Calculating Mean, Variance and Autocovariance Functions of Time Series

Question

I'm having trouble finding the mean, variance and autocovariance functions of a time series function. Looked around and couldn't find a problem like this. Image attached

Vsubt = 1/q * summation Xsub(t-j)

^much clearer in the image though. Thanks for the help!

Image of the problem

Answer 1

You need to know things about $X_k$ and $X_kX_m$ for all values of $k$ and $m$. Assuming that is available then: $E(V_t)=\frac{1}{q}\sum_{j=1}^qE(X_{t-j})$ and $E(V_tV_s)=\frac{1}{q^2}\sum_{j=1}^q\sum_{i=1}^qE(X_{t-j}X_{s-i})$. The mean is $E(V_t)$, the variance is $E(V_t^2)-(E(V_t))^2$ and the autocovariance is $E(V_tV_s)-E(V_t)E(V_s)$.