Calculating the Mean and Autocovariance Function of a Piecewise Time Series

Question

I'm taking an introductory class to time series and this was a practice problem assigned for the first week (not homework).

Consider the following time series ($w_t$ is iid. Normal($0, 1$)):

$x_t = w_t$ for $t = 1, 3, 5, 7,...$ and

$x_t = \frac{1}{\sqrt{2}}(w_{t-1}^2 -1)$ for $t = 2, 4, 6, 8,...$

Find the mean function and autocovariance function for $x_t$. Are $x_1$ and $x_2$ identically distributed?

For the mean function, I'm stuck on finding the expectation of $w_{t-1}^2$. This is what I've got so far:

$\mu_{x_t} = E(x_t$) = $0$, if $t$ is an odd number (since the expectation of white noise is $0$, right?), and $\mu_{x_t} = E(\frac{1}{\sqrt{2}}(w_{t-1}^2 -1))$ = $\frac{1}{\sqrt{2}}E(w_{t-1}^2) - E(\frac{1}{\sqrt{2}})$, if $t$ is an even number. For the even part of the mean function, is the expectation of $w_{t-1}^2$ simply $0$? I think $w_{t-1}^2$ is just white noise, but I don't know if I'm right.

For the autocovariance function, are there 3 possible cases? $t$ can be odd or even. In either case, if $h = 0, 2, 4, 6$... (i.e. even number), then the autocovariance would simply be the variance. So, for cases where $h$ is odd, the autocovariance would be Cov$(x_t, x_{t + h})$ = Cov$(x_{t + h}, x_t)$. Am I on the right track with this line of thinking? If so, then the covariance would be Cov$(w_t, \frac{1}{\sqrt{2}}(w_{t-1}^2 -1)$ $=$ Cov$(w_t, \frac{1}{\sqrt{2}}(w_{t-1}^2))$ $-$ Cov$(w_t, w_t)$, right?

Any guidance on this problem and verification (or correction) of my line of thinking would be very helpful. Thank you.

Answer 1

Clearly for odd $t$:$$E(x_t)=E(w_t)=0$$and for even $t$:$$E(x_t)=\dfrac{1}{\sqrt 2}E(w_{t-1}^2-1)=\dfrac{1}{\sqrt 2}(\sigma_{w_{t-1}}^2-1)=\dfrac{1}{\sqrt 2}(1-1)=0$$Also:$$C(t_1,t_2)=E((x_{t_1}-E(x_{t_1}))(x_{t_2}-E(x_{t_2})))=E(x_{t_1}x_{t_2})$$ according to definition for distinct $t_1$ and $t_2$, $x_{t_1}$ and $x_{t_2}$ are dependent only if $t_1$ is even and $t_2=t_1+1$ or $t_2$ is even and $t_1=t_2+1$. Assume the former case. Then $t_1=t$ is even and $t_2=t+1$ is odd so:$$C(t,t+1)=E(x_tx_{t+1})=E(x_tx_{t+1})=\dfrac{1}{\sqrt 2}E(w_t^3-w_t)=0$$So the covariance function is zero for distinct $t_1$ and $t_2$ and for $t_1=t_2=t$ we have:$$C(t,t)=E(x^2_t)$$for odd $t$:$$C(t,t)=E(w_t^2)=1$$and for even $t$:$$C(t,t)=\dfrac{1}{2}E(w^4_{t-1}-2w^2_{t-1}+1)=\dfrac{1}{2}(E(w^4_{t-1})-1)=\dfrac{1}{2}(3-1)=1$$and we finally obtain:$$C(t_1,t_2)=\delta[t_1-t_2]$$where $\delta[n]=1$ for $n=0$ and zero elsewhere.

For the second question we have $$x_1=w_1$$ and $$x_2=\dfrac{1}{\sqrt 2}(w_1^2-1)$$ so $x_1$ can vary in $(-\infty,\infty)$ but $x_2$ can vary in $[-\dfrac{1}{\sqrt 2},\infty)$ so they can't be identically distributed.