Weak but Not Strong Asymptotic Testability

Senhan Yao

Weak but Not Strong Asymptotic Testability

Statistics Theory 2026-07-29 v1

Authors: Senhan Yao

Abstract

We construct two fixed disjoint families $H_0,H_1$ of stationary ergodic binary process distributions for which a weakly asymptotically consistent test exists, but no strongly asymptotically consistent test exists. The construction combines a synchronizing binary suspension code, an independent i.i.d marker process, and countably many independent slowly switching two-state Markov chains. In particular, this disproves the asymptotic-consistency branch of a conjecture of Ryabko.

Cite

@article{arxiv.2607.26476,
  title  = {Weak but Not Strong Asymptotic Testability},
  author = {Senhan Yao},
  journal= {arXiv preprint arXiv:2607.26476},
  year   = {2026}
}