In this paper, we examine the role of the switching period on the stochastic stability of jump linear systems. More specifically, we consider a jump linear system in which the state matrix switches every m time steps randomly within a finite set of realizations, without a memory of past switching instances. Through the computation of the Lyapunov exponents, we study delta-moment and almost sure stability of the system. For scalar systems, we demonstrate that almost sure stability is independent of m, while delta-moment stability can be modulated through the selection of the switching period. For higher-dimensional problems, we discover a richer influence of m on stochastic stability, quantified in an almost sure and a delta-moment sense. Through the detailed analysis of an archetypical two-dimensional problem, we illustrate the existence of disconnected windows of opportunity where the system is asymptotically stable. Outside of these windows, the system is unstable, even though it switches between two Schur-stable state matrices. (C) 2018 Elsevier Ltd. All rights reserved.