Large Deviations with Diminishing Rates

The theory of large deviations for jump Markov processes has been generally proved only when jump rates are bounded below, away from zero (Dupuis and Ellis, 1995, The large deviations principle for a general class of queueing systems I. Trans. Amer. Math. Soc. 347 2689–2751; Ignatiouk-Robert, 2002, Sample path large deviations and convergence parameters. Ann. Appl. Probab. 11 1292–1329; Shwartz and Weiss, 1995, Large Deviations for Performance Analysis, Chapman-Hall). Yet, various applications of interest do not satisfy this condition. We describe several classes of models where jump rates diminish to zero in a Lipschitz continuous way. Under appropriate conditions, we prove that the sample path large deviations principle continues to hold. Under our conditions, the rate function remains an integral over a local rate function, which retains its standard representation.