Ibragimov–Iosifescu conjecture for φ-mixing sequences

Ibragimov–Iosifescu conjecture for φ-mixing sequences in probability theory is the collective name for 2 closely related conjectures by Ildar Ibragimov and ro:Marius Iosifescu.

Conjecture
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 $$S_n:=\sum_{j=1}^nX_j$$ 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 $$\phi_X(n)\to 0$$.

Reformulated:

Suppose $$X:=(X_k, k \in {\mathbf Z})$$ is a strictly stationary sequence of random variables such that $$EX_0 = 0, \ EX_0^2 < \infty$$ and $$ES_n^2 \to \infty$$ as $$n \to \infty$$ (that is, such that it has finite second moments and $$\operatorname{Var}(X_1 + \ldots + X_n) \to \infty$$ as $$n \to \infty$$).

Per Ibragimov, under these assumptions, if also $$X$$ is $$\phi$$-mixing, then a central limit theorem holds. Per a closely related conjecture by Iosifescu, under the same hypothesis, a weak invariance principle holds. Both conjectures together formulated in similar terms:

Let $$\{X_n\}_n$$ be a strictly stationary, centered, $$\phi$$-mixing sequence of random variables such that $$EX^2_1 < \infty$$ and $$\sigma^2_n \to \infty$$. Then per Ibragimov $$S_n / \sigma_n \overset{W}{\to} N(0, 1)$$, and per Iosifescu $$S_{[n1]} / \sigma_n \overset{W}{\to} W$$. Also, a related conjecture by Magda Peligrad states that under the same conditions and with $$\phi_1 < 1$$, $$\overset{\sim}{W}_n \overset{W}{\to} W$$.