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Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes
2.	Creator	Author's name, affiliation, country	Wolfgang König; BRIMS, HP Labs
2.	Creator	Author's name, affiliation, country	Neil O'Connell; BRIMS, HP Labs
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Wishart and Laguerre ensembles and processes, eigenvalues as diffusions, non-colliding squared Bessel processes.
3.	Subject	Subject classification	15A52, 60J65, 62E10.
4.	Description	Abstract	Let $A(t)$ be an $n\times p$ matrix with independent standard complex Brownian entries and set $M(t)=A(t)^*A(t)$. This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process. The purpose of this note is to remark that, assuming $n > p$, the eigenvalues of $M(t)$ evolve like $p$ independent squared Bessel processes of dimension $2(n-p+1)$, conditioned (in the sense of Doob) never to collide. More precisely, the function $h(x)=\prod_{i < j}(x_i-x_j)$ is harmonic with respect to $p$ independent squared Bessel processes of dimension $2(n-p+1)$, and the eigenvalue process has the same law as the corresponding Doob $h$-transform. In the case where the entries of $A(t)$ are real Brownian motions, $(M(t))_{t > 0}$ is the Wishart process considered by Bru (1991). There it is shown that the eigenvalues of $M(t)$ evolve according to a certain diffusion process, the generator of which is given explicitly. An interpretation in terms of non-colliding processes does not seem to be possible in this case. We also identify a class of processes (including Brownian motion, squared Bessel processes and generalised Ornstein-Uhlenbeck processes) which are all amenable to the same $h$-transform, and compute the corresponding transition densities and upper tail asymptotics for the first collision time.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2001-08-31
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1040
10.	Identifier	Digital Object Identifier	10.1214/ECP.v6-1040
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 6
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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