Sum of strictly stationary sequence

Question

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence strictly stationary real random variables and $k\in\mathbb{N}$. Does \begin{align*} \sum_{i=1}^{n}X_i\end{align*} have the same distribution as \begin{align*} \sum_{i=1+k}^{n+k}X_i?\end{align*} Thanks!

Answer 1

From the definition of strict stationarity, the joint distribution of (X1,...,Xn) is the same as the joint distribution of (X1+k,...,Xn+k), so the application of the same continuous function (say f) to both will yield random variables/vectors with the same distribution.

Let the continuous function f: R^n --> R be given by f(X1,...,Xn) = X1 + X2 + ... + Xn.

Apply f to both random vectors and you're done.