Large deviations for correlated random variables described by a matrix product ansatz

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.