__Abstract:__

We develop a method to approximate the moments of a discrete-time stochastic polynomial system.
Our method is built upon Carleman linearization with truncation. Specifically, we take a stochastic
polynomial system with finitely many states and transform it into an infinite-dimensional system with
linear deterministic dynamics, which describe the exact evolution of the moments of the original polynomial
system. We then truncate this deterministic system to obtain a finite-dimensional linear system,
and use it for moment approximation by iteratively propagating the moments along the finite-dimensional
linear dynamics across time. We provide efficient online computation methods for this propagation
scheme with several error bounds for the approximation. Our result also shows that precise values of
certain moments can be obtained when the truncated system is sufficiently large. Furthermore, we
investigate techniques to reduce the offline computation load using reduced Kronecker power.
Based on the obtained approximate moments and their errors, we also provide probability bounds for the
state to be outside of given hyperellipsoidal regions. Those bounds allow us to conduct probabilistic
safety analysis online through convex optimization. We demonstrate our results on a logistic map with
stochastic dynamics and a vehicle dynamics subject to stochastic disturbance.

__Key words:__

Stochastic systems, nonlinear systems, probabilistic safety analysis, moment computation, Carleman linearization