LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Lepski scale selection transfers fixed-scale covariance bounds

Statement

Let $Σ \neq 0$ , $σ_{0} > 0$ , $m \geq 1$ , $β > 1$ and $0 < σ_{\min} \leq σ_{0} \leq σ_{\max}$ . Use the finite grid and agreement rule of Adaptive truncated covariance estimator, with measurable matrix estimates on that grid. Put $L_{g} = \log_{2} (2 σ_{\max} / σ_{\min}), J = {0, \dots, ⌈ L_{g} ⌉ - 1}, \overset{ˉ}{j} = ⌈ \log_{2} (σ_{0} / σ_{\min}) ⌉, N_{+} = ⌈ L_{g} ⌉ - \overset{ˉ}{j} .$ On any event on which every $j \geq \overset{ˉ}{j}$ satisfies $∥ {\hat{Σ}}_{j} - Σ ∥ \leq 3 σ_{j} \sqrt{β / m}$ , the selected estimate satisfies $∥ {\hat{Σ}}^{*} - Σ ∥ < 18 σ_{0} \sqrt{β / m}$ . If the individual failures are at most $q_{j}$ , its success probability is at least $1 - \sum_{j \geq \overset{ˉ}{j}} q_{j}$ . For the prescribed same-sample estimator, when the hypotheses of Fixed-scale matrix-truncated covariance deviation under finite matrix variance hold at every required scale, its individual failure bounds $q_{j} = 5 p e^{- β}$ give success probability at least $1 - 5 p ⌈ L_{g} ⌉ e^{- β}$ . The shared-event bound below counts the common pilot failure once.

More sharply, suppose a common event $E$ has $P (E^{c}) \leq δ$ , and for each $j \geq \overset{ˉ}{j}$ an event $G_{j}$ has $P (G_{j}^{c}) \leq h e^{- β}$ , with the stated fixed-scale bound valid on $E \cap G_{j}$ . Then success is at least $1 - δ - h N_{+} e^{- β}$ . No independence between scales or the pilot and covariance estimates is required. The exact grid cardinality is $⌈ L_{g} ⌉$ , not the generally false unrounded bound $∣ J ∣ \leq L_{g}$ .

Facts & Assumptions

Given: The bracket, positive finite grid scales, matrix estimates and deterministic agreement rule in the statement, with the stated probability assumptions when invoked.

[F1]

The scale grid is $σ_{j} = 2^{j} σ_{\min}$ ; the agreement rule selects the smallest index whose estimate is within $6 σ_{k} \sqrt{β / m}$ of every larger-scale estimate. Adaptive truncated covariance estimator.

[F2]

The finite agreement rule has a valid index since its largest grid index satisfies every comparison vacuously. The adaptive truncated covariance procedure is well-defined on its declared domain.

[F3]

Under its trace-aware sample condition, the prescribed pilot in Fixed-scale matrix-truncated covariance deviation under finite matrix variance has fixed-scale failure probability at most $5 p e^{- β}$ for each scale $σ \geq σ_{0}$ .

Proof

1.1givenF1F2algebra

The grid condition is $0 \leq j < L_{g}$ , so it has exactly $⌈ L_{g} ⌉$ elements. The first scale at or above $σ_{0}$ is $\overset{ˉ}{j}$ , and $σ_{\overset{ˉ}{j}} < 2 σ_{0} \leq 2 σ_{\max}$ puts it in the grid. Thus exactly $N_{+}$ required scales are $j \geq \overset{ˉ}{j}$ , with $N_{+} \geq 1$ .

2.1givenF1step 1.1algebra

On the joint fixed-scale event, for every $k > \overset{ˉ}{j}$ the triangle inequality gives $∥ {\hat{Σ}}_{k} - {\hat{Σ}}_{\overset{ˉ}{j}} ∥ \leq 3 (σ_{k} + σ_{\overset{ˉ}{j}}) \sqrt{β / m} \leq 6 σ_{k} \sqrt{β / m}$ . Hence $\overset{ˉ}{j}$ is admissible and the selected $j^{*} \leq \overset{ˉ}{j}$ . If they differ, the selection rule gives $∥ {\hat{Σ}}_{j^{*}} - {\hat{Σ}}_{\overset{ˉ}{j}} ∥ \leq 6 σ_{\overset{ˉ}{j}} \sqrt{β / m}$ ; if equal this distance is zero. Adding the error at $\overset{ˉ}{j}$ yields $∥ {\hat{Σ}}^{*} - Σ ∥ \leq 9 σ_{\overset{ˉ}{j}} \sqrt{β / m} < 18 σ_{0} \sqrt{β / m}$ .

3.1F3givenstep 1.1step 2.1algebra∎

A finite union bound costs at most $\sum_{j \geq \overset{ˉ}{j}} q_{j}$ for individual failure bounds; under [F3], $q_{j} = 5 p e^{- β}$ gives the stated $5 p ⌈ L_{g} ⌉ e^{- β}$ bound. For a shared event $E$ , the complement of $E \cap ⋂_{j \geq \overset{ˉ}{j}} G_{j}$ has probability at most $δ + h N_{+} e^{- β}$ ; the common failure is counted once. Applying step 2.1 proves all probability assertions.

Depends on

Used by

Adaptive truncated covariance has a finite matrix-variance operator bound Theorem

Dependency tree · two levels

16 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

Stanislav Minsker and Xiaohan Wei, Estimation of the Covariance Structure of Heavy-Tailed Distributions, Theorem 2.1 and §4.3 (standard reference, not scraped)