$\mathcal{I}$-Cesaro Summability of a Sequence of Order $\alpha$ of Random Variables in Probability

Year 2018, Volume: 1 Issue: 2, 157 - 161, 25.12.2018

Ömer Kişi , Erhan Güler

https://doi.org/10.33401/fujma.480808

Cited By: 4

Abstract

In this paper, we define four types of convergence of a sequence of random variables, namely, $\mathcal{I}$-statistical convergence of order $ \alpha $, $\mathcal{I}$-lacunary statistical convergence of order $\alpha $, strongly $\mathcal{I}$-lacunary convergence of order $\alpha $ and strongly $ \mathcal{I}$-Cesaro summability of order $\alpha $ in probability where $ 0<\alpha <1$. We establish the connection between these notions.

Keywords

Probability, Lacunary, Ideal convergence

References

• [1] H. Fast, Sur la convergence statistique, Coll. Math., 2 (1951), 241-244.
• [2] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313.
• [3] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139–150.
• [4] J. A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160(1) (1993), 43-51.
• [5] P. Kostyrko, T. Salat, W. Wilezynski, I-Convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
• [6] P. Das, E. Savas¸, S. Ghosal, On generalization of certain summability methods using ideals, Appl. Math. Lett., 24 (2011), 1509–1514.
• [7] U. Ulusu, E. Dundar, Asymptotically $\mathcal{I-}$-Cesaro equivalence of sequences of sets, Univers. J. Math. Appl., 1(2) (2018), 101-105.
• [8] R. C¸ olak, Statistical convergence of order a, Modern methods in analysis and its applications, Anamaya Pub., New Delhi, India, (2010), 121-129.
• [9] H. Şengöl, M. Et, On lacunary statistical convergence of order a, Acta Math. Sci., 34B(2) (2014), 473–482.
• [10] P. Das, E. Savas, On I-statistical and I-lacunary statistical convergence of order a, Bull. Iranian Math. Soc., 40(2) (2014), 459-472.
• [11] S. Ghosal, Statistical convergence of a sequence of random variables and limit theorems, Appl. Math., 4(58) (2013), 423–437.
• [12] S. Ghosal, $\mathcal{I-}$statistical convergence of a sequence of random variables in probability, Afrika Mat., http://dx.doi.org/10.1007/s13370-013-0142-x.
• [13] S. Ghosal, Sl -statistical convergence of a sequence of random variables, J. Egypt. Math. Soc., (2014), http://dx.doi.org/10.1016/j.joems.2014.03.007.
• [14] S. Ghosal, Statistical convergence of order a in probability, Arab J. Math. Sci., 21 (2015), 253-265.
• [15] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375.