We investigate different turnpike phenomena of generalized discrete-time
stochastic linear-quadratic optimal control problems. Our analysis is based on
a novel strict dissipativity notion for such problems, in which a stationary
stochastic process replaces the optimal steady state of the deterministic
setting. We show that from this time-varying dissipativity notion, we can
conclude turnpike behaviors concerning different objects like distributions,
moments, or sample paths of the stochastic system and that the distributions of
the stationary pair can be characterized by a stationary optimization problem.
The analytical findings are illustrated by numerical simulations