MMN-3624

Deferred Cesaro statistical convergence of martingale sequence and Korovkin-type approximation theorems

  • H. M. Srivastava, Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada, harimsri@math.uvic.ca
  • Bidu Bhusan Jena, Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India, bidumath.05@gmail.com
  • Susanta Kumar Paikray, Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India, skpaikray_math@vssut.ac.in

Abstract

In the present paper, we introduce and study the concepts of statistical convergence and statistical summability for martingale sequences of random variables via deferred Ces\`{a}ro mean. We then establish an inclusion theorem concerning the relation between these two beautiful and potentially useful concepts. Also, based upon our proposed notions, we state and prove new Korovkin-type approximation theorems with algebraic test functions for a martingale sequence over a Banach space. Moreover, we demonstrate that our theorems effectively extend and improve most (if not all) of the previously existing results (in statistical and classical versions). Finally, by using the generalized Bernstein polynomials, we present an illustrative example of a martingale sequence in order to demonstrate that our established theorems are stronger than their traditional and statistical versions.


Vol. 23 (2022), No. 1, pp. 443-456
DOI: 10.18514/MMN.2022.3624


Download: MMN-3624