Detecting clustering in binary sequences

We consider the following question: given a sequence X₁, . . . , X_n of binary values, how likely is it that the sequence was the output of n i.i.d. Bernoulli trials? And if it was not, can we detect the presence of clustering - increased local density on smaller consecutive intervals - in a reliable way? In this paper we propose a relatively simple statistic Ȓ, the sum of the reciprocal run lengths in the sequence, as a first step towards meeting this goal, and show that it can detect a wide range of clustering with relatively high probability.