Statistics Seminars: Convex hulls of planar random walks with drift
21 January 2013 14:00 in CM221
On each of n unsteady steps, a drunken gardener drops a seed.
Once the flowers have bloomed, what is the minimum length of fencing
required to enclose the garden?
Denote by L(n) the length of the perimeter of the convex hull of n steps
of a planar random walk whose increments have finite second moment and
non-zero mean. Snyder and Steele showed that L(n)/n converges almost
surely to a deterministic limit, and proved an upper bound on the
variance Var[L(n)] = O(n).
I will describe recent work with Chang Xu (Strathclyde) in which we show
that Var[L(n)]/n converges, and give a simple expression for the limit,
which is non-zero for walks outside a certain degenerate class. This
answers a question of Snyder and Steele. Furthermore, we prove a central
limit theorem for L(n) in the non-degenerate case.
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