I am having a very hard time understanding Strict Sense Stationary Random Processes(SSSRP). One of the examples I am given has $X[n]$ being a SSSRP. We then have $Y[n] = X[n]^2$. Does this make $Y[n]$ SSS?
I originally thought yes, because $Y[n+t] = X[n+t]^2$ which would make it SSS, correct?
(Homework)
Assume that the sequence $(X_n)_{n\geqslant 1}$ is strictly stationary. This means that for each integer $d$, the $d$-uple $(X_1,\dots,X_d)$ and $(X_2,\dots,X_{d+1})$ have the same distribution, that is, if $B$ is a Borel subset of $\mathbb R^d$, then $$\mathbb P\{(X_1,\dots,X_d)\in B\}=\mathbb P\{(X_2,\dots,X_{d+1})\in B\}.$$ Define the map $f\colon\mathbb R^d\to\mathbb R^d$ given by $f(x_1,\dots,x_d)=(x_1^2,\dots,x_d^2)$. Using the definition of stationary with the Borel set $f^{-1}(B)$ that $$\mathbb P\{(X_1^2,\dots,X_d^2)\in B\}=\mathbb P\{(X_2^2,\dots,X_{d+1}^2)\in B\},$$ which proves that the sequence $(X_n^2)_{n\geqslant 1}$ is strictly stationary.
We can generalize, proving that $(h(X_n))_{n\geqslant 1}$ is a strictly stationary sequence for any $h\colon\mathbb R\to\mathbb R$ Borel-measurable.