  We establish new conditions for obtaining uniform bounds on the moments of
discrete-time stochastic processes. Our results require a weak negative drift
criterion along with a state-dependent restriction on the sizes of the one-step
jumps of the processes. The state-dependent feature of the results make them
suitable for a large class of multiplicative-noise processes. Under the
additional assumption of Markovian property, new result on ergodicity has also
been proved. There are several applications to iterative systems, control
systems, and other dynamical systems with state-dependent multiplicative noise,
and we include illustrative examples to demonstrate applicability of our
results.
