The Extended Large Deviation Principle for the Trajectories of a Compound Renewal Process

Abstract

We study a homogeneous compound renewal process (c.r.p.) \(Z(t) \). It is assumed that the elements of the sequence that rules the process satisfy Cramér’s moment condition \([{\bf C}_0] \). We consider the family of processes

$$ z_T(t):=\frac 1xZ(tT),\enspace \enspace 0\le t\le 1,$$

where \(x=x_T\sim T \) as \(T\to \infty \). Conditions are proposed under which the extended large deviation principle holds for the trajectories \( z_T\) in the space \((\mathbb {V},\rho B) \) of functions with bounded variation, endowed with Borovkov’s metric. If the trajectories of the process \(Z(t) \) are monotone with probability 1 then, under the same condition, we prove the classical trajectory large deviation principle.