Ibragimov's central limit conjecture for $\phi$-mixing sequences

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(X_j)_{j\in\mathbb Z}$ be real-valued random variables on $(\Omega,\mathcal A,\mathbb P)$, strictly stationary, meaning their joint law is invariant under every integer shift. Assume $\mathbb E X_0=0$ and $\mathbb E X_0^2<\infty$. Put $\mathcal A_{-\infty}^{m}=\sigma(X_j:j\le m)$ and $\mathcal A_{m+n}^{\infty}=\sigma(X_j:j\ge m+n)$, and define \[\phi(n)=\sup_{m\in\mathbb Z}\sup_{\substack{A\in\mathcal A_{-\infty}^{m},\ \mathbb P(A)>0\\B\in\mathcal A_{m+n}^{\infty}}}|\mathbb P(B\mid A)-\mathbb P(B)|.\] The sequence is $\phi$-mixing if $\phi(n)\to0$. Let $S_n=\sum_{j=1}^n X_j$ and $s_n^2=\operatorname{Var}(S_n)$.

Statement recorded in the supplied source.

If the sequence is $\phi$-mixing and $s_n^2\to\infty$, must \[\frac{S_n}{s_n}\ \xrightarrow{\ d\ }\ \mathcal N(0,1)\qquad(n\to\infty)\] hold? Here convergence is in distribution to the standard normal law. No mixing rate or positive lower bound on $s_n^2/n$ is assumed. [S2]

Short English statement

Does every centered stationary phi-mixing sequence with finite second moment and diverging partial-sum variance satisfy the variance-normalized central limit theorem?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10100012 (AMR-100-0012). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. C. Bradley, Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions, Probability Surveys 2 (2005), pp. 107--144, Section 2.3 (Ibragimov conjecture), and definitions (1.2), (2.2). https://arxiv.org/abs/math/0511078v1.