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 Cesáro convergence of order γ, weighted modulus Sαβ-convergence of order γ, and weighted modulus Nαβ-convergence of order γ, have been introduced and their basic interrelations also have been investigated.