Show that the zero-mean exchangeable Gaussian process in Sect. 15.3.3 with covariances

\[
\operatorname{cov}\left(Y_{r}, Y_{s}ight)=\sigma_{0}^{2} \delta_{r s}+\sigma_{1}^{2}
\]

has a dynamic or sequential representation beginning with \(Y_{0}=0\) followed by

\[
Y_{n+1}=\frac{n \theta \bar{Y}_{n}}{1+n \theta}+\sigma_{0} \sqrt{1+\theta /(1+n \theta)} \epsilon_{n+1}
\]

for \(n \geq 0\). Here \(\theta=\sigma_{1}^{2} / \sigma_{0}^{2}\) is the variance ratio, and \(\epsilon_{1}, \ldots\) are independent standard normal variables.