In this article we elaborately study certain characteristics of the set of all $\mathcal{I}$-lacunary statistical convergence sequences over various sequence spaces. Earlier results of some authors were mainly concerned regarding the closeness property of the sets: set of all bounded statistically convergent sequences, set of all bounded statistically convergent sequences of order $\alpha,$ set of all bounded $\mathcal{I}$-convergent sequences over the space $\ell^\infty$ ($\ell^\infty$- endowed with the sup-norm). Our approach is to examine different behaviors of the set of all $\mathcal{I}$-lacunary statistical convergence sequences over different sequence spaces. Finally we are able to impose some condition(s) over sequence spaces that turn the set of all $\mathcal{I}$-lacunary statistical convergence sequences to be a closed set.

n this work, our attention is focused on $\lambda$- double almost statistical convergence of weight $h$. Mainly we work $\lambda$- double almost statistical convergence of weight $h$ and strong $\lambda$- double almost summability of weight $h$.