4
votes
0answers
95 views

Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion

Is there a way to solve analytically the Fredholm integral equation of the second kind $$ \int_0^{100} K(s, t) f(s) ds = \lambda f(t) $$ where the kernel has the piecewise 'linear' form \begin{align} ...
0
votes
1answer
105 views

On the expected value of a random integral:

Is it possible to find the expected value of $u(t)$ in terms of the following information: $$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$ where: $X_s$ is a wide sense stationary process with known ...
5
votes
1answer
256 views

Is this inverted integral transform valid?

I have the following transform: $$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$ with the following conditions: $f(x)$ and $F(y)$ must ...
2
votes
1answer
175 views

Determining the asymptotic behavior of random matrices with vanishing ratio dimensions

Consider an $N\times K$ random matrix $X$ (defined on a probability space $(Ω,F,μ)$) with i.i.d. entries having zero mean and variance $1/K$. There are a lot of results regarding the asymptotic ...
0
votes
0answers
76 views

Expectation and Stieltjes transformation

I need to find the expectation of $\ln (x-\epsilon) $ with respect to a probability distribution $\mathbb{P}(x)$. A direct evaluation seems very difficult as the expression for $ \mathbb{P}(x)$ is ...
3
votes
1answer
1k views

Intuitive understanding of the Stieltjes transform

I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does. The gist of my work is that I have an $N\times N$ true covariance ...
5
votes
1answer
350 views

Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise. On ...