ON THE INCREMENTS OF THE DOUBLY STOCHASTIC POISSON PROCESSES

Let {X (t) }t greater-than-or-equal-to 0 be a doubly stochastic Poisson process, of the form X (t) = N (LAMBDA(t)), where {N (t) }t greater-than-or-equal-to 0 is a standard Poisson process and { LAMBDA (t) }t greater-than-or-equal-to 0 is a process with independent and stationary increments, independent of N (0). We study the asymptotic behaviour of the maximal increments of X (t), of the form L (T, K) = sup {X (t + K) - X (t) } when K = K(T) satisfies suitable 0 less-than-or-equal-to t less-than-or-equal-to T-K conditions of growth and regularity.