Consider a random process $X_n$ that may be non i.i.d. Assume that for $X_n$ the following equality holds \begin{equation} \sup_{\ell \geq 1}\text{ess} \sup \lim_{n\rightarrow \infty} \mathbb{P} \left(\frac{1}{n}\sum\limits_{i= (\ell -1)n +1}^{\ell n} X_i <\alpha \bigg|\ X_1,\dots,X_{(\ell-1)n}\right) =0. \end{equation} I am trying to investigate under which conditions the following equation holds:
\begin{equation} \lim_{n\rightarrow \infty} \sup_{\ell \geq 1}\text{ess} \sup \mathbb{P} \left(\frac{1}{n}\sum\limits_{i= (\ell -1)n +1}^{\ell n} X_i <\alpha \bigg|\ X_1,\dots,X_{(\ell-1)n}\right) =0. \end{equation} When the limit is outside then the $\ell$ that attains the supremum depends on $n$ and I am not sure if the result holds or how to rigorously show it. Could someone intuitively explain what is the core difference between those two equations and what assumptions might be needed to proceed to prove the second equation? Any help would be greatly appreciated. Thanks!