PDF of cross correlation of filtered bi-variate Gaussian

Question

The probability distribution function (PDF) of sample Pearson correlation coefficient for bi-variate Gaussian is available in the literature. For example see : http://mathworld.wolfram.com/CorrelationCoefficientBivariateNormalDistribution.html

Now, my problem is this: I have bi-varite Gaussian random variables say $w[n]$ and $z[n]$ with correlation coefficient $\rho_{wz}$ and individual variances $\sigma_1^2$ and $\sigma_2^2$, respectively. Now consider the AR(1) model with $$x[n]=\alpha x[n-1]+w[n] \\ y[n]=\beta y[n-1]+z[n]$$

1. Are $x[n]$ and $y[n]$ bi-variate Gaussian?
2. What is the PDF of the correlation coefficient of that bi-variate Gaussian?

Any help or pointers to references will be appreciated. Thanks

My Try:

We can write the above processes as $$x[n]=\sum_{i=0}^{n}\alpha^{n-i}w[i] = \mathbf{\alpha}_n^T\mathbf{w}_n$$ $$y[n]=\sum_{i=0}^{n}\beta^{n-i}z[i] =\mathbf{\beta}_n^T\mathbf{z}_n$$

Clearly $x[n]$ and $y[n]$ are Gaussian with mean 0 and variances $\sigma_x^2=\frac{1-(\alpha^2)^{(n+1)}}{1-\alpha^2}$ and $\sigma_y^2=\frac{1-(\beta^2)^{(n+1)}}{1-\beta^2}$, respectively. Are $x[n]$ and $y[n]$ bi-variate Gaussian? Let us consider the covariance between $x[n]$ and $y[n]$ given by

\begin{equation} \begin{split} \mathbb{E}[x[n]y[n]] &= \mathbb{E}[x[n]y[n]] \\ &= \mathbb{E}[\mathbf{\alpha}_n^T\mathbf{w}_n \mathbf{\beta}_n^T\mathbf{z}_n] \\ &= \mathbb{E}[\mathbf{\alpha}_n^T\mathbf{w}_n \mathbf{z}_n^T\mathbf{\beta}_n]\\ &= \mathbf{\alpha}_n^T \mathbb{E}[\mathbf{w}_n \mathbf{z}_n^T]\mathbf{\beta}_n\\ \end{split} \end{equation} where the matrix $$\mathbb{E}[\mathbf{w}_n \mathbf{z}_n^T]= \begin{bmatrix} \rho_{wz} \sigma_1\sigma_2 & \rho_{wz} \sigma_1\sigma_2 & \rho_{wz}\sigma_1\sigma_2 & \dots & \rho_{wz} \sigma_1\sigma_2\\ \rho_{wz} \sigma_1\sigma_2 & \rho_{wz}\sigma_1\sigma_2 & \rho_{wz} \sigma_1\sigma_2& \dots & \rho_{wz}\sigma_1\sigma_2 \\ \vdots & & & & \vdots \\ \rho_{wz} \sigma_1\sigma_2 & \rho_{wz} \sigma_1\sigma_2& \rho_{wz} \sigma_1\sigma_2& \dots & \rho_{wz}\sigma_1\sigma_2 \end{bmatrix} = \rho_{wz} \sigma_1\sigma_2 \begin{bmatrix} 1 & 1 & 1 & \dots & 1\\ \vdots & & & & \vdots \\ 1 & 1& 1& \dots & 1 \end{bmatrix}$$

Now, \begin{equation} \begin{split} \mathbb{E}[x[n]y[n]] &= \mathbf{\alpha}_n^T \mathbb{E}[\mathbf{w}_n \mathbf{z}_n^T]\mathbf{\beta}_n\\ & = \rho_{wz}\sigma_1\sigma_2 \frac{1-\alpha^{(n+1)}}{1-\alpha} \frac{1-\beta^{(n+1)}}{1-\beta} \\ \rho_{xy}& = \rho_{wz}\frac{\sigma_1}{\sigma_x} \frac{\sigma_2}{\sigma_y} \frac{1-\alpha^{(n+1)}}{1-\alpha} \frac{1-\beta^{(n+1)}}{1-\beta} \end{split} \end{equation}

Can anyone verify the above math. statements?

My new Try:

Answer 1

For the first equation, bring the $X$ terms over to the left hand side. Write the same equation for $ n , n-1, n-2 $ etc. Multiply each equation starting from the $X[n-1] $ equation by $ \alpha, \alpha^2, \alpha^3$ etc. Add the equations vertically. On the left hand side there will be telescoping cancellations and you will be left with the $ X[n] $ and $X[1] $ terms.On the right hand side you will have a sum of bivariate Gaussian variables which is also bivariate Gaussian. You could use the moment generating function idea to see what parameters it has and then refer back to your link to see what the PDF of the correlation coefficient is.

Answer 2

There should be $ \alpha\beta $ terms.

Using the moment generating function result for $ W $ and $ Z $ bivariate Gaussian, namely

$ E(exp(sW+tZ)) = exp(1/2(\sigma_1^2s^2 +2\rho\sigma_1\sigma_2st+ \sigma_2^2t^2)) $

we can get that

$ E(exp(s{\alpha}W+t{\beta}Z)) = exp(1/2(\alpha^2\sigma_1^2s^2 +2\rho\alpha\beta\sigma_1\sigma_2st+ \beta^2\sigma_2^2t^2)) $

Replace $ W $ and $ Z $ by the sums you have above. Then use something like

$ E(exp(s(W[1] + W[2] +...)+t(Z[1] + Z[2] + ...))) = E(exp(sW[1]+tZ[1]))E(exp(sW[2]+tZ[2]))E(exp(...))...$ $ = exp(...)exp(...)exp(...)...= exp(...) $

You can then add the coefficients of the $ s $ $ t $ and $ st $ exponents in the moment generating function.

This assumes that $ (W[1],Z[1]), (W[2], Z[2]), ... $ are independent random variables.

The advantage of using the moment generating function is that it proves that the result is bivariate Gaussian.