Generalized weighted random convergence in probability

The definition of weighted statistical convergence was first introduced by Karakaya & Chishti (2009) and later on Mursaleen et al. (2012) modified the definition of this concept. Using the modified definition, Edely et al. (2013) had researched further in approximation theory for periodic functions and Belen & Mohiuddine (2013) had further extended to the new-definition of weighted λ-statistical convergence. But some problems are still there, so the definition of weighted λ-statistical convergence is needed to modify. In this paper, we will introduce some new constraints which will make the definition of weighted λ-statistical convergence is more useful by using the definition of αβ-statistical convergence. Using it and independently, some newly developed concepts of the convergence of a sequence of random variables in probability, namely, weighted modulus αβ-statistical convergence of order γ, weighted modulus αβ-strong Cesaro convergence of order γ, weighted modulus S α β -convergence of order γ, and weighted modulus N α β -convergence of order γ, have been introduced and their basic interrelations also have been investigated.

[1]  E. Savaş,et al.  ON $I$-STATISTICALLY PRE-CAUCHY SEQUENCES , 2014 .

[2]  Hüseyin Aktuglu,et al.  Korovkin type approximation theorems proved via αβ-statistical convergence , 2014, J. Comput. Appl. Math..

[3]  E. Savaş,et al.  ON SOME DOUBLE ALMOST LACUNARY SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS , 2005 .

[4]  Mohammad Mursaleen,et al.  $\lambda$-statistical convergence , 2000 .

[5]  H. Fast,et al.  Sur la convergence statistique , 1951 .

[6]  S. Ghosal.  Weighted statistical convergence of order α and its applications , 2016 .

[7]  M. Gürdal,et al.  ON I-CONVERGENCE OF DOUBLE SEQUENCES IN THE TOPOLOGY INDUCED BY RANDOM 2-NORMS , 2013 .

[8]  Mohammad Mursaleen,et al.  Weighted statistical convergence and its application to Korovkin type approximation theorem , 2012, Appl. Math. Comput..

[9]  Syed Abdul Mohiuddine,et al.  Generalized weighted statistical convergence and application , 2013, Appl. Math. Comput..

[10]  J. A. Fridy,et al.  ON STATISTICAL CONVERGENCE , 1985 .

[11]  M. Gürdal,et al.  Extremal A-statistical limit points via ideals , 2014 .

[12]  Allen R. Freedman,et al.  SOME CESARO-TYPE SUMMABILITY SPACES , 1978 .

[13]  P. Das,et al.  Statistical convergence of order α in probability , 2015, 1605.05555.

[14]  I. J. Schoenberg.  The Integrability of Certain Functions and Related Summability Methods , 1959 .

[15]  Mohammad Mursaleen,et al.  Approximation for periodic functions via weighted statistical convergence , 2013, Appl. Math. Comput..

[16]  Sanjoy Ghosal.  $$\mathcal{I }$$-statistical convergence of a sequence of random variables in probability , 2014 .

[17]  I. J. Maddox.  Sequence spaces defined by a modulus , 1986, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  P. Das,et al.  Restricting statistical convergence , 2012 .

[19]  Ahmet Sahiner,et al.  Ideal convergence in n-normal spaces and some new sequence spaces via n-norm , 2014 .

[20]  Tibor Šalát,et al.  On statistically convergent sequences of real numbers , 1980 .

[21]  J. A. Fridy,et al.  Lacunary statistical convergence. , 1993 .

[22]  P. Heywood.  Trigonometric Series , 1968, Nature.

[23]  Vatan Karakaya,et al.  WEIGHTED STATISTICAL CONVERGENCE , 2009 .

[24]  E. Savaş,et al.  Some σ-double sequence spaces defined by Orlicz function , 2006 .

[25]  R. Çolak,et al.  Statistical convergence of order a , 2011 .

[26]  S. Ghosal.  Statistical convergence of a sequence of random variables and limit theorems , 2013 .

[27]  M. Gürdal,et al.  On Ideal Convergent Sequences in 2-Normed Spaces , 2006 .

[28]  J. S. Connor,et al.  THE STATISTICAL AND STRONG p-CESARO CONVERGENCE OF SEQUENCES , 1988 .

[29]  C. Sençimen.  Statistical Convergence in Probability for a Sequence of Random Functions , 2013 .