Size-n bootstrap from the empirical law

Input. Let $n\ge 1$, let $(\mathcal{Z},\mathcal{F})$ be the declared measurable space of sampling units, and fix the observed ordered tuple $z=(z_1,\dots ,z_n)$. Let $P_n$ be its empirical probability measure from Empirical measure of observed units and The empirical measure is a probability measure. Declare a sample sigma-algebra ${\mathcal{F}}_n$ on the ordered tuple space ${\mathcal{Z}}^n$ and provide a statistic $T_n:({\mathcal{Z}}^n,{\mathcal{F}}_n)\to ({\mathbb{R}}^q,{\mathcal{B}}_q)$, 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.

Procedure. Draw indices $I_1,\dots ,I_n$ independently and uniformly from $[n]$, so repeated indices are allowed. Set $Z_j^{\ast }:=z_{I_j}$ for $j=1,\dots ,n$. Return the bootstrap tuple $(Z_1^{\ast },\dots ,Z_n^{\ast })$ and the replicate $T_n(Z_1^{\ast },\dots ,Z_n^{\ast })$. Generate further replicates by repeating the index draws with fresh randomness. These are resampling units only when the tuple entries satisfy the unit contract in Sampling units and the resampling-unit contract.

The conditional IID law of the draws is proved in Empirical-law bootstrap draws are conditionally IID ↗. This algorithm does not claim that its conditional law consistently approximates any sampling law; such validity requires separate hypotheses and a separate theorem.