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We consider a general linear program in standard form whose right-hand side constraint vector is subject to random perturbations. This defines a stochastic linear program for which, under general conditions, we characterize the fluctuations of the corresponding empirical optimal solution by a central limit-type theorem. Our approach relies on the combinatorial nature and the concept of degeneracy inherent in linear programming, in strong contrast to well-known results for smooth stochastic optimization programs. In particular, if the corresponding dual linear program is degenerate the asymptotic limit law might not be unique and is determined from the way the empirical optimal solution is chosen. Furthermore, we establish consistency and convergence rates of the Hausdorff distance between the empirical and the true optimality sets. As a consequence, we deduce a limit law for the empirical optimal value characterized by the set of all dual optimal solutions which turns out to be a simple consequence of our general proof techniques. Our analysis is motivated from recent findings in statistical optimal transport that will be of special focus here. In addition to the asymptotic limit laws for optimal transport solutions, we obtain results linking degeneracy of the dual transport problem to geometric properties of the underlying ground space, and prove almost sure uniqueness statements that may be of independent interest.