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Conditional bootstrap delta method for Hadamard differentiable maps

Statement

Let $E, F$ be separable Banach spaces, $E_{0} \subseteq E$ a closed linear subspace, $θ \in E$ , and $D \subseteq E$ a neighbourhood of $θ$ . Let $t_{n} > 0$ decrease to zero, let $Z_{n} \in E_{0}$ be data-measurable and tight, and let $Z_{n}^{*} \in E_{0}$ be bootstrap random elements such that $θ + t_{n} Z_{n}$ and $θ + t_{n} (Z_{n} + Z_{n}^{*})$ lie in $D$ with probability tending to one. Supply weak-convergence metrics on probability laws on $E_{0}$ and $F$ as in Conditional weak convergence in probability, with measurable distances for the conditional laws below. Assume:

$L^{*} (Z_{n}^{*})$ converges conditionally weakly in probability to the fixed law of $G \in E_{0}$ , as in Conditional weak convergence in probability;
$Z_{n}^{*}$ is conditionally tight in probability: for each $ϵ, η > 0$ , there is a deterministic compact $K \subset E_{0}$ such that ${lim sup}_{n} \Pr {\Pr^{*} (Z_{n}^{*} \notin K) > ϵ} < η$ ;
$D$ is Borel and $Ψ : D \to F$ is Borel measurable and Hadamard differentiable at $θ$ , tangentially to $E_{0}$ , with continuous linear derivative $Ψ_{θ}^{'}$ as in Hadamard differentiability tangentially to a subspace.

Then

$L^{*} ⁣ (t_{n}^{- 1} {Ψ (θ + t_{n} (Z_{n} + Z_{n}^{*})) - Ψ (θ + t_{n} Z_{n})})$

converges conditionally weakly in probability to the law of $Ψ_{θ}^{'} (G)$ . Define the displayed bootstrap difference to be $0 \in F$ whenever either argument lies outside $D$ ; the assumed vanishing probability of this event makes this convention asymptotically immaterial.

Facts & Assumptions

Given: The spaces, sequences, conditional convergence, conditional tightness, domain and differentiability assumptions in the Statement.

[F1]

Hadamard differentiability tangentially to a subspace: Along every $t_{m} \to 0$ and $h_{m} \to h \in E_{0}$ , the difference quotient ${Ψ (θ + t_{m} h_{m}) - Ψ (θ)} / t_{m}$ converges to $Ψ_{θ}^{'} (h)$ , where $Ψ_{θ}^{'}$ is continuous and linear.

[F2]

Conditional weak convergence in probability: Conditional weak convergence means that a metric generating weak convergence between the conditional bootstrap law and the fixed limiting law tends to zero in probability.

[F3]

Portmanteau theorem: Weak convergence of Borel probabilities is equivalent to the closed-set upper bound ${lim sup}_{n} μ_{n} (C) \leq μ (C)$ for every closed set $C$ .

[F4]

Continuous mapping theorem: If $μ_{n} \Rightarrow μ$ and a measurable map $g$ is continuous $μ$ -almost everywhere, then the image laws satisfy $g_{*} μ_{n} \Rightarrow g_{*} μ$ .

[F5]

Subsequence characterization of convergence in probability: A scalar sequence tends to zero in probability exactly when every subsequence has a further subsequence tending to zero almost surely.

Proof

Given: The hypotheses and facts in the Statement and Facts & Assumptions.

1.1F1givenchoose

For every compact $K \subset E_{0}$ , $\sup_{h \in K} {∥ t_{n}^{- 1} {Ψ (θ + t_{n} h) - Ψ (θ)} - Ψ_{θ}^{'} (h) ∥}_{F} ⟶ 0$ . If this failed, some subsequence would have $h_{n} \in K$ with the displayed norm bounded below by a positive constant. Compactness gives a further subsequence $h_{n} \to h \in K$ ; [F1] applied to that subsequence makes the norm tend to zero, a contradiction.

2.1F1givenstep 1.1algebra

Write $T_{n}^{*}$ for the bootstrap difference with the stated convention, and $R_{n}^{*} = T_{n}^{*} - Ψ_{θ}^{'} (Z_{n}^{*})$ . Fix $δ, ρ, η > 0$ . Tightness gives a compact $K_{0}$ with ${lim sup}_{n} \Pr (Z_{n} \notin K_{0}) < η$ ; conditional tightness gives compact $K_{1}$ with ${lim sup}_{n} \Pr {\Pr^{*} (Z_{n}^{*} \notin K_{1}) > ρ / 2} < η$ . Because $D$ is a neighbourhood of $θ$ , the arguments associated to the bounded compact sets $K_{0}$ and $K_{0} + K_{1}$ lie in $D$ for all sufficiently large $n$ . Step 1.1 on these two compact sets and linearity show uniformly there that $∥ R_{n}^{*} ∥ < δ$ . Since $Z_{n}$ is fixed under conditioning, $\Pr^{*} (∥ R_{n}^{*} ∥ > δ) \leq 1_{{Z_{n} \notin K_{0}}} + \Pr^{*} (Z_{n}^{*} \notin K_{1})$ eventually. Thus the limit superior of $\Pr {\Pr^{*} (∥ R_{n}^{*} ∥ > δ) > ρ}$ is at most $2 η$ . Letting $η ↓ 0$ proves $\Pr^{*} (∥ R_{n}^{*} ∥ > δ) \to 0$ in probability for every $δ > 0$ .

3.1F2F3F4F5step 2.1∎

Start with any subsequence. The nonnegative scalar $\min (1, d_{w} (L^{*} (Z_{n}^{*}), L (G))) + \sum_{j \geq 1} 2^{- j} \Pr^{*} (∥ R_{n}^{*} ∥ > 1 / j)$ tends to zero in probability: truncate the sum, use step 2.1 for its finitely many terms, and bound the tail by its geometric sum. By [F5], extract a further subsequence along which this scalar tends to zero almost surely, so the input metric distance and every remainder probability do also. Fix an outcome in their common probability-one event. The continuous mapping theorem [F4] gives weak convergence of the conditional laws of $Ψ_{θ}^{'} (Z_{n}^{*})$ . For every closed $C \subset F$ and $δ > 0$ , $\Pr^{*} (T_{n}^{*} \in C) \leq \Pr^{*} (∥ R_{n}^{*} ∥ > δ) + \Pr^{*} (Ψ_{θ}^{'} (Z_{n}^{*}) \in C^{δ})$ , where $C^{δ} = {y : d (y, C) \leq δ}$ is closed. Apply [F3] and let $δ ↓ 0$ ; continuity from above of the fixed limiting probability gives the closed-set bound for $T_{n}^{*}$ . Thus [F3] gives weak convergence on this outcome, hence convergence to zero of the supplied output weak-metric distance. A final application of [F5] proves convergence of that distance in probability along the full sequence.

Remarks

This theorem is conditional on the bootstrap limit and tightness hypotheses. It does not prove those hypotheses for a particular resampling algorithm.

Depends on

Used by

Conditional K-sample block-bootstrap validity under a joint CLT Theorem

Dependency tree · two levels

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Sources

D. Pilavakis, E. Paparoditis, and T. Sapatinas, Moving block and tapered block bootstrap for functional time series with an application to the K-sample mean problem, Bernoulli 25 (2019), Remark 2.1 (standard reference, not scraped)