Convolute White noise

Question

In the famous paper Higdon (2002) http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.5356&rep=rep1&type=pdf

It is stated that a Gaussian process is established by convolving a convolving a gaussian white noise process $x(s)$ with a smoothing kernel $k(s)$. Like the one in the figure below

$$z(s)=\int_{S}^{} \! k(u-s) x(u).du \ \ \text{where } s\in R $$

White noise is discontinuous and Riemann integration cannot be used. What are the asusmptions here ? Can anyone help me understand the intuition

Link to Pic

Answer 1

The first paper and book will answer your question will answer your question

1) http://www.sciencedirect.com/science/article/pii/S0378375897001626

2) Yaglom, A.M., 1987. Correlation Theory of Stationary and Related Random Functions I: Basic Results. Springer, New York.

PS: You also might want to look at stable linear filters which poses the same question

Answer 2

The Higdon paper defines Gaussian white noise as i.i.d. normal distributions (of zero mean) on a discrete lattice of points, and takes a limit as the lattice spacing goes to zero and the variance of the distributions decreases like the square root of the lattice distance. This is a lot like ways to define Brownian motion.

As to whether the paper properly treats the limit as $d \to 0$, I can't verify that myself but the paper's referees would presumedly have caught any unjustified steps.