How does one give a simulation of a Gaussian process $(X_t)$ (zero mean, variance 1) over the real line with covariance $\Gamma(t)=e^{-\vert t \vert/2}?$
Since this is an AR1 process I can simulate over partitions of arbitrarily large intervals $[-T,T]$, and use this as an approximation to the simulation over the real line, however, I am not looking for an approximation I would like the simulation over the real line.
The spectal density is $g(\omega)=\frac{1}{\pi(\omega^2+\frac{1}{4})}$. I don't know what to do with $g$ once I take the Fourier transform of $X$. How can I simulate $X_t$ over $\mathbb{R}$?