In this paper, following a very recent and new approach of Aizpuru et al. (Quaest. Math. 37:525-530, 2014), we further generalize a concept of α-density to that of f α -density, where f is an unbounded modulus and 0 < α ≤ 1. As a consequence, we

We have introduced and studied a new concept off-lacunary statistical convergence, wherefis an unbounded modulus. It is shown that, under certain conditions on a modulusf, the concepts of lacunary strong convergence with respect to a modulusfandf-lacunary statistical convergence are equivalent on bounded sequences. We further characterize thoseθfor whichSθf=Sf, whereSθfandSfdenote the sets of allf-lacunary statistically convergent sequences andf-statistically convergent sequences, respectively. A general description of inclusion between two arbitrary lacunary methods off-statistical convergence is given. Finally, we give anSθf-analog of the Cauchy criterion for convergence and a Tauberian theorem forSθf-convergence is also proved.

We have generalized the notion of statistical boundedness by introducing the concept of -statistical boundedness for scalar sequences where is an unbounded modulus. It is shown that bounded sequences are precisely those sequences which arestatistically bounded for every unbounded modulus . A decomposition theorem for -statistical convergence for vector valued sequences and a structure theorem for -statistical boundedness have also been established.

In this paper, we generalized the Wijsman statistical convergence of closed sets in metric space by introducing the f -Wijsman statistical convergence these of sets, where f is an unbounded modulus. It is shown that the Wijsman convergent sequences are precisely those sequences which are f -Wijsman statistically convergent for every unbounded modulus f . We also introduced a new concept of Wijsman strong Cesàro summability with respect to a modulus, and investigate the relationships between the f -Wijsman statistically convergent sequences and the Wijsman strongly Cesàro summable sequences with respect to f .

The main object of this paper is to introduce and study a new concept of f-Wijsman lacunary statistical convergence of sequences of sets, where f is an unbounded modulus. The definition of Wijsman lacunary strong convergence of sequences of sets is extended to a definition of Wijsman lacunary strong convergence with respect to a modulus for sequences of sets and it is shown that, under certain conditions on a modulus f, the concepts of Wijsman lacunary strong convergence with respect to a modulus f and f-Wijsman lacunary statistical convergence are equivalent on bounded sequences. We further characterize those θ for which , where and denote the sets of all f-Wijsman lacunary statistically convergent sequences and f-Wijsman statistically convergent sequences, respectively.