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

Umm the link says a continuous white noise process. (section 2 paragraph 1 just before equation1) — Kitter Catter, Sep 02 '16 at 22:55
@KitterCatter what is a continuous gausssian white noise process, how is it defined ? and how does it differ from typical gaussian white noise ? thanks — raK1, Sep 02 '16 at 22:56
Looking it up: https://en.wikipedia.org/wiki/White_noise#Continuous-time_white_noise Essentially it follows the normal rules of white noise, but rather than a Kronecker Delta in the expectation value you end up with a dirac delta. — Kitter Catter, Sep 02 '16 at 23:00
@KitterCatter I have read the wikipedia page, I cant understand how the dirac delta function is able to construct integerability — raK1, Sep 02 '16 at 23:02
Worth checking: "Convolution process with gaussian white noise" https://math.stackexchange.com/q/1911580/532409 — Quillo, Feb 25 '23 at 16:11

score 0 · Answer 1 · answered Sep 02 '16 at 23:16

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.

Isnt this the way to define a continuous white noise ? – raK1 Sep 02 '16 at 23:19 — raK1, Sep 02 '16 at 23:19

score 0 · Accepted Answer · answered Sep 03 '16 at 01:02

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

Convolute White noise

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