In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following:
Let $(X_n,n\in\Bbb N)$ be a strictly stationary $\phi$-mixing sequence, for which $\mathbb E(X_0^2)<\infty$ and $\operatorname{Var}(S_n)\to +\infty$. Then $S_n/\sqrt{\operatorname{Var}S_n}$ (where $S_n:=\sum_{j=1}^nX_j$) is asymptotically normally distributed.
$\phi$-mixing coefficients are defined as $$\phi_X(n):=\sup(|\mu(B\mid A)-\mu(B)|, A\in\mathcal F^m, B\in \mathcal F_{m+n},m\in\Bbb N ),$$ where $\mathcal F^m$ and $\mathcal F_{m+n}$ are the $\sigma$-algebras generated by the $X_j$, $j\leqslant m$ (respectively $j\geqslant m+n)$, and $\phi$-mixing means that $\phi_X(n)\to 0$.
My question here :Is there a connection between uniform law of large number and Ibragimov's conjecture ? Thank you for any kind of help.