  In the regime where the parameter beta is proportional to the reciprocal of
the system size, it is known that the empirical distribution of Gaussian beta
ensembles (resp.\ beta Laguerre ensembles) converges to a probability measure
of associated Hermite polynomials (resp.\ associated Laguerre polynomials).
Gaussian fluctuations around the limit have been known as well. This paper aims
to study a dynamical version of those results. More precisely, we study beta
Dyson's Brownian motions and beta Laguerre processes and establish LLNs and
CLTs for their moment processes in the same regime.
