Originated from applications in signal processing, random evolution,
telecommunications, risk management, financial engineering,
and manufacturing systems, two-time-scale Markovian systems have
drawn much attention. This talk discusses asymptotic
expansions of solutions to the forward equations, scaled and unscaled
occupation measures, approximation error bounds, and associated
switching diffusion processes. Controlled dynamic systems will also
be mentioned.

Consider a simple symmetric random walk $S$ and another random walk $S'$ whose $k$th increments are the $k$-fold product of the first $k$ increments of $S$.
The random walks $S$ and $S'$ are strongly dependent. Still the 2-dimensional walk $(S, S')$, properly rescaled, converges to a two dimensional Brownian motion. The goal of this talk is to present the proof of this fact, and its generalizations. Based on joint works with K. Hamza and S. Meng.