# Must i.i.d. stochastic process be stationary?

I have learned that, for an i.i.d. random process $\{X_k\}$, $k=1, 2, \ldots$, $$\frac{1}{n}\sum_{k=1}^n{X_k}\longrightarrow EX_1$$ in probability sense.

For a stationary random process $\{X_k\}$, $k=1, 2, \ldots$, $$\frac{1}{n}\sum_{k=1}^n{X_k}\longrightarrow EX_1$$ with probability $1$.

However, someone told me that for an i.i.d. random process, its sample mean converges to expectation value with probability $1$ as $n$ grows.

I am confused about the fact between WLLN and SLLN. Isn't there any problem that I admit that i.i.d. random processes are also having stationarity?

• This proposition is true if $\operatorname{var}(X_1)<\infty$. However, if the distribution is $\displaystyle \left(\text{constant} \cdot \frac{dx}{1+x^2}\right)$ then it is false. $\qquad$ – Michael Hardy Oct 26 '16 at 0:46
• @MichaelHardy Ah, thank you for supplement. I omitted finite variance. And, do you mean that the distribution refers to CDF of $X$?? – Danny_Kim Oct 26 '16 at 3:07
• The density is $x\mapsto \dfrac{\text{constant}}{1+x^2}$; therefore the distribution is $\left( \text{constant} \cdot \dfrac{dx}{1+x^2}\right)$ and the cumulative distribution function is $x\mapsto \dfrac 1 \pi\cdot \arctan x. \qquad$ – Michael Hardy Oct 26 '16 at 3:14

A strict(strong)-sense stationary process $\{X_t\}$ is one whose joint distributions for any set of times $t_1,\ldots,t_k$, that $F_X(t_1,\ldots,t_k) = F_X(t_1+\tau,\ldots,t_k+\tau)$ for any $\tau$.