  We study the large deviations of sums of correlated random variables
described by a matrix product ansatz, which generalizes the product structure
of independent random variables to matrices whose non-commutativity is the
source of correlations. We show with specific examples that different large
deviation behaviors can be found with this ansatz. In particular, it is
possible to construct sums of correlated random variables that violate the Law
of Large Numbers, the Central Limit Theorem, as well as sums that have
nonconvex rate functions or rate functions with linear parts or plateaux.
