Some New Types of Convergence Definitions for Random Variable Sequences

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DOI: https://doi.org/10.28924/2291-8639-20-2022-42
Published: Aug 24, 2022
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Saadettin Aydın, Some New Types of Convergence Definitions for Random Variable Sequences, Int. J. Anal. Appl., 20 (2022), 42.

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Saadettin Aydın

Abstract

In this paper, we introduce the concepts of invariant convergence in probability, statistically invariant convergence in probability, invariant convergence almost surely, invariant convergence in distribution and invariant convergence in Lp-norm for sequences of random variables. Also, we investigate some inclusion relations.

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References

  1. L. Breiman, Probability, SIAM, 1992.
  2. P. Brockwell, R. Davis, Time Series: Theory and Methods, Springer, 1991.
  3. J.S. Connor, The Statistical and Strong p-Cesàro of Sequences, Analysis, 8 (1988), 47–63. https://doi.org/10.1524/anly.1988.8.12.47.
  4. J.S. Connor, On Strong Matrix Summability with Respect to a Modulus and Statistical Convergence, Can. Math. Bull. 32 (1989), 194–198. https://doi.org/10.4153/cmb-1989-029-3.
  5. H. Fast, Sur la Convergence Statistique, Colloq. Math. 2 (1951), 241–244. http://eudml.org/doc/209960.
  6. J.A. Fridy, On Statistical Convergence, Analysis, 5 (1985), 301–313. https://doi.org/10.1524/anly.1985.5.4.301.
  7. J.A. Fridy, C. Orhan, Lacunary Statistical Convergence, Pac. J. Math. 160 (1993), 43–51.
  8. S. Ghosal, Statistical Convergence of a Sequence of Random Variables and Limit Theorems, Appl. Math. 58 (2013), 423–437. https://doi.org/10.1007/s10492-013-0021-7.
  9. S. Ghosal, I-statistical convergence of a sequence of random variables in probability, Afr. Mat. 25 (2013), 681–692. https://doi.org/10.1007/s13370-013-0142-x.
  10. S. Ghosal, Sλ-Convergence of a Sequence of Random Variables, J. Egypt. Math. Soc. 23 (2015), 85–89. https://doi.org/10.1016/j.joems.2014.03.007.
  11. S. Ghosal, Weighted Statistical Convergence of Order α and Its Applications, J. Egypt. Math. Soc. 24 (2016), 60–67. https://doi.org/10.1016/j.joems.2014.08.006.
  12. P.G. Hoel, S.C. Port, C.J. Stone, Introduction to Probability Theory, Houghton Mifflin Company, 1971.
  13. G.G. Lorentz, A Contribution to the Theory of Divergent Sequences, Acta Math. 80 (1948), 167–190. https://doi.org/10.1007/bf02393648.
  14. I.J. Maddox, A New Type of Convergence, Math. Proc. Camb. Phil. Soc. 83 (1978), 61–64. https://doi.org/10.1017/s0305004100054281.
  15. M. Mursaleen, Matrix Transformation Between Some New Sequence Spaces, Houston J. Math. 9 (1983), 505–509.
  16. M. Mursaleen, On Infinite Matrices and Invariant Means, Indian J. Pure Appl. Math. 10 (1979), 457–460.
  17. M. Mursaleen, O.H.H. Edely, On the Invariant Mean and Statistical Convergence, Appl. Math. Lett. 22 (2009), 1700–1704. https://doi.org/10.1016/j.aml.2009.06.005.
  18. F. Nuray, E. Savaş, Invariant Statistical Convergence and A-Invariant Statistical Convergence, Indian J. Pure Appl. Math. 25 (1994), 267–274.
  19. E. Savaş, Some Sequence Spaces Involving Invariant Means, Indian J. Math. 31 (1989), 1–8.
  20. E. Savaş, Strongly σ-Convergent Sequences, Bull. Calcutta Math. 81 (1989), 295–300.
  21. R.A. Raimi, Invariant Means and Invariant Matrix Methods of Summability, Duke Math. J. 30 (1963), 81–94. https://doi.org/10.1215/s0012-7094-63-03009-6.
  22. S. Ross, A First Course in Probability, Macmillan Publishing Company, 2nd ed., 1984.
  23. P. Schaefer, Infinite Matrices and Invariant Means, Proc. Amer. Math. Soc. 36 (1972), 104–110. https://doi.org/10.1090/s0002-9939-1972-0306763-0.
  24. T. Šalát, On Statistically Convergent Sequences of Real Numbers, Math. Slovaca, 30 (1980), 139–150. http://dml.cz/dmlcz/136236.
  25. I.J. Schoenberg, The Integrability of Certain Functions and Related Summability Methods, Amer. Math. Mon. 66 (1959), 361–775. https://doi.org/10.1080/00029890.1959.11989303.
  26. H. Steinhaus, Sur la Convergence Ordinaire et la Convergence Asymptotique, Colloq. Math. 2 (1951), 73–74.
  27. D. Williams, Probability with Martingales, Cambridge University Press, 1991.