A new version of uniform integrability via power series summability methods

M. Ordóñez Cabrera   ; A. Rosalsky   ; M. Ünver   ; A. Volodin

Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 1, pp. 115-133

Résumé

Uniform integrability is an interesting concept in probability theory and functional analysis since it plays an important role in establishing laws of large numbers. In the literature, there are several versions of uniform integrability. Some are defined with the help of matrix summability methods, such as the Cesàro matrix, or more general methods. In this paper, we introduce a new version of uniform integrability via power series summability methods. We investigate the relationships of this new concept with some previous concepts and give $L_1$- and $L_2$-convergence results for the laws of large numbers.

Détail

Keywords: uniform integrability, power series summability method
Mots-clés : $L_1$-convergence.

Bibliographie

[1] K. L. Chung, A course in probability theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2001, xviii+419 pp. | MR | Zbl

[2] T. K. Chandra, “Uniform integrability in the Cesàro sense and the weak law of large numbers”, Sankhyā Ser. A, 51:3 (1989), 309–317 | MR | Zbl

[3] M. Ordóñez Cabrera, “Convergence of weighted sums of random variables and uniform integrability concerning the weights”, Collect. Math., 45:2 (1994), 121–132 | MR | Zbl

[4] M. Ordóñez Cabrera, A. I. 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., 305:2 (2005), 644–658 | DOI | MR | Zbl

[5] T.-C. Hu, M. Ordóñez Cabrera, A. I. Volodin, “Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights”, Statist. Probab. Lett., 51:2 (2001), 155–164 | DOI | MR | Zbl

[6] R. Giuliano Antonini, T.-C. Hu, Y. Kozachenko, A. Volodin, “An application of $\varphi$-subgaussian technique to Fourier analysis”, J. Math. Anal. Appl., 408:1 (2013), 114–124 | DOI | MR | Zbl

[7] A. D. Gadjiev, C. Orhan, “Some approximation theorems via statistical convergence”, Rocky Mountain J. Math., 32:1 (2002), 129–138 | DOI | MR | Zbl

[8] D. Söylemez, M. Ünver, “Korovkin type theorems for Cheney–Sharma operators via summability methods”, Results Math., 72:3 (2017), 1601–1612 | DOI | MR | Zbl

[9] 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 (2018), 53–65 | MR | Zbl

[10] M. Ünver, “Abel transforms of positive linear operators”, 11th international conference of numerical analysis and applied mathematics (ICNAAM 2013) (Rhodes, 2013), AIP Conf. Proc., 1558, no. 1, Amer. Inst. Phys., Melville, NY, 2013, 1148–1151 | DOI

[11] E. Tas, T. Yurdakadim, “Approximation by positive linear operators in modular spaces by power series method”, Positivity, 21:4 (2017), 1293–1306 | DOI | MR | Zbl

[12] W. Balser, M. Miyake, “Summability of formal solutions of certain partial differential equations”, Acta Sci. Math. (Szeged), 65:3-4 (1999), 543–551 | MR | Zbl

[13] W. Balser, Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, Springer-Verlag, New York, 2000, xviii+299 pp. | DOI | MR | Zbl

[14] J. Boos, Classical and modern methods in summability, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2000, xiv+586 pp. | MR | Zbl

[15] H. Fast, “Sur la convergence statistique”, Colloq. Math., 2:3-4 (1951), 241–244 | DOI | MR | Zbl

[16] G. Di Maio, L. D. R. Kočinac, “Statistical convergence in topology”, Topology Appl., 156:1 (2008), 28–45 | DOI | MR | Zbl

[17] M. Unver, M. K. Khan, C. Orhan, “$A$-distributional summability in topological spaces”, Positivity, 18:1 (2014), 131–145 | DOI | MR | Zbl

[18] H. Çakalli, M. K. Khan, “Summability in topological spaces”, Appl. Math. Lett., 24:3 (2011), 348–352 | DOI | MR | Zbl

[19] M. Ünver, “Abel summability in topological spaces”, Monatsh. Math., 178:4 (2015), 633–643 | DOI | MR | Zbl

[20] P. Das, “Some further results on ideal convergence in topological spaces”, Topology Appl., 159:10-11 (2012), 2621–2626 | DOI | MR | Zbl

[21] G. H. Hardy, Divergent series, Oxford, Clarendon Press, 1949, xvi+396 pp. | MR | MR | Zbl

[22] R. E. Powell, S. M. Shah, Summability theory and its applications, Van Nostrand Reinhold Co., London–New York, 1972, ix+178 pp. | Zbl

[23] E. A. Feinberg, P. O. Kasyanov, Y. Liang, “Fatou's lemma for weakly converging measures under the uniform integrability condition”, Teoriya veroyatn. i ee primen., 64:4 (2019), 771–790 | DOI | MR | Zbl

[24] R. Giuliano Antonini, M. Ünver, A. Volodin, “On the concept of $A$-statistical uniform integrability and the law of large numbers”, Lobachevskii J. Math., 40:12 (2019), 2034–2042 | DOI | MR | Zbl

[25] M. Ordóñez Cabrera, A. Rosalsky, M. Ünver, A. Volodin, “On the concept of $B$-statistical uniform integrability of weighted sums of random variables and the law of large numbers with mean convergence in the statistical sense”, TEST, 30:1 (2021), 83–102 | DOI | MR | Zbl

[26] M. Ünver, C. Orhan, “Statistical convergence with respect to power series methods and applications to approximation theory”, Numer. Funct. Anal. Optim., 40:5 (2019), 535–547 | DOI | MR | Zbl