Universality of the Airy limit for cointegration CCA statistics

Prove that the Airy_1 edge-limit theorem for the modified CCA-based cointegration statistic—obtained under Gaussian VAR(1) errors with specific de-trending and de-meaning (Procedure for modified cointegration)—extends to much wider generality, including non-Gaussian errors and broader preprocessing schemes, thereby establishing universality of the limiting Airy_1 law.

Background

For VAR(1) with Gaussian errors and a specific detrending/demeaning procedure, the largest CCA-based cointegration statistics converge to the Airy_1 point process, enabling precise critical values. This result was proved with assumptions that may be technical but useful for analysis.

The authors expect this limit to hold under much broader conditions; however, a proof covering those general settings is not yet available as of 2024.

References

We expect Theorem \ref{Theorem_Cointegration_Airy} to extend to much wider generality, but this was not proven as of 2024.

— Canonical Correlation Analysis: review (2411.15625 - Bykhovskaya et al., 23 Nov 2024) in Discussion following Theorem on the Airy limit for cointegration; footnote after Theorem “Cointegration_Airy” (Chapter 5)