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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | import OAI.Probability.InvariantIsing.Pressure.RandomPressureBound
namespace OAI
/-! The complete in-probability, L1 and expectation clauses of Proposition gen:random. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology
namespace InvariantIsing
theorem random_pressure_limit
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
(hpub : PanchenkoTalagrandFieldPairInput)
{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
(eig : (N : ℕ) → Ω → Fin N → ℝ) (Y : ℕ → Ω → ℝ)
(heig : ∀ N, Measurable (eig N)) (hY : ∀ N, Measurable (Y N))
(H : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (H N)]
[∀ N, (H N).IsMulRightInvariant]
(hlaw : ∀ N, ConditionalFieldOrbitLaw P (fun ω => (eig N ω,fun _ => 0)) (Y N) (H N))
(ν : ProbabilityMeasure ℝ) (a b : ℝ)
(hcompact : IsCompact (ν : Measure ℝ).support)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(hweak : TendstoInMeasure P (fun k ω => LevyProkhorov.ofMeasure
(empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1) ω))) atTop
(fun _ => LevyProkhorov.ofMeasure ν))
(hexcess : TendstoInMeasure P (fun k ω => spectralExcess (eig (k+1) ω) a b)
atTop (fun _ => 0))
(hUI : UniformIntegrable (fun k => Y (k+1)) 1 P) :
TendstoInMeasure P (fun k => Y (k+1)) atTop
(fun _ => (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) ∧
Tendsto (fun k => eLpNorm (fun ω => Y (k+1) ω-
(variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) 1 P) atTop (𝓝 0) ∧
Tendsto (fun k => ∫ ω, Y (k+1) ω ∂P) atTop
(𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by
have hp := random_pressure_tendsto_in_measure hhaar hgauss hpub P eig Y heig hY H hlaw
ν a b hcompact hbound ha hb hweak hexcess
exact ⟨hp,pressure_L1_of_probability_and_uniformIntegrability P _ _ hUI hp⟩
end InvariantIsing
end
end OAI
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