Uniform subsampling without replacement

Input. Let $n\ge 2$, let $(\mathcal{Z},\mathcal{F})$ be the declared measurable space of unit records, and fix observed records $z_1,\dots ,z_n\in \mathcal{Z}$ whose resampling unit satisfies Sampling units and the resampling-unit contract. Declare a sample sigma-algebra ${\mathcal{F}}_b$ on the ordered tuple space ${\mathcal{Z}}^b$, choose $1\le b<n$, and provide a statistic $T_b:({\mathcal{Z}}^b,{\mathcal{F}}_b)\to ({\mathbb{R}}^q,{\mathcal{B}}_q)$, where $q\ge 1$ is finite and ${\mathcal{B}}_q$ is the declared target sigma-algebra; "statistic" has the measurable-map meaning in Statistics and estimators.

Write $[n]:=\{1,\dots ,n\}$ for the indexing set; the finite subset law on these indices is established in Uniform laws on finite subset families.

Procedure. Draw $I$ uniformly from the finite family ${\mathcal{I}}_{n,b}:=\{J\subseteq [n]:\mathrm{\mid }J\mathrm{\mid }=b\}$, without replacement; list its indices in increasing order; and return the subsample $(z_i{)}_{i\in I}$ and statistic value $T_b((z_i{)}_{i\in I})$. The uniform law is well-defined by Uniform laws on finite subset families.

This is subsampling without replacement. It is different from drawing $b$ indices independently with replacement. The procedure defines a finite law only; validity requires a separate theorem for the data-generating design and asymptotic regime.