Asymptotically J-Lacunary statistical equivalent of order α for sequences of sets

This paper presents the following definition which is a natural combination of the definition for asymptotically equivalent of order α, where 0 α ≤ 1, J-statistically limit, and J-lacunary statistical convergence for sequences of sets. Let (X, ρ) be a metric space and θ be a lacunary sequence. For any non-empty closed subsets Ak, Bk ⊆ X such that d(x, Ak) 0 and d(x, Bk) 0 for each x ∈ X, we say that the sequences {Ak} and {Bk} are Wijsman asymptotically J-lacunary statistical equivalent of order α to multiple L, where 0 α ≤ 1, provided that for each ε 0 and each x ∈ X, 1 {r ∈ N : |{k ∈ I : |d(x; A , B ) − L| ) £}| ) δ} ∈ I, r SL α (denoted by {Ak} θ (IW ) ∼ {Bk}) and simply asymptotically J-lacunary statistical equivalent of order α if L = 1. In addition, we shall also present some inclusion theorems. The study leaves some interesting open problems.