constant variance for strictly stationary process

Question

How to prove mathematically that variance is constant for strictly stationary process? Given that distribution function is time invariant. It is intuitive but not sure where to start to prove it.

Answer 1

In general: the mean of a random variable is given by $\mu=\int_{-\infty}^\infty xdF(x)$, where $F(x)$ is the distribution function. Similarly the variance is defined by $\sigma^2=\int_{-\infty}^\infty (\mu-x)^2dF(x)$. Since $F(x)$ is the same for all the elements of the process, $\mu$ and $\sigma^2$ will be constant.