The Local Principle Of Large Deviations For Compound Poisson Process With Catastrophes

The continuous time Markov process considered in this paper be-longs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle.Motivation and historical remarks. First, we recall that the so-called random processes with resettings has recently reappeared in the context of random search models, demographic models, biological and chemical models. See, for example, Bhat, De Bacco and Redner (2016), Evans and Majumdar (2011), Eule and Metzger (2016), Gupta, Majumdar and Schehr (2014), Montero and Villarroel (2016) and the references therein. Informally, a random process with resettings is constructed as follows. We modify a Markov process (i.e., a random walk, a birth-and-death process, a diffusion, etc.) by introducing the jumps (also called the reset points or resettings) to a fixed state (the origin). The jumps happen at the arrival times of a renewal process, independent from the original Markov process. In the above-cited papers, the time intervals between resettings can be deterministic, or they can have an exponential distribution (i.e., a Poisson process), Weibull distribution or the distribution which depends on the current state of the process. The cited works investigate the questions concerning the stationary distribution of random processes with resettings, its limiting behavior, and various properties of its trajectories. Importantly, random processes with resettings are a subclass of random processes with catastrophes that emerged in the 1970s and 80s (see, for example, Di Crescenzo et al. (2008),The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle.