Finite samples of a declared arrangement: the CHSH estimator, its concentration and the limits of a finite-sample verdict

Adopt a declared spin correlation arrangement $(V\widehat{\otimes }V,\rho ,a_0,a_1,b_0,b_1)$ of Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models with its outcome laws ${\nu }_{ij}^{\rho }(\epsilon ,\delta )$, correlations $E_{ij}=E_{ij}({\nu }^{\rho })$ and CHSH functional $S=S({\nu }^{\rho })$. Let $n\ge 1$ and declare the following sampling model (the statistical model of clause (ii) of the preparation and apparatus postulate P7, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions): on a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ let $({\mathrm{\Sigma }}_{ijk},{\mathrm{T}}_{ijk}{)}_{\begin{array}{c}i,j\in \{0,1\}\\ k\in \{1,\dots ,n\}\end{array}}$ be $4n$ random elements with values in $\mathrm{\Sigma }\times \mathrm{\Sigma }$ that are mutually independent and have the declared laws $\mathbb{P}(({\mathrm{\Sigma }}_{ijk},{\mathrm{T}}_{ijk})=(\epsilon ,\delta ))={\nu }_{ij}^{\rho }(\epsilon ,\delta )\text{for every }i,j,k,\epsilon ,\delta$ (Pairwise and mutual independence of finite-valued random variables, Random elements and real random variables). Define the sample correlations and the sample CHSH functional ${\widehat{E}}_{ij}:=\frac{1}{n}{\sum }_{k=1}^n{\mathrm{\Sigma }}_{ijk}{\mathrm{T}}_{ijk},\widehat{S}:={\widehat{E}}_{00}+{\widehat{E}}_{01}+{\widehat{E}}_{10}-{\widehat{E}}_{11}\mathrm{.}$ Then:

(a) $\mathbb{E}[{\widehat{E}}_{ij}]=E_{ij}$ and $\mathrm{Var}({\widehat{E}}_{ij})\le 1\mathrm{/}n$ for all $(i,j)$;

(b) $\mathbb{E}[\widehat{S}]=S$, $\mathrm{Var}(\widehat{S})\le 4\mathrm{/}n$, and for every $\epsilon >0$ $\mathbb{P}(\mathrm{\mid }\widehat{S}-S\mathrm{\mid }\ge \epsilon )\le \frac{4}{n{\epsilon }^2};$

(c) threshold consequences: if the declared model satisfies $S\le 2-\epsilon$ for some $\epsilon >0$, then $\mathbb{P}(\widehat{S}>2)\le 4\mathrm{/}(n{\epsilon }^2)$; if the declared model is the singlet arrangement with $S=-2\sqrt{2}$ and $\epsilon :=2\sqrt{2}-2$, then $\mathbb{P}(\widehat{S}\ge -2)\le 4\mathrm{/}(n{\epsilon }^2)$. Moreover, for every declared acceptance criterion $C\subseteq (\mathrm{\Sigma }\times \mathrm{\Sigma }{)}^{4n}$ that rejects at least one outcome sequence $(x_{ijk})$ with ${\prod }_{i,j,k}{\nu }_{ij}^{\rho }(x_{ijk})>0$, the failure probability of the criterion is at least ${\prod }_{i,j,k}{\nu }_{ij}^{\rho }(x_{ijk})>0$;

(d) scope: the independence, the identical distribution and the tables ${\nu }^{\rho }$ are declared model assumptions of the sampling model, not consequences of P1-P6; clauses (a)-(c) assert no measured count, no p-value, no error rate, no power, no required sample size and no empirical conclusion.

• Finite-sample threshold crossings. At the displayed singlet settings, each of the four possible outcome pairs has strictly positive probability. A sequence with product signs $+1$ at the first three setting pairs and $-1$ at the last therefore has positive probability and gives $\widehat{S}=4\ge -2$. For a representable model with $S<2$, a crossing $\widehat{S}>2$ has positive probability only when its declared tables support at least one such sequence, as specified by clause (c); a deterministic table need not support one. The displayed Chebyshev upper bounds remain valid independently of that support condition. More generally, if two specified finite-sample models both assign positive probability to the same record, that record is possible under both and cannot deductively exclude either; a record assigned probability zero by one specified ideal model can exclude that model conditional on its sampling and apparatus assumptions. Thus this item makes no blanket claim that finite samples cannot distinguish all model pairs. No optimality, power or sample-size assertion is made.
• No inference is drawn. The clauses are probabilities inside the declared sampling model. They do not compute a p-value, do not attach a confidence level and do not assert anything about a real experiment; the preparation, detection and selection qualifications remain as stated in Measurement-model boundaries of the singlet-local comparison: what the no-go excludes and what it does not.