This article addresses the problem of detecting, with high probability, the presence of a large class of history-dependent false data injection actuator attacks in cyber-physical systems (CPSs) modeled as stochastic linear time-invariant systems. Among the primary contributions is the introduction of a new detection algorithm or scheme that leverages the fundamental idea of separating or classifying two classes of state trajectories: The first class comprises state trajectories generated by the CPS under attack situations, while the second class includes state trajectories generated under nominal conditions. The existence and non-existence of such a separator are studied and these results are asymptotic. We then establish finite-time guarantees associated with the detection algorithm or scheme. A numerical example is provided to demonstrate the theory.