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CATEGORIES:Probability
SUMMARY:A polynomial expansion for Brownian motion and the
associated fluctuation process - Karen Habermann
(Warwick)
DTSTART;TZID=Europe/London:20220905T150000
DTEND;TZID=Europe/London:20220905T160000
UID:TALK178460AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/178460
DESCRIPTION:We start by deriving a polynomial expansion for Br
ownian motion expressed in terms of shifted Legend
re polynomials by considering Brownian motion cond
itioned to have vanishing iterated time integrals
of all orders. We further discuss the fluctuations
for this expansion and show that they converge in
finite dimensional distributions to a collection
of independent zero-mean Gaussian random variables
whose variances follow a scaled semicircle. We th
en link the asymptotic convergence rates of approx
imations for Brownian L\\'evy area which are based
on the Fourier series expansion and the polynomia
l expansion of the Brownian bridge to these limit
fluctuations. We close with a general study of the
asymptotic error arising when approximating the G
reen's function of a Sturm-Liouville problem throu
gh a truncation of its eigenfunction expansion\, b
oth for the Green's function of a regular Sturm-Li
ouville problem and for the Green's function assoc
iated with the classical orthogonal polynomials.
LOCATION:MR9\, Centre for Mathematical Sciences
CONTACT:Jason Miller
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