  We study stochastic team problems with static information structure where we
assume controllers have linear information and quadratic cost but allow the
noise to be from a non-Gaussian class. When the noise is Gaussian, it is well
known that these problems admit linear optimal controllers. We show that for
such linear-quadratic static teams with any log-concave noise, if the length of
the noise or data vector becomes large compared to the size of the team and
their observations, then linear strategies approach optimality for `most'
problems. The quality of the approximation improves as length of the noise
vector grows and the class of problems for which the approximation is
asymptotically not exact approaches a set of measure zero. We show that if the
optimal strategies for problems with log-concave noise converge pointwise, they
do so to the (linear) optimal strategy for the problem with Gaussian noise. And
we derive an asymptotically tight error bound on the difference between the
optimal cost for the non-Gaussian problem and the best cost obtained under
linear strategies.
