In this paper we present, for sequences of sets, the concept of generalized asymptotically $\mathcal I$-lacunary statistical equivalent of order $\alpha$, $(0<\alpha\leq1)$, to multiple $L$, where $\mathcal I$ is an ideal of the subset of $\mathbb N$. In addition to this definition, inclusion theorems are also presented. The study leaves some interesting open problems.