  We present a systematic study of moment evolution in multidimensional
stochastic difference systems, focusing on characterizing systems whose
low-order moments diverge in the neighborhood of a stable fixed point. We
consider systems with a simple, dominant eigenvalue and stationary, white
noise. When the noise is small, we obtain general expressions for the
approximate asymptotic distribution and moment Lyapunov exponents. In the case
of larger noise, the second moment is calculated using a different approach,
which gives an exact result for some types of noise. We analyze the dependence
of the moments on the system's dimension, relevant system properties, the form
of the noise, and the magnitude of the noise. We determine a critical value for
noise strength, as a function of the unperturbed system's convergence rate,
above which the second moment diverges and large fluctuations are likely.
Analytical results are validated by numerical simulations. We show that our
results cannot be extended to the continuous time limit except in certain
special cases.
