LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Measurability of the finite-bootstrap CDF supremum

Statement

Let $n, d \geq 1$ , let $(X_{1}, \dots, X_{n})$ be a measurable $(R^{d})^{n}$ -valued random vector, and fix $a \in R^{d}$ . For each finite index tuple $ω = (i_{1}, \dots, i_{n}) \in I_{n} = {1, \dots, n}^{n}$ , define the projected centered bootstrap-mean value $R_{n, ω} = \sqrt{n} a^{T} ⁣ (\frac{1}{n} \sum_{ℓ = 1}^{n} X_{i_{ℓ}} - {\overset{ˉ}{X}}_{n}), {\overset{ˉ}{X}}_{n} = \frac{1}{n} \sum_{i = 1}^{n} X_{i} .$ The finite-index conditional bootstrap CDF is $F_{n}^{*} (t) = n^{- n} \sum_{ω \in I_{n}} 1 {R_{n, ω} \leq t} .$ For every fixed continuous cumulative distribution function $G : R \to [0, 1]$ , the supremum $D_{n} = \sup_{t \in R} ∣ F_{n}^{*} (t) - G (t) ∣$ is a measurable real random variable.

Facts & Assumptions

Given: A measurable sample vector, finite $n, d \geq 1$ , fixed $a$ , the finite-index bootstrap in the statement, and a fixed continuous CDF $G$ .

[F1]

The bootstrap index set has $n^{n}$ tuples, each with mass $n^{- n}$ , and the conditional law is their finite uniform average. IID empirical bootstrap sample and mean statistic The set $A^{B}$ of functions $B \to A$ between finite sets is finite, with $∣ A^{B} ∣ = ∣ A ∣^{∣ B ∣}$

[F2]

Finite coordinate sums and scalar multiples of measurable real-valued maps are measurable; measurable indicators of measurable events and finite lattice maxima are measurable. A measurable function between measurable spaces The Borel sigma-algebra of a topological space An indicator function is measurable exactly when its set is measurable Arithmetic and lattice operations preserve measurability whenever they are defined

[F3]

The CDF $G$ is nondecreasing, and by assumption it is continuous. Cumulative distribution function of a real random variable

[F4]

A continuous real-valued map is Borel measurable. A continuous map has Borel preimages of Borel sets

Proof

1.1F1F3givenalgebra

Fix an outcome and let $u_{1} < \dots < u_{m}$ be the distinct values among the finite list $(R_{n, ω})_{ω \in I_{n}}$ . Write $F_{n}^{*} (u_{j}^{-}) = n^{- n} # {η : R_{n, η} < u_{j}}$ . On $[u_{j}, u_{j + 1})$ , $F_{n}^{*}$ is constant at $F_{n}^{*} (u_{j})$ and $G$ is nondecreasing, so the absolute difference is bounded by the larger of its two endpoint errors, $∣ F_{n}^{*} (u_{j}) - G (u_{j}) ∣$ and $∣ F_{n}^{*} (u_{j + 1}^{-}) - G (u_{j + 1}) ∣$ . For $t < u_{1}$ , the error is at most $∣ F_{n}^{*} (u_{1}^{-}) - G (u_{1}) ∣$ ; for $t \geq u_{m}$ , it is at most $∣ F_{n}^{*} (u_{m}) - G (u_{m}) ∣$ . Each left-limit error is approached as $t ↑ u_{j}$ , and each right-value error occurs at $u_{j}$ . Therefore $D_{n} = \max_{ω \in I_{n}} \max {∣ F_{n}^{*} (R_{n, ω}) - G (R_{n, ω}) ∣, ∣ F_{n}^{*} (R_{n, ω}^{-}) - G (R_{n, ω}) ∣}$ , where $F_{n}^{*} (r^{-}) = n^{- n} # {η : R_{n, η} < r}$ .

2.1F1F2F3F4step 1.1algebra∎

For each fixed tuple $ω$ , $R_{n, ω}$ is a finite linear combination of measurable coordinates, hence measurable by [F2]. Thus each comparison event ${R_{n, η} \leq R_{n, ω}}$ or ${R_{n, η} < R_{n, ω}}$ is measurable by applying [F2] to the measurable difference and its Borel half-line. Consequently $F_{n}^{*} (R_{n, ω})$ and $F_{n}^{*} (R_{n, ω}^{-})$ are measurable finite sums; $G (R_{n, ω})$ is measurable by [F4]. The finite maximum in step 1.1 is measurable by [F2], so $D_{n}$ is measurable.

Depends on

Used by

Conditional IID bootstrap CLT for an empirical law Theorem

Dependency tree · two levels

41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

University of Washington STAT 581, Chapter 8: Bootstrap and Jackknife Estimation of Sampling Distributions, §2 (standard reference, not scraped)
Bradley Efron and Robert Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Behaviormetrika 12 (1985), §§1–2 (standard reference, not scraped)