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We propose a method to compute an approximation of the moments of adiscrete-time stochastic polynomial system. We use the Carleman linearizationtechnique to transform this finite-dimensional polynomial system into aninfinite-dimensional linear one. After taking expectation and truncating theinduced deterministic dynamics, we obtain a finite-dimensional lineardeterministic system, which we then use to iteratively compute approximationsof the moments of the original polynomial system at different time steps. Weprovide upper bounds on the approximation error for each moment and show that,for large enough truncation limits, the proposed method precisely computesmoments for sufficiently small degrees and numbers of time steps. We use ourproposed method for safety analysis to compute bounds on the probability of thesystem state being outside a given safety region. Finally, we illustrate ourresults on two concrete examples, a stochastic logistic map and a vehicledynamics under stochastic disturbance.

Moment Propagation of Discrete-Time Stochastic Polynomial Systems using Truncated Carleman Linearization