Analytically tractable asymptotics for TV in the inhomogeneous product case

Derive a simple, analytically tractable asymptotic expression for the total variation distance TV(⊗_{i=1}^n P_i, ⊗_{i=1}^n Q_i) for sequences of probability measures (P_i) and (Q_i) that may vary with i (inhomogeneous case), analogous to the Chernoff-information-based asymptotic known for the homogeneous case.

Background

The introduction contrasts the well-understood asymptotics of TV(P{⊗ n}, Q{⊗ n}) for i.i.d. (homogeneous) products—governed by Chernoff information—with the lack of an analogous, simple formula for heterogeneous (non-identically distributed) products.

The paper’s main result provides a nonasymptotic lower bound comparing heterogeneous and homogenized products but does not supply a general asymptotic characterization for the inhomogeneous setting, which the authors explicitly note is not known.

References

In particular, whereas the total variation distance between homogeneous products is fairly well understood --- it is known Corollary 16.2 that asymptotically, $ \TV(P{\otimes n},Q{\otimes n}) \approx 1-\exp(-nC(P,Q)) $, where $C(P,Q) = -\inf_{\lambda\in[0,1]}\log\int(} P){1-\lambda}(} Q){\lambda} $ is the {\em Chernoff information} --- no such simple, analytically tractable asymptotic is known for the inhomogeneous case.

— A homogenization principle for total variation  (2604.03882 - Kontorovich, 4 Apr 2026) in Introduction, first paragraph