On Excess-time Correlated Cumulative Processes

Abstract

In this paper a class of correlated cumulative processes, B s (t) = ∑N(t)i=1 H s (X i )X i , is studied with excess level increments X i ⩾s, where {N(t), t ⩾0} is the counting process generated by the renewal sequence T n , T n and X n are correlated for given n, H s (t) is the Heaviside function and s⩾0 is a given constant. Several useful results, for the distributions of B s (t), and that of the number of excess (non-excess) increments on (0, t) and the corresponding means, are derived. First passage time problems are also discussed and various asymptotic properties of the processes are obtained. Transform results, by applying a flexible form for the joint distribution of correlated pairs (T n , X n ) are derived and inverted. The case of non-excess level increments, X i < s, is also considered. Finally, applications to known stochastic shock and pro-rata warranty models are given.