Final answer:
I.i.d. normally distributed random variables xt ∼ N (0, σ2) are both strictly stationary and weakly stationary. Strictly stationary because their joint distributions are invariant with time shifts, and weakly stationary since mean and autocovariances are constant. The Central Limit Theorem implies that sums of such variables approach normality as sample size increases.
Step-by-step explanation:
The question asks whether a sequence of i.i.d. normally distributed random variables xt ∼ N (0, σ2) is strictly stationary and/or weakly stationary. A process is strictly stationary if the joint distribution of any set of terms xt1,...,xtk is the same as the joint distribution of the terms xt1+h,...,xtk+h for all t1,...,tk, and for every integer h. Since all the variables in the sequence have the same distribution by being identically distributed, this condition is met. Hence, the process is strictly stationary.
On the other hand, a process is weakly stationary (or second-order stationary) if its mean and all its autocovariances do not change over time. As each xt has a mean of 0 and a variance of σ2 which are constant, and since all covariances are zero due to independence, this process is also weakly stationary.
Central Limit Theorem
The Central Limit Theorem (CLT) states that, given a sufficiently large sample size n, the sample means (and sums) of an independent and identically distributed random variable will tend to be normally distributed, regardless of the original distribution of the variable.
In the context of this question, if we consider a sum of such variables (ΣX), as n increases, the distribution of the sum will approximate a normal distribution according to EX ~ N[(n)(µx), (√n)(σx)], where µx is the mean and σx is the standard deviation of the original distribution of X.