Weak convergence on randomly deleted sets

Suppose t(1), t(2),... are the arrival times of units into a system. The kt h entering unit, whose magnitude is X-k and lifetime L-k, is said to be 'ac tive' at time t if I(t(k) < t(k) + L-k) = I-k,I-t = 1. The size of the acti ve population at time t is thus given by A(t) = Sigma(k greater than or equ al to 1) I-k,I-t. Let V-t denote the vector whose coordinates are the magni tudes of the active units at time t, in their order of appearance in the sy stem. For n greater than or equal to 1, suppose lambda(n) is a measurable f unction on n-dimensional Euclidean space. Of interest is the weak limiting behaviour of the process lambda*(t) whose value is lambda(m) (V-t) or 0, ac cording to whether A(t) = m > 0 or A(t) = 0.