  The fundamental lemma by Jan C. Willems and co-workers, which is deeply
rooted in behavioral systems theory, has become one of the supporting pillars
of the recent progress on data-driven control and system analysis. This
tutorial-style paper combines recent insights into stochastic and
descriptor-system formulations of the lemma to further extend and broaden the
formal basis for behavioral theory of stochastic linear systems. We show that
series expansions -- in particular Polynomial Chaos Expansions (PCE) of
$L^2$-random variables, which date back to Norbert Wiener's seminal work --
enable equivalent behavioral characterizations of linear stochastic systems.
Specifically, we prove that under mild assumptions the behavior of the dynamics
of the $L^2$-random variables is equivalent to the behavior of the dynamics of
the series expansion coefficients and that it entails the behavior composed of
sampled realization trajectories. We also illustrate the short-comings of the
behavior associated to the time-evolution of the statistical moments. The paper
culminates in the formulation of the stochastic fundamental lemma for linear
(descriptor) systems, which in turn enables numerically tractable formulations
of data-driven stochastic optimal control combining Hankel matrices in
realization data (i.e. in measurements) with PCE concepts.
