Let [NA(t), t≥0] and [NB(t), t≥0] be two counting processes with independent interarrivals Ai and Bi, respectively, i ∈ + ≡ [1,2,… ], and let [Xi, i ∈ +] and [Yi, i ∈ +] be two sequences of independent non-negative random variables. Consider the two compound processes [U(t), t ≥ 0] and [V(t), t ≥ 0] defined as U(t) = Xi, t ≥ 0, and V(t) = Yi, t ≥ 0. Then it is well known (and not hard to verify) that if the Ai's are independent of the Xi's, and if the Bi's are independent of the yi's, and if Ai ≥st Bi and Xi ≤st Yi, for all i ∈ +, then [U(t), t ≥ 0] ≤st [V(t), t ≥ 0]. In this paper we provide conditions under which this stochastic order relation holds even if Ai ≤st Bi and/or Xi ≥st Yi for all i ∈ +. In fact, we derive the stochastic order relationships among processes that are much more general than the cumulative ones described above. For example, it is not necessarily assumed that the Ai's are independent of the Xi's or that the Bi's are independent of the Yi's. Examples that illustrate the applications of the theory are included.