Proposition: Finite-grid coverage and compact extension

Suppose a compact rectangle has a finite grid $\Gamma$ of $M$ points such that each point of the rectangle has a grid point within $h$ in each coordinate. At each grid point estimate the stationary and quench quantities with $N$ repetitions as in Specification spec:sampling.

With probability at least

all the two types of mean estimates have errors at most $r$. On this event,

A certified score-expectation bias $b$ for each of the two protocols adds $2b$. A calibration-failure probability $\eta$ adds $\eta$ to the total failure probability.

Proof. A score has range length $2v_n$. Hoeffding's inequality gives

Apply a union bound to $2M$ estimates. Their difference has error at most $2r$. Lemma lem:modulus extends the bound from the grid to the rectangle. Add the deterministic biases and calibration failure by the triangle inequality and a union bound Hoeffding.