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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the explicit form of the density of Brownian excursion local time


Author: G. Hooghiemstra
Journal: Proc. Amer. Math. Soc. 84 (1982), 127-130
MSC: Primary 60J65; Secondary 60J55
MathSciNet review: 633293
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Abstract: Let $ \underline W _0^ + (t)$, $ 0 \leqslant t \leqslant 1$, denote Brownian excursion and let $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{l} _0^ + (\upsilon )$, $ \upsilon \geqslant 0$, be its local time at level $ \upsilon $. Starting from a representation of the density of $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{l} _0^ + (\upsilon )$ as a complex integral we derive an explicit form of this density, written as an infinite series involving the $ n$-fold convolution of known densities. Finally the result is used as an alternative check of Knight's result on the same topic.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0633293-5
Article copyright: © Copyright 1982 American Mathematical Society