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Hybrid jump linear systems (HJLS's) were recently introduced to study the fundamental properties of supervisory control systems. In this paper, their mean square (MS) stability is analyzed through a modified lifting technique adapted from the literature on Markov jump linear systems. The original technique tests whether rho(A), the spectral radius of a second moment transition matrix A (which contains the transition probabilities of the Markov chain driving the jump linear system), is less than 1. Here, the test is adapted to yield two new sufficient MS stability conditions. The first condition applies to jump linear systems driven by general finite-state stochastic processes. It requires computing and testing the spectral radius of the matrix AMthetas tilde, which is not a second moment transition matrix, using estimated upper bounds of the transition probabilities associated with the processes driving the system. The second condition applies to a large subclass of HJLS's and makes use of the Markov kernel associated with HJLS's to test rho(AMz), a particular instance of rho(AMthetas tilde). These results are illustrated through the stability analysis of an AFTI-F16 aircraft deployed on a computer platform equipped with an advanced error recovery mechanism.