Almost surely uniform convergence using ergodic Theorem

Question

I have a question rearding the almost surely uniform convergence using ergodic Theorem.

Let $\left\lbrace X_t\right\rbrace_t$ be a stationnary ergodic process. By the ergodic theorem, we have as $m\to\infty$ $$\dfrac{1}{m} \sum_{t=1}^m X_t\longrightarrow \mathbb{E} X_1\;\;\; a.s. $$

Let $\lfloor x\rfloor$ denote the flooring perator, i.e., the largest integer smaller than or equal to $x$.

My question is, by the above convergence how can we prove the following convergence, as $m\to\infty$ $$\sup\limits_{\tau\in\Pi}| \dfrac{1}{\lfloor m\tau\rfloor}\sum_{t=1}^{\lfloor m\tau\rfloor}X_t-\mathbb{E} X_1|\longrightarrow 0\;\;\;a.s $$ with $\Pi\subset [0,1]$? Thank you very much in advance for your help.

Answer 1

It works for $\Pi=(a,1]$ with $0<a<1$. Indeed, if $\tau\in (a,1]$, then $\lfloor m\tau\rfloor$ necessarily belongs to $\left\{\lfloor ma\rfloor,\dots,m\right\}$ hence $$ \sup\limits_{\tau\in\Pi}\left| \dfrac{1}{\lfloor m\tau\rfloor}\sum_{t=1}^{\lfloor m\tau\rfloor}X_t-\mathbb{E} X_1\right| \leqslant \max_{\lfloor ma\rfloor\leqslant \ell\leqslant m}\left|\frac 1\ell\sum_{t=1}^\ell X_t-\mathbb E\left[X_1\right]\right|\leqslant \sup_{\ell\geqslant\lfloor ma\rfloor}\left|\frac 1\ell\sum_{t=1}^\ell X_t-\mathbb E\left[X_1\right]\right|, $$ which goes to $0$ almost surely by the ergodic theorem.

As pointed out by Stefan, the wanted convergence does not hold if $\Pi=[0,1]$.