# White noise stricly stationary proof

How do you show that the white noise process is strictly stationary?

Let's consider the i.i.d. white noise process $$a_t$$: \begin{align} E[a_t] &= 0\\ Var[a_t] &= \sigma_a^2 \end{align}

The weak stationarity is obvious. But I don't know how to show the strict stationarity. Basically I would like to show that for all $$t_1, \dots, t_n, k \in \mathbb{Z}$$, the joint distribution of $$a_{t_1},\dots, a_{t_n}$$ is the same as $$a_{t_1 -k},\dots, a_{t_n-k}$$.

• Hi: the joint distribution of the $n$ $X_{i}$ random variables at time $t_{n}$, ( the $n$ element vector ) is multivariate normal with mean zero and covariance matrix with diagonal elements $\sigma_{a}^2$ and zero everywhere else. So, you just need to show that the same thing is true for the $n$ random variables at time $t_{k+n}$ (i.e: the $n$ vector $X_{t+k+1}, X_{t+k+2}, \ldots, X_{t+k+n}$. ). I hope that helps. – mlofton Apr 16 '19 at 2:20

• Every random variable in the process/time series has the same distribution, and so for any given distinct integers $$t_1, t_2, \ldots, t_n$$ and any $$k \in \mathbb Z$$, the $$2n$$ random variables $$A_{t_1}, A_{t_2}, \ldots, A_{t_n}, A_{t_1-k}, A_{t_2-k}, \ldots, A_{t_n-k}$$ all have the same distribution.
• The random variables $$A_{t_1}, A_{t_2}, \ldots, A_{t_n}$$ are independent, and so their joint distribution is just the product of their $$n$$ identical marginal distributions. Similarly, $$A_{t_1-k}, A_{t_2-k}, \ldots, A_{t_n-k}$$ are also independent and since the have the same marginal distributions as the $$A_{t_1}, A_{t_2}, \ldots, A_{t_n}$$, the joint distribution of $$A_{t_1-k}, A_{t_2-k}, \ldots, A_{t_n-k}$$, which is the product of their marginal distributions, is the same as the joint distribution of $$A_{t_1}, A_{t_2}, \ldots, A_{t_n}$$.