Cheney—Sharma operators via summability methods,” Results Math. 72, 1601–1612 (2017).
N. L. Braha, “Some properties of new modified Szász-Mirakyan operators in polynomial weight spaces via power summability methods,” Bull. Math. Anal. Appl. 10(3), 53–65 (2018).
M. Ünver, “Abel transforms of positive linear operators,” AIP Conf. Proc. 1558, 1148–1151 (2013).
Ö. G. Atlihan, M. Ünver, and O. Duman, “Korovkin theorems on weighted spaces: revisited,” Period. Math. Hungar. 75, 201–209 (2017).
E. Tas and T. Yurdakadim, “Approximation by positive linear operators in modular spaces by power series method,” Positivity 21, 1293–1306 (2017).
W. Balser and M. Miyake, “Analysis-summability of formal solutions of certain partial differential equations,” Acta Sci. Math. 65, 543–552 (1999).
W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations (Springer, New York, 2000).
A. Zygmund, Trigonometric Series, 2nd ed. (Cambridge University Press, Cambridge, 1959), Vol. 1.
H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloq. Math. 2(1), 73–74 (1951).
H. Fast, “Sur la convergence statistique,” Colloq. Math., No. 2, 241–244 (1951).
I. J. Schoenberg, “The integrability of certain functions and related summability methods,” Am. Math. Mon. 66, 361–775 (1959).
T. Šalát, “On statistically convergent sequences of real numbers,” Math. Slov. 30, 139–150 (1980).
H. I. Miller, “A measure theoretical subsequence characterization of statistical convergence,” Trans. Am. Math. Soc. 347, 1811–1819 (1995).
T. Šalát V. and V. Toma, “A classical Olivier’s theorem and statistical convergence,” Ann. Math. Blaise Pascal 10, 305–313 (2003).
T. Yurdakadim, M. K. Khan, H. I. Miller, and C. Orhan, “Generalized limits and statistical convergence,” Mediterr. J. Math. 13, 1135–1149 (2016).
V. Kadets, A. Leonov, and C. Orhan, “Weak statistical convergence and weak filter convergence for unbounded sequences,” J. Math. Anal. Appl. 371, 414–424 (2010).
M. Máčaj and T. Šalát, “Statistical convergence of subsequences of a given sequence,” Math. Bohem. 126, 191–208 (2001).
A. Faisant, G. Grekos, and V. Toma, “On the statistical variation of sequences,” J. Math. Anal. Appl. 306, 432–439 (2005).
J. Connor, “On strong matrix summability with respect to a modulus and statistical convergence,” Canad. Math. Bull. 32, 194–198 (1989).
J. A. Fridy and H. I. Miller, “A matrix characterization of statistical convergence,” Analysis 11, 59–66 (1991).
M. K. Khan and C. Orhan, “Matrix characterization of A-statistical convergence,” J. Math. Anal. Appl. 335, 406–417 (2007).
P. Kostyrko, W. Wilczyński, and T. Šalát, “I-convergence,” Real Anal. Exchange 26, 669–686 (2000).
O. Duman, “A Korovkin type approximation theorems via \({\mathcal I}\)-convergence,” Czechosl. Math. J. 57, 367–375 (2007).
V. Baláž, J. Gogola and T. Visnyai, “I (q)c -convergence of arithmetical functions,” J. Number Theory 183, 74–83 (2018).
R. C. Buck, “The measure theoretic approach to density,” Am. J. Math. 68, 560–580 (1946).
R. C. Buck, “Generalized asymptotic density,” Am. J. Math. 75, 335–346 (1953).
J. A. Fridy, “On statistical convergence,” Analysis 5, 301–313 (1985).
E. Kolk, “Matrix summability of statistically convergent sequences,” Analysis 13, 77–83 (1993).
A. R. Freedman and J. J. Sember, “Densities and summability,” Pacif. J. Math. 95, 293–305 (1981).
H. Cakalli and M. K. Khan, “Summability in topological spaces,” Appl. Math. Lett. 24, 348–352 (2011).
J. A. Fridy, “Statistical limit points,” Proc. Am. Math. Soc. 118, 1187–1192 (1993).
P. Kostyrko, M. Máčaj, T. Šalát, and O. Strauch, “On statistical limit points,” Proc. Am. Math. Soc. 129, 2647–2654 (2001).
P. Kostyrko, M. Máčaj, T. Šalát, and M. Sleziak, “I-convergence and extremal I-limit points,” Math. Slov. 55, 443–464 (2005).
S. Pehlivan and M. A. Mamedov, “Statistical cluster points and turnpike,” Optimization 48, 91–106 (2000).
J. Fridy and C. Orhan, “Statistical limit superior and limit inferior,” Proc. Am. Math. Soc. 125, 3625–3631 (1997).
K. Demirci, “A-statistical core of a sequence,” Demonstr. Math. 33, 343–354 (2000).
M. Kucukaslan and M. Altınok, “A-statistical supremum-infimum and A-statistical convergence,” Azerb. J. Math. 4, 2221–9501 (2014).
J. Boos, Classical and Modern Methods in Summability (Oxford Univ. Press, Oxford, 2000).
K. L. Chung, A Course in Probability Theory, 3rd ed. (Academic, San Diego, CA, 2001).
P. A. Meyer, Probability and Potentials (Blaisdell, Waltham, MA, 1966).
S. Ghosal, “Statistical convergence of a sequence of random variables and limit theorems,” Appl. Math. 58, 423–437 (2013).
S. Ghosal, “Weighted statistical convergence of order α and its applications,” J. Egypt. Math. Soc. 24, 60–67 (2016).
Funding
The research of M. Ünver was done while he was visiting University of Regina, Canada and the research has been supported by The Scientific and Technological Research Council of Turkey (TÜBITAK) Grant 1059B191800534. A part of research of A. Volodin has been done during the visit to the University of Pisa in December 2016. The authors would like to express their appreciation to the INDAM for supporting this visit.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Submitted by A.M. Elizarov
Rights and permissions
About this article
Cite this article
Giuliano Antonini, R., Ünver, M. & Volodin, A. On the Concept of A-Statistical Uniform Integrability and the Law of Large Numbers. Lobachevskii J Math 40, 2034–2042 (2019). https://doi.org/10.1134/S1995080219120035
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080219120035