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Yes, you can define properly the first expectation, see e.g. PlanetMath Then you have with your notations $$ \mathbb E_{X_i : i\in\mathbb N}\int_{[0,1]^k}\inf_{i\in\mathbb N}\|X_i-y\|^2dy\leq \math …

I know that the limiting eigenvalue distribution satisfies a variational principle (see e.g. the joint work with A. Kuijlaars Large Deviations for a Non-Centered Wishart Matrix) from which you may der …

I came up with this operation after playing with determinantal point processes: Given two probability densities $f,g$ defined on some measurable space with reference measure $\mu$, set $$ f\star g(x …

By symmetry you're looking at the probability that the maximal eigenvalue is smaller than some number. Explicit inequalities for such events can be obtained by using the tridiagonal representation for …

It seems that what you are looking for is the notion of "abstract Wiener space", which provides a rigorous setting for putting a white noise on various spaces of functions. Long story short, you defin …

I'm less enthusiastic than the previous answers concerning how your question is easily solvable: What tells the HKPV algorithm mentioned above is that you can exactly sample a determinantal point proc …

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the podiu …

If you accept non-commutative random variables as "other random processes $(X_1,\ldots,X_n)$" then, as Jon Bannon's comment suggests, free probability could provide interesting examples [and I don't t …

Just an heuristic answer: The joint eigenvalue distribution of a GOE random matrix is given by $$ \frac{1}{Z_n}\prod_{i<j}|x_i-x_j|\prod_{i=1}^ne^{-x_i^2/2}dx_i. $$ Thus, $E_{GOE}(|\det(y+A)|e^{-x(tr …

Consider a probability measure $\mu$ on, let's say, $\mathbb R$. Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ? I agree this question is too vague, an …

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well define …

For an absolutely continuous finite Borel measure $\mu(dx)=f(x)dx$ on $\mathbb R$, if $\mu(\mathbb R)\neq 1$ then $f$ is sometimes called the "intensity" of the measure.

Consider $N$ real random particles $x_1,\cdots, x_N$ distributed according to a density $\rho(x_1,\ldots,x_N)$ with respect to the Lebesgue measure on $\mathbb R^N$, which is assumed to be invariant u …

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with law $\mu$. The Sanov Theorem then states that the empirical measures $$ \mu^N =\frac{1}{N} \sum _{n=1}^N\delta _{X_n} $$ sat …

We denote $\varphi:\mathbb R^2\rightarrow\mathbb R$ the addition of real numbers, and $\varphi_*:M_1(\mathbb R^2)\rightarrow M_1(\mathbb R)$ the induced push-forward map (where $M_1(\Delta)$ stands fo …