  In this paper we study reachability verification problems of stochastic
discrete-time dynamical systems over the infinite time horizon. The
reachability verification of interest in this paper is to certify specified
lower and upper bounds of the reachability probability, with which the system
starting from a designated initial set will enter a desired target set
eventually. Existing sufficient conditions for reachability verification over
the infinite time horizon are established based on the Doob's
non-supermartingle inequality, which are often restrictive. Recently, a set of
equations was proposed in [18], to which the solution is able to characterize
the exact reachability probability. Inspired by this result, we in this paper
propose sufficient conditions for reachability verification based on relaxing
this equation. These sufficient conditions are shown to be weaker than the
state-of-the-art ones, consequently being capable of providing more
opportunities for verifying the reachability property successfully.
