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Then $M_t:=e^{-\\lambda_0t}\\langle\\phi_0, X_t\\rangle$ is a positive martingale. Let $M_\\infty$ be the limit of $M_t$. It is known (see, J. Appl. Probab. 46 (2009), 479--496) that $M_\\infty$ is non-degenerate iff the $L\\log L$ condition is satisfied. 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