On Asymptotic Behaviour of Probabilities of Small Deviations for Compound Cox Processes

The asymptotic behaviour of probabilities of small deviations has been investigated for various classes of stochastic processes. Asymptotics of probabilities of small deviations for sums of independent random variables and homogeneous processes with independent increments were found in Mogul’skii [1], Borovkov and Mogul’skii [2] and references therein. Note that the last class of processes includes stable, Poisson and compound Poisson processes. Various results for Gaussian processes and references may be found in surveys by Ledoux [3], Li and Shao [4] and Lifshits [5]. Further results were obtained for various stochastic processes, generated by sums of random numbers of independent random variables. In this case, the number of summands is a stochastic process which is usually independent with the summands. Increments of such the processes may be dependent. These processes are called compound processes. Now we give definitions for some of them. Let X,X1, X2, . . . be a sequence of independent, identically distributed random variables. Put Sn = X1 +X2 + · · ·+Xn for n 1, S0 = 0. Let ν(t) be a standard Poisson process, independent with the sequence {Xk}. Then the stochastic process η(t) = Sν(λt), λ > 0, is called a compound Poisson process. If δ(t) is a renewal process, independent with {Xk}, then ζ(t) = Sδ(t) is called a compound renewal process. When the renewal times have an exponential distribution, the compound renewal process coincides with the compound Poisson process. Small deviations of the renewal and compound renewal processes has been studied in Frolov, Martikainen, Steinebach [6]. Now we turn to the definition of the compound Cox process. We start with the definition of a Cox process which we borrow from the paper of Embrechts and Klüppelberg [7]. Let Λ(t), t > 0, be a random measure, i.e. a.s. (almost surely) Λ(0) = 0, Λ(t) < ∞ for all t > 0 and Λ(t) has non-decreasing trajectories. Assume that Λ(t) does not depend on the standard Poisson process ν(t). The point process N(t) = ν(Λ(t)) is called a Cox process. If, in particular, the trajectories of Λ(t) are continuous a.s., then for every