Expectation of absolute value of stationary time series

Question

Let $Y_t$ be a stochastic process (time series). We consider stationarity as follows:

$Y_t$ is said to stationary if the mean $\mu_t = \mathbb{E}(Y_t)$ is constant (given $\mathbb{E}|Y_t|<\infty$) and the autocovariance function $\text{Cov}(Y_t, Y_{t+k})$ depends only on the lag $k$.

I am wondering if this stationarity property of a given process $Y_t$ has an implication on the expectation of its absolute value : $\mathbb{E}|Y_t|$.

Is this also constant ? Can we quantify the difference $\mathbb{E}|Y_t| - \mathbb{E}(Y_t)$ ? To be precise I want to find an upper bound for the sum : $$\sum_{i=0}^{\infty} \beta^{2i} \mathbb{E}\left(|Y_{t-i}| - \ Y_{t-i} \right),$$ for a given $\beta$ such that $|\beta|<1$.

Thanks a lot.

Answer 1

More over, as far as I know second order stationary process is different from weakly stationary process. May be this helps.

Answer 2

Well, obviously $E|Y_{t-i}|\leqslant\sqrt{E(Y_{t-i}^2)}=\sqrt{\sigma^2+\mu^2}$, where $\sigma^2=\mathrm{Cov}(Y_t,Y_t)$ and $\mu=E[Y_t)$ do not depend on $t$, hence $$ \sum_{i\geqslant0}\beta^{2i}E(|Y_{t-i}|-Y_{t-i})\leqslant\sum_{i\geqslant0}\beta^{2i}\left(\sqrt{\sigma^2+\mu^2}-\mu\right)=\frac{\sqrt{\sigma^2+\mu^2}-\mu}{1-\beta^2}. $$