Central limit theorem under phi-mixing and finite second moments
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Does every strictly stationary centered φ-mixing sequence with finite second moments and divergent partial-sum variance satisfy a central limit theorem after normalization by its standard deviation?
Why it matters and what counts as progress
Why it matters: This asks whether mixing and finite variance alone suffice, beyond positive results requiring an extra variance condition.
What counts as progress: A proof under these exact hypotheses, or a sequence whose normalized partial sums fail to approach the standard normal law, even along a subsequence.
How the problem developed
- 1962 · Limit theorem [1]
Ildar Ibragimov Developed central limit theorems for stationary mixing processes under summability assumptions, establishing the classical setting for later questions about weaker φ-mixing hypotheses.
- October 2026 · Reported counterexample [2]
Tomasz Adamczewski Posted a stationary φ-mixing sequence with finite second moments for which normalized sums fail the proposed central limit behavior along a subsequence; the repository provides a formal proof artifact.
AI contribution
Lean-checked candidate against the φ-mixing CLT
GPT-6 Astra constructed a strictly stationary, centered φ-mixing sequence with finite second moments and divergent partial-sum variance. Along a subsequence, its variance-normalized sums converge to zero in probability rather than to a standard normal law.
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