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The Ibragimov–Iosifescu conjecture for φ-mixing sequences

Ibragimov conjectured that if (Xn)(X_n) is a strictly stationary, φ\varphi-mixing sequence withEX0=0,EX02<∞, \mathbb E X_0=0,\qquad \mathbb E X_0^2<\infty, andσn2=Var⁡(Sn)→∞,Sn=∑j=0n−1Xj, \sigma_n^2=\operatorname{Var}(S_n)\to\infty,\qquad S_n=\sum_{j=0}^{n-1}X_j, thenSnσn⇒N(0,1). \frac{S_n}{\sigma_n}\Rightarrow N(0,1). GPT-6 Astra constructs a counterexample satisfying all these hypotheses for which, along a subsequence nj→∞n_j\to\infty,Snjσnj→0 \frac{S_{n_j}}{\sigma_{n_j}}\to 0 in probability. Hence the conjectured central limit theorem fails.

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Stationary stochastic processes
Posed by
I. A. Ibragimov
Year posed
1971
Years open
55y
Solved
2026-09-05
Model
GPT-6 Astra (pre-release)
Vendor
OpenAI
Collaborators
Tom Adamczewski
Verification
Lean-checked, statement unaudited
Publication
Announced
Significance
40 / 100
Disclosed cost
$261
Wikipedia
No dedicated article

What was actually shown

Astra constructs a strictly stationary real processXt=ξt+K(ξt−1,ξt−2,…), X_t=\xi_t+K(\xi_{t-1},\xi_{t-2},\ldots), where the innovations ξt\xi_t are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and KK is bounded, continuous, and odd. The process is φ\varphi-mixing, centered, square-integrable, and satisfiesVar⁡(Sn)→∞. \operatorname{Var}(S_n)\to\infty. Nevertheless there are times nj→∞n_j\to\infty such thatSnjVar⁡(Snj)→0 \frac{S_{n_j}}{\sqrt{\operatorname{Var}(S_{n_j})}}\to0 in probability. Therefore the normalized sums cannot converge in distribution to N(0,1)N(0,1).

The same example also rules out Iosifescu's stronger weak invariance-principle conjecture, since Brownian convergence would imply the CLT at time 11.

What the AI did

GPT-6 Astra autonomously constructed the counterexample and wrote its Lean proof. The run lasted about 12 hours and 1054 turns. The core idea is to build a stationary causal process from Gaussian-smoothed rare spikes plus a bounded nonlinear feedback correction. The feedback suppresses ordinary fluctuations along selected times, while extremely rare large spikes keep Var⁡(Sn)→∞\operatorname{Var}(S_n)\to\infty. Astra also proves that the resulting process remains genuinely φ\varphi-mixing by constructing a bounded causal inverse with square-summable tail variation.

Verification

Lean-checked, statement unaudited. Checked here on 6 September 2026 from a clone of tadamcz/phi-mixing-clt at 8e08498: 13,047 lines, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; the repository's recorded verifier accepted the disproof with the default kernel and only the three standard axioms. The statement, however, was produced by Epoch's AI-autoformalized 'wikipedia' run, not by a human-curated repository, and this site has not audited the formal definitions of strict stationarity and the φ-mixing coefficient against the sources beyond reading the docstring, which is careful about conventions. The repository's own audit finds no mismatch, but that audit is machine-written. No probabilist outside the run has read the 12,900-line construction. Hence Candidate.

Sources

Submitted by VibeGene on

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