If $\{S_n\},\{T_n\}$ are two families of random variables such that $S_n, T_n$ are identically distributed. Then, is it possible that $\limsup_{n\to\infty} S_n$ and $\limsup_{n\to \infty} T_n$ are identically distributed?
It seems possible for if $\{S_n\},\{T_n\}$ are collections of independent random variables. This is because let $A_m:=\{\omega:S_m(\omega)\le c\}, B_m:=\{\omega:T_m(\omega)\le c\}$ for fixed $c\in\mathbb R$, and so $\mathbb P(A_m)=\mathbb P(B_m)$.
we have that $$\mathbb P\left(\left\{\omega:\limsup_{n\to\infty} S_n(\omega)\le c\right\}\right)=\mathbb P\left(\bigcap_{n=1}^\infty\bigcup_{m=n}^\infty A_m\right)=\lim_{N\to\infty} \mathbb P\left(\bigcup_{m=N}^\infty A_m\right)=\lim_{N\to\infty}\lim_{M\to\infty}\mathbb P\left(\bigcup_{m=N}^M A_m\right),$$ and the last probability measure can be computed by splitting it into inclusion exclusion principle which by their independences the two are the same, and so they are identically distributed?
Can we drop one (or both) of the independence(s) to get the result? Is this approach okay?