pr.probability - What are the big problems in probability theory?

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following:

Let $(X_n,ninBbb N)$ be a strictly stationary $phi$-mixing sequence, for which $mathbb E(X_0^2)<infty$ and $operatorname{Var}(S_n)to +infty$. Then $S_n:=sum_{j=1}^nX_j$ is asymptotically normally distributed.

$phi$-mixing coefficents are defined as

$$phi_X(n):=sup(|mu(Bmid A)-mu(B)|, Ainmathcal F^m, Bin mathcal F_{m+n},minBbb N ),$$
where $mathcal F^m$ and $mathcal F_{m+n}$ are the $sigma$-algebras generated by the $X_j$, $jleqslant m$ (respectively $jgeqslant m+n)$, and $phi$-mixing means that $phi_X(n)to 0$.

It was posed in Ibragimov and Linnik paper in 1965.

Peligrad showed the result holds with the assumption $liminf_{nto +infty}n^{-1}operatorname{Var}(S_n)>0$. It also holds when $mathbb Elvert X_0rvert^{2+delta}$ is finite for some positive $delta$ (Ibragimov, I think).

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