Lacunary Statistical Summability

The sequence x is said to be Sθ-convergent to L if for each ϵ > 0, limr (1/hr){the number of k ∈ Ir: |xk − L| ≥ ϵ} = 0, where θ = {kr} is an increasing sequnece of integers such that k0 = 0, hr ≔ kr − kr − 1 → ∞ as r → ∞ and Ir ≔ (kr − 1, kr]. In this paper we define the Sθ-analog of the Cauchy criterion for convergence and show that tt is equivalent to Sθ-convergence. Also, Sθ-convergence is compared to other summability methods, and it is shown that the Sθ method can not be included by any matrix method. In addition, a Tauberian theorem for Sθ-convergence is given