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

By W. König, Neil O'Connell. 2001

132 citations recorded by the source corpus.

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 $h(x)=\prod_{i 0}$ 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.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/85ea677adad1863d4158780a3a2c47e81ff0abc8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Ordered Random Walks (2006, unknown): “cription in terms of non-colliding random processes, the Wishart processes, which are based on squared Bessel processes in place of Brownian motions. These processes and some few more were studied in [KO01]. Non-colliding Brownian motions on the circle were investigated in [HW96], asymptotic questio”
Universality of local eigenvalue statistics for some sample covariance matrices (2005, background): “Then it was also proven in [24] that, for any point z, (3.”
Random matrices, non-colliding processes and queues (2002, unknown): “It is well-known, and easy to check using the Karlin-MacGregor formula (see, for example, [35]) that lim E dy) = E dy) , is a normalisation constant.”
Multidimensional Yamada-Watanabe theorem and its applications to particle systems (2011, methodology): “ x, h(x) = 1 and b(x) = 2δ > 0, when the equation (3.8) is the SDE for the complex Wishart process, called also a Laguerre process. This process and its eigenvalues were studied by König-O’Connell [18] and Demni [5]. 3.3. Collision time. In this subsection we show that under some mild conditions ”
A Representation for Non-Colliding Random Walks (2002, background): “For related work on non-colliding di usions and random matrices, see [4, 9, 18, 22], and references therein.”
Some Properties of the Wishart Processes and a Matrix Extension of the Hartman-Watson Laws † (2004, background): “In [18], König and O’Connell consider the eigenvalues of the Laguerre process (defined as in (2.”