  The use of random sampling in decision-making and control has become popular
with the ease of access to graphic processing units that can generate and
calculate multiple random trajectories for real-time robotic applications. In
contrast to sequential optimization, the sampling-based method can take
advantage of parallel computing to maintain constant control loop frequencies.
Inspired by its wide applicability in robotic applications, we calculate a
sampling complexity result applicable to general nonlinear systems considered
in the path integral method, which is a sampling-based method. The result
determines the required number of samples to satisfy the given error bounds of
the estimated control signal from the optimal value with the predefined risk
probability. The sampling complexity result shows that the variance of the
estimated control value is upper-bounded in terms of the expectation of the
cost. Then we apply the result to a linear time-varying dynamical system with
quadratic cost and an indicator function cost to avoid constraint sets.
