# Red Noise (aside exercise)

Consider the first-order autoregressive process: $X_{t+1} = \rho X_t + \sqrt{1-\rho^2}Z_t$

Why the funny term at the end? This comes from normalizing the variance,

$\begin{multline} E( (\rho X_t + \sqrt{1-\rho^2} Z_t)^2 ) = \\ E (\rho^2 X_t^2) + E(2 \rho X_t \sqrt{1-\rho^2}) Z_t ) + \\ E( X_t^2 Z_t^2 (1-\rho^2) ) \end{multline}$

by independence:

$= \rho^2 E(X_t^2) + 2 \rho \sqrt{1-\rho^2} E( X_t ) E( Z_t ) + E( X_t^2) E(Z_t^2) (1-\rho^2)$

and as $$E(Z_t) = 0$$ and $$E(Z_t^2) = 1$$ then

$= \rho^2 E(X_t^2) + E( X_t^2) \sigma^2 (1-\rho^2) = E(X_t^2)$

and the variance is stationary, $$E((X_{t+1})^2) = E((X_{t})^2)$$

A nice exercise connects this to the continuous time result,

$dX = -\gamma X dt + dW$

Where the variance is again reduced by the correlation coefficient, $$E(X^2) = 1/(2\gamma)$$.