Compound Poisson approximation with association or negative association via Stein's method

On magic factors in Stein’s method for compound Poisson approximation

“…From this, it is easy to see that θ 2 < 2θ 1 for all n and k if q k < 2/3, in which case we are in a position to apply Corollary 3. Similarly, 2θ 3 < 3θ 2 in this case, so we expect Corollary 3 to yield a condition on U weaker than that imposed by (7). Now, straightforward calculations show that, in the notation of Corollary 3,…”

“…Choosing k = 1 in Theorem 2, the condition δ 1 > 0 is easily shown to be equivalent to (7). The condition δ 2 > 0 is actually stronger than (7).…”

“…The condition δ 2 > 0 is actually stronger than (7). However, for k ≥ 3 we obtain a condition which will, in certain situations, be weaker than (7). In Section 3, our applications will mainly employ the case k = 3.…”

“…which is positive provided that p = 1/2. For comparison, (7) is valid only under the stronger condition that p < 1/4, in which case (8) gives M…”

“…Our generalization of (7), by modifying the inequality to include θ j for j > 1, is presented in Section 2. Applications to runs and reliability systems are then given in Section 3.…”