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The integral of the supremum process of Brownian motion
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2009 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 46, no 2, 593-600 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study the integral of the supremum process of standard   Brownian motion. We present an explicit formula for the moments of the   integral (or area) A(T) covered by the process in the time interval [0,   T]. The Laplace transform of A(T) follows as a consequence. The main   proof involves a double Laplace transform of A(T) and is based on   excursion theory and local time for Brownian motion.

Place, publisher, year, edition, pages
2009. Vol. 46, no 2, 593-600 p.
Keyword [en]
Brownian motion, supremum process, local time, Brownian areas
National Category
URN: urn:nbn:se:uu:diva-98029DOI: 10.1239/jap/1245676109ISI: 000267854000021OAI: oai:DiVA.org:uu-98029DiVA: diva2:173189
Available from: 2009-02-05 Created: 2009-02-05 Last updated: 2010-07-19Bibliographically approved
In thesis
1. The Maximum Displacement for Linear Probing Hashing
Open this publication in new window or tab >>The Maximum Displacement for Linear Probing Hashing
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study the standard probabilistic model for hashing with linear probing. The main purpose is to determine the asymptotic distribution for the maximum displacement. Depending on the ratio between the number of items and the number of cells, there are several cases to consider. Paper I solves the problem for the special case of almost full hash tables. That is, hash tables where every cell but one is occupied. Paper II completes the analysis by solving the problem for all remaining cases. That is, for every case where the number of items divided by the number of cells lies in the interval [0,1].

The last two papers treat quite different topics. Paper III studies the area covered by the supremum process of Brownian motion. One of the main theorems in Paper I is expressed in terms of the Laplace transform of this area. Paper IV provides a new sufficient condition for a collection of independent random variables to be negatively associated when conditioned on their total sum. The condition applies to a collection of independent Borel-distributed random variables, which made it possible to prove a Poisson approximation that where essential for the completion of Paper II.

Place, publisher, year, edition, pages
Uppsala: Universitetsbiblioteket, 2009. viii+20 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 63
Probabilistic analysis of algorithms, hashing, linear probing, negative dependence, Brownian motion.
National Category
urn:nbn:se:uu:diva-9545 (URN)978-91-506-2050-4 (ISBN)
Public defence
2009-02-27, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15
Available from: 2009-02-05 Created: 2009-02-05Bibliographically approved

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