Large Deviation Principles for Trajectories of Compound Renewal Processes. I

Abstract

The present paper continues studies of large deviation principles for compound renewal processes that were started in [A. A. Borovkov, Asymptotic Analysis of Random Walking. Fast Decreasing Increment Distributions, Fizmatlit, Moscow, 2013 (in Russian)], [A. A. Borovkov and A. A. Mogul'skii, Siberian Math. J., 56 (2015), pp. 28--53]. The main subject of this research is probabilities of large deviations of the trajectories of compound renewal processes. The paper consists of two parts. In part I, under some condition on the distribution of the process, we obtain the so-called first partial local large deviation principle for the trajectories of a compound renewal process. In part II, under an alternative condition, we obtain the second partial local large deviation principle. Under additional conditions, we also obtain the “total local” and “total integral” large deviation principles for compound renewal processes.