This is a reminder that there is a Statistics seminar at UMIST today,
Wednesday 16th December. Details are given below.
The seminar will be followed by tea and biscuits. All those interested are
welcome to attend.
(UMIST) 2.15p.m., Room O10 of the Mathematics and Social Sciences Building
at UMIST.
Rate of Growth of the Overshoot.
Professor Ross Maller,
Department of Mathematics, University of Western Australia.
Abstract. Boundary crossing probabilities for random walks are of interest
in sequential
analysis, finance, and other applications. I'll discuss some recent
results obtained with
Phil Griffin relating to the overshoot of the boundary of a strip. More
precisely, if T(r) is
the first time at which a random walk S_n escapes from the region [-r,
r], conditions for
|S_{T(r)}|/r to converge to 1, either in probability (weakly) or almost
surely (strongly), as r
tends to infinity, will be given. (We refer to this as the "stability"
of S_{T(r)}.
The almost sure characterisation turns out to be extremely simple to
state: we have
|S_{T(r)}|/r converging to 1 a.s. if and only if E(X^2) < infinity and
E(X)=0 or
0<|E(X)|<E|X|<infinity. To prove this we relate the stability of
S_{T(r)} to certain
dominance properties of the maximum partial sum over its maximal
increment, and to
certain new kinds of "stochastic compactness" of the random walk, which
I will also briefly
discuss.
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