Ibragimov–Iosifescu conjecture

The conjecture that a stationary, bounded sequence with c6-mixing dependence satisfies a central limit theorem whenever its mixing coefficients tend to zero.

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Central limit theorem: A limit theorem stating that suitably normalized sums converge in distribution to a normal law. This is the convergence the conjecture predicts for normalized sums.
Mixing coefficient: A numerical measure of dependence between separated parts of a stochastic process. The conjecture’s condition is stated through the decay of this family of quantities.
Ildar Ibragimov: A Soviet and Russian probabilist known for foundational results on limit theorems for dependent random variables. His work established central limit criteria for stationary sequences with mixing dependence.
Counterexample: An example that disproves a universal mathematical claim by satisfying its assumptions while violating its conclusion. A counterexample would settle the conjecture negatively while preserving boundedness and vanishing c6-mixing.
c6-mixing: A dependence condition measuring how well events in a future sigma-algebra can be predicted from a past sigma-algebra. The conjecture assumes these coefficients vanish, without requiring a summable rate.
Sigma-algebra: A collection of events closed under complements and countable unions, used to formalize available information. Mixing coefficients compare information represented by past and future sigma-algebras.
Vladimir Iosifescu: A Romanian probabilist whose research included limit theorems for dependent variables and stochastic processes. The conjecture bears his name alongside Ibragimov’s in this line of probability research.
Mixing rate: The speed at which a sequence of dependence coefficients approaches zero. The conjecture’s distinctive challenge is the absence of a required rate of decay.
Strong mixing: A dependence condition requiring events separated in time to become asymptotically independent in a uniform absolute-probability sense. It is a weaker mixing notion, so the conjecture tests the strength needed for a central limit theorem.
Weak convergence: Convergence of probability distributions tested against bounded continuous functions. The central limit theorem asserts this kind of convergence for normalized sums.