Consider a Poisson process with arrival rate lambda arrivals per unit time. Given a window of time W and a total of k events, What is the upper bound of the probability that no three events happen in that window. Said another way, if events are numbered 1,2,...,k; what is the upper bound of Prob(gap(1,3) >= W and gap(2,4) >= W and gap(3,5) >= W and ... gap(k-2,k) >= W).

The problem in estimating this for me has been that while each successive gap is distributed exponentially, gap(1,3) and gap(2,4) are dependent. So, my best upper bound so far is obtained by just completely ignoring half the gaps:

Prob(gap(1,3) > W and gap(2,4) > w and gap(3,5) > W and ... gap(k-2,k) > W)
< Prob(gap(1,3) > W and gap(3,5) > W and ....)
~ (Prob(gap(1,3) > W) ^ (k-1)/2
= exp(-lambda*W)^(k-1)/2

Is there a better upper bound?