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I need some help to understand what professor Terence Tao means in this part of "Topics in random matrix theory".

pg 160

I'm having a hard time to undertand this function ESD. How is possible for it to be a distribution function, a probability measure and a random variable at the same time? If someone knows something about, please, I need to understand the details here...I'm a bit confused. If is not ask too much, I would like to see explicitly the definitions of all objects that T. Tao mentions in this page.

PS: This problem is related to this one Eigenvalues of a Random Matrix . What I need most is an explanation about how the ESD of a random matrix works with a ESD of an real matrix (a random matrix evaluated). Please take a look at the comments of that link.

Thank you.

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  • $\begingroup$ The eigenvalues of the matrix are random, so if you put a $\delta$ at each of the eigenvalues, you get a random measure. The distribution function is the integral of this measure. $\endgroup$ Feb 7, 2014 at 18:55

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A random variable is, by definition, a measurable function on a probability space, $(\Omega,\Sigma,\Pr)$. The range of the measurable function just has to be something on which it makes sense to talk about an appropriate sigma-field.

For example, a real valued random variable is one that is measurable with respect to any Borel set. That is, it is a function $X:\Omega\to\mathbb R$ so that $X^{-1}(A) \in \Sigma$ for any Borel set $A$.

A Banach space random variable is defined similarly, except that you have to worry about which topology you are going to use to generate the Borel sets on the Banach space. Do you use the norm topology, or the weak topology, or something else? For this reason they make the added restriction that the range of the random variable stays in a separable subset of the Banach space almost surely - this has the effect that all these topologies produce the same Borel sets.

Now Tao has a random variable that takes its value in the Banach space of measures on $\mathbb R$. So each $\omega \in \Omega$ produces a matrix $M$, and that matrix produces a measure on $\mathbb R$. That measure on $\mathbb R$ happens to be a probability measure itself (i.e. non-negative with total variation $1$), and in fact picks each eigenvalue with a probability proportional to its multiplicity.

Issues of measurability are more complex, because the range is most definitely not separable. But it does seem to me that it is measurable with respect to the weak* topology on $\mathcal M(\mathbb R)$ induced by its duality from $\mathcal C_0(\mathbb R)$.

In any case, if you ignore the issues of what it means to be measurable, each $\omega\in\Omega$ produces a probability measure on $\mathbb R$. So we can call it a random measure.

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