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Rough statistical convergence of a sequence of random variables in probability

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Abstract

In this paper, following the works of Phu (Numer Funct Anal Optimiz 22:201–224, 2001), Aytar (Numer Funct Anal Optimiz 29(34):291–303, 2008) and Ghosal (Appl Math 58(4):423–437, 2013) we make a new approach to extend the application area of rough statistical convergence usually used in sequence of real numbers to the theory of probability distributions. The introduction of these concepts in probability of rough statistical convergence, rough strong Cesàro summable, rough lacunary statistical convergence, rough \(N_\theta \)-convergence, rough \(\lambda \)-statistical convergence and rough strong \((V,\lambda )\)-summable generalize the convergence analysis to accommodate any form of distribution of random variables. Among these six concepts in probability only three convergences are distinct- rough statistical convergence, rough lacunary statistical convergence and rough \(\lambda \)-statistical convergence where rough strong Cesàro summable is equivalent to rough statistical convergence, rough \(N_\theta \)-convergence is equivalent to rough lacunary statistical convergence, rough strong \((V,\lambda )\)-summable is equivalent to rough \(\lambda \)-statistical convergence. Basic properties and interrelations of above mentioned three distinct convergences are investigated and make some observations about these classes and in this way we show that rough statistical convergence in probability is the more generalized concept than usual rough statistical convergence.

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Acknowledgments

Thankful to respected referee for his/her careful reading of the paper and several valuable suggestions which has improved the quality and presentation of the paper substantially.

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Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, West Bengal, India

    Pratulananda Das

  2. Department of Mathematics, Kalyani Government Engineering College, Kalyani, Nadia, 741235, West Bengal, India

    Sanjoy Ghosal

  3. Department of Mathematics, Mandra Aswini Kumar High School, Murshidabad, 742161, West Bengal, India

    Avishek Ghosh

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  1. Pratulananda Das
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  2. Sanjoy Ghosal
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  3. Avishek Ghosh
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Correspondence to Sanjoy Ghosal.

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Das, P., Ghosal, S. & Ghosh, A. Rough statistical convergence of a sequence of random variables in probability. Afr. Mat. 26, 1399–1412 (2015). https://doi.org/10.1007/s13370-014-0295-2

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