# Area under Gaussian sample path curve

Dear All, We know that the coastline paradox is related with fractal dimension of a curve. Now I want to know how to estimate the area under a sample path curve of a Gaussian process:

Given a stationary centered Gaussian process $X(t), t\in[0,1]$ with covariance function $r(h)=Cov(X(t),X(t+h))$, how to estimate $$S=\int_0^1 X^2(t)\ dt\quad ?$$

Any reference for this problem? Thanks.

Updata: Here what I mean by estimation of the integral, which is a random variable, is actually about the sample property instead of distribution property. That is given a sample path of the Gaussian process, how to estimate the area under the corresponding curve, which has nothing to do with the randomness?

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$$E(S)=E\left(\int_0^1 X^2(t) dt\right)=\int_0^1 E\bigl(\; X(t)^2\;\bigr) dt =\int_0^1 r(0) dt =r(0).$$