The Ibragimov–Iosifescu conjecture for φ-mixing sequences

Precise statement

Ibragimov conjectured that if $(X_n)$ is a strictly stationary, $\varphi$-mixing sequence with $$ \mathbb E X_0=0,\qquad \mathbb E X_0^2<\infty, $$ and $$ \sigma_n^2=\operatorname{Var}(S_n)\to\infty,\qquad S_n=\sum_{j=0}^{n-1}X_j, $$ then $$ \frac{S_n}{\sigma_n}\Rightarrow N(0,1). $$ GPT-6 Astra constructs a counterexample satisfying all these hypotheses for which, along a subsequence $n_j\to\infty$, $$ \frac{S_{n_j}}{\sigma_{n_j}}\to 0 $$ in probability. Hence the conjectured central limit theorem fails.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-6 Astra (pre-release)

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 $\operatorname{Var}(S_n)\to\infty$. Astra also proves that the resulting process remains genuinely $\varphi$-mixing by constructing a bounded causal inverse with square-summable tail variation.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

lean checked statement unaudited

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.

Correctness: supported · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-09-05

   Github

   Astra constructs a strictly stationary real process $$ X_t=\xi_t+K(\xi_{t-1},\xi_{t-2},\ldots), $$ where the innovations $\xi_t$ are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and $K$ is bounded, continuous, and odd. The process is $\varphi$-mixing, centered, square-integrable, and satisfies $$ \operatorname{Var}(S_n)\to\infty. $$ Nevertheless there are times $n_j\to\infty$ such that $$ \frac{S_{n_j}}{\sqrt{\operatorname{Var}(S_{n_j})}}\to0 $$ in probability. Therefore the normalized sums cannot converge in distribution to $N(0,1)$. The same example also rules out Iosifescu's stronger weak invariance-principle conjecture, since Brownian convergence would imply the CLT at time $1$.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• The source status is candidate and must not be represented as solved.
• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.