Source: J. Appl. Probab. Volume 47, Number 2
(2010), 367-377.
We introduce and motivate the study of (n+1) x r arrays X
with Bernoulli entries Xk,j and independently
distributed rows. We study the distribution of
Sn =
∑j=1r∑k=1n
Xk,jXk+1,j,
which denotes the number of consecutive pairs of successes (or runs of length
2) when reading the array down the columns and across the rows. With the case
r = 1 having been studied by several authors, and permitting some
initial inferences for the general case r > 1, we examine various
distributional properties and representations of Sn
for the case r = 2, and, using a more explicit analysis, the case of
multinomial and identically distributed rows. Applications are also given in
cases where the array X arises from a Pólya sampling scheme.
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