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Abstract:
Suppose n arcs, each of length a, are placed at random on the edge of a
circle of circumference 1. The measure of the circumference that is covered
will be a random variable with a very intriguing asymptotic distribution as
the number of arcs grows while decreasing their length so that the
probability of complete coverage remains fixed. I will review historical
background on coverage problems including (1) a manuscript fragment
discovered in 1994 at the former estate of Buffon, who created the field of
geometrical probability with his needle problem, (2) a duality due to Ailam
that links the k-th moment when n arcs are placed to the
n-th moment when k
arcs are placed, and (3) Stevens' formula for the probability that the
circle is covered. Then I will present my own work, including (1) an exact
characterization of the distribution of the amount covered by providing the
moments and cumulative distribution function in closed form, (2) an
application of this distribution to a time-series problem in which the goal
is to discover whether apparent multiple sine waves can be distinguished
from background noise (extending Fisher's test for a single frequency), and
(3) the emergence of the previously unpublished noncentral chi-squared
distribution with zero degrees of freedom as the asymptotic coverage
distribution.
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