Manuscript Oai227-A@1 · Oai227-A.tex · lemma · 7f45f607

Lemma: Mixing-to-correlation implication, with its scale

Stage 1: finite identities and the five-stage audit › A mixing bound does not close these gates · lines 638–658 · proof included or linked

Oai227-A@1#lem:mixing-comparison, Oai227-A@1#eq:tv-comparison

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For fixed $n, J$ , define

G_{ℓ} (σ) = \frac{1}{n} i \sum f_{i} (σ) (P_{ℓ}^{J} f_{i}) (σ) .

Then $∥ G_{ℓ} ∥_{\infty} \leq 1$ , and

∣ B_{n, J} (s, t) - A_{n, J} (t) ∣ \leq 2 v_{n} d_{TV} (U_{n} P_{T_{n} s}^{J}, π) .

Proof. The semigroup sends functions bounded by one to functions bounded by one. The two autocorrelations are $v_{n}$ times the expectation of $G_{T_{n} t}$ under the two displayed laws. The total-variation expectation bound gives the result.

Displayed formulas in this passage

Oai227-A@1#eq:tv-comparison · line 645
$∣ B_{n, J} (s, t) - A_{n, J} (t) ∣ \leq 2 v_{n} d_{TV} (U_{n} P_{T_{n} s}^{J}, π) .$
unlabeled · line 641
$G_{ℓ} (σ) = \frac{1}{n} i \sum f_{i} (σ) (P_{ℓ}^{J} f_{i}) (σ) .$

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section: A mixing bound does not close these gates