Weighted modulus Sθ-convergence of order α in probability

Let θ={kr}r∈N∪{0} be a lacunary sequence, ϕ be a modulus function and {tn}n∈N be a sequence of real numbers such that tn>δ,∀n∈N (where δ is a fixed positive real number) and Tn=t1+t2+⋯+tn (where n∈N and T0=0). A sequence of random variables {Xn}n∈N is said to be weighted modulus Sθ-convergent of order α in probability (where 0<α≤1) to a random variable X (like Ghosal(2014)) if for any ε,δ>0,limr→∞1(Tkr−Tkr−1)α|{k∈(Tkr−1,Tkr]:tkϕ(P(|Xk−X|≥ε))≥δ}|=0.The results are applied to build the probability distribution for weighted modulus Nθ-convergence of order α. Also these methods are compared with the convergence of weighted modulus statistical convergence of order α and weighted modulus strong Cesàro convergence of order α respectively. If limsupr→∞TkrTkr−1α<∞, then weighted modulus Sθ-convergence of order α in probability implies weighted modulus statistical convergence of order α in probability and weighted modulus Nθ-convergence of order α implies weighted modulus strong Cesàro convergence of order α in probability except the condition limsupr→∞TkrTkr−1α=∞. So our main objective is to interpret the above exceptional condition and produce a relational behavior of above mentioned four convergences. This is also used to prove the uniqueness of limit value of weighted lacunary statistical convergence and improve the definition of weighted lacunary statistical convergence.