Counterexamples below the logarithmic mixing threshold

Determine in which subregion of the parameter regime <gamma<2alpha, under mixing decay rho(n) approximately proportional to alpha(log n)^{-1} and moment condition E[X_0^2(log^+|X_0|)^gamma]<infinity, a counterexample to the central limit theorem can be constructed.

Background

The paper establishes the central limit theorem when the logarithmic moment exponent satisfies gammagreater than or equal to2alpha, while Bradley's construction yields failure in another parameter region. The authors identify the intervening range as unresolved and ask where counterexamples can actually be constructed.

This problem concerns the sharp boundary between validity and failure of the central limit theorem for stationary rho-mixing sequences with logarithmic mixing decay. Resolving it would determine the precise parameter threshold in the gap left between the paper's positive result and known counterexamples.

References

Since Corollary \ref{cor:clt:log:mixing} ensures that the CLT holds in the region \gamma \ge 2\alpha, a natural subsequent question is: in which subspace of \gamma < 2\alpha can one construct a counterexample to the CLT ($see Figure \ref{fig:parameter_regions} (A)$).

— Can we further improve the central limit theorem for stationary $ρ$-mixing sequences $?$  (2609.04638 - Matsui, 4 Sep 2026) in Remark 2.1(i), immediately following Corollary 2.2; Figure 1(A)