I have question regrading the below statement.
Let $X_1,X_2,\dots,X_n$ be a stationary sequence taking values in $\mathbf{Z}$ with $E|X_i|<\infty$. Let $S_n = X_1+\dots+X_n$.Then, $$\lim\sup_{n\rightarrow\infty}(\max_{1\leq k\leq n}|S_k|/n)=\lim\sup_{n\rightarrow\infty}(\max_{K\leq k \leq n}|S_k|/n)\leq (\max_{k\geq K}|S_k|/k)$$ If $E(X_1|\mathcal{I})=0$,then by ergodic theorem, $S_n/n\rightarrow 0$. Therefore as $K\uparrow\infty$,the righthand side goes to zero.
In the above statement, how does $\lim\sup_{n\rightarrow\infty}(\max_{1\leq k\leq n}|S_k|/n)=\lim\sup_{n\rightarrow\infty}(\max_{K\leq k \leq n}|S_k|/n)$ hold? Because $K$ is finite? But then how come $K\uparrow\infty$ is used?
For any fixed $k$, the value $|S_k|/n$ goes to zero if $n$ goes to infinity. Therefore you can just ignore $k$ in the $\max$. Do this $K$ times and you get the inequality as written. Only for finite $K$ of course.
After the inequality is proved, you look at what happens if you increase $K$. The right side goes does arbitrarily close to zero. And the left side is independent of $K$. Therefore the left side has to be equal to zero.
You DONT take the $K\to\infty$ limit in the middle expression.