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Finite moment probability and rank split

Lax342547.MomentTails · concepts/Lax342547/MomentTails.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    Finite Markov bounds and the low-rank/high-rank decomposition preserve the actual sample masses and explicit channel rank exponent.

    Concept map
    15 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 finite_moment_markov proven

    2 rank_moment_split proven

    Lean source view on GitHub

    1import Lax342547.FiniteMoments
    2
    3/-!
    4---
    5title: Finite moment probability and rank split
    6type: lemma
    7---
    8Finite Markov bounds and the low-rank/high-rank decomposition preserve the actual sample masses and explicit channel rank exponent.
    9-/
    10
    11namespace Lax342547.MomentTails
    12
    13open Lax342547.RelativeEntropy Lax342547.RetainedImages
    14open scoped BigOperators
    15
    16axiom finite_moment_markov {C : Type} [Fintype C] (ν : C → ℝ) (f : C → ℝ)
    17 (a : ℝ) (n : ℕ) (hν : ∀ c, 0 ≤ ν c) (ha : 0 < a) :
    18 cellMass ν (fun c => a ≤ |f c|) ≤ (∑ c, ν c*|f c|^n)/a^n
    19
    20axiom rank_moment_split {Ω : Type} [Fintype Ω] (μ : Ω → ℝ)
    21 (rank : Ω → ℕ) (h R : ℕ) (hμ : Probability μ) :
    22 (∑ x, μ x/(2 : ℝ)^(h*rank x)) ≤
    23 cellMass μ (fun x => rank x < R)+1/(2 : ℝ)^(h*R)
    24
    25end Lax342547.MomentTails
    26
    Show ProofShow Proof
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    From Mathlib

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