Let $\text{GP}(\mu(x),k(x,x^{\prime})$ be a gaussian process. Here, $\mu$ is the mean function. Typically, $\mu$ equals to $0$. $k$ is the kernel function.

Can you define a convolution of the gaussian process, $\int_{x} \text{GP}\left(\mu(x), k(x,x^{\prime})\right)f(x)dx$?

Here, $f$ is the $L_2$ function.

Intuitively, this is a limitation of gaussian random variables, $\int_{x} \text{GP}\left(\mu(x), k(x,x^{\prime})\right)f(x)dx \approx \sum_{i=0}^{N} f(x_i)X_{x_i}\approx N\left(\int_{x} \mu(x) f(x)dx, \int_{x,x^\prime}k(x,x^\prime)f(x)f(x^\prime)dxdx^\prime\right)$

This intuition is true? And, how to formulate it? Maybe, this is a stochastic integral. What field in math should I study?


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