Summary: This paper presents the following definition which is a natural combination of the definition for asymptotically equivalent of order \(\alpha\), where \(0 < \alpha \leqslant 1\), \(\mathcal{J}\)-statistically limit, and \(\mathcal{J}\)-lacunary statistical convergence for sequences of sets. Let \((X, \rho)\) be a metric space and \(\theta\) be a lacunary sequence. For any non-empty closed subsets \(A_k, B_k \subseteq X\) such that \(d(x,A_k) > 0\) and \(d(x, B_k) > 0\) for each \(x \in X\), we say that the sequences \(\{A_k\}\) and \(\{B_k\}\) are Wijsman asymptotically \(\mathcal{J}\)-lacunary statistical equivalent of order \(\alpha\) to multiple \(L\), where \(0 < \alpha \leqslant 1\), provided that for each \(\varepsilon > 0\) and each \(x \in X\), \[\{r\in \mathbb{N}: \frac{1}{h^\alpha_r}|\{k\in I_r: |d(x;A_k,B_k)-L|\geqslant\varepsilon\}|\geqslant\delta\}\in \mathcal{J},\] (denoted by \(\{A_k\}\overset{s\frac{1}{\theta}(\mathcal{J}_W)^\alpha}{\sim}\{B_k\}\)) and simply asymptotically \(\mathcal{J}\)-lacunary statistical equivalent of order \(\alpha\) if \(L = 1\). In addition, we shall also present some inclusion theorems. The study leaves some interesting open problems.