In the present paper, we give a new characterization of unconditional convergent series and give some new versions of the Orlicz-Pettis theorem via FQ σ-family and a natural family F with the separation property S 1 through w R p-summability which may be considered as a generalization of the well-known strong p-Cesàro summability. Among other results, we obtain a new (weak) compactness criteria for the summing operator.