Let $X_n$, $Y_n$ be two sequences of random variables. Suppose for each $n$, $X_n$ has the same distribution as $Y_n$. Can I conclude that $\limsup_n X_n$ has the same distribution as $\limsup_n Y_n?$
The reason I was thinking about this question is another question that I have in the following question. Now I summarised the key obstacle that I have.
Take $(X_n)$ to be i.i.d. copies of Bernoulli variables, i.e. $\mathbb{P}(X_n=1)=\mathbb{P}(X_n=-1)=1/2$. Then let $Y_0=Y_1=Y_2=\dots$ be a Bernoulli variable (let's say independant from the $X_i$).
For each $n$, $X_n$ and $Y_n$ have the same distribution. However, $\limsup{X_n}$ is $1$ almost surely while $\limsup{Y_n}=Y_0$ is a Bernoulli variable.