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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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianEmpiricalWeak
namespace OAI
/-! Continuity of finite empirical spectral measures in their actual eigenvalues. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped Topology
namespace InvariantIsing
theorem continuous_empiricalSpectralLaw {N : ℕ} (hN : 0 < N) :
Continuous (fun eig : Fin N → ℝ => empiricalSpectralLaw hN eig) := by
apply ProbabilityMeasure.continuous_iff_forall_continuous_integral.mpr
intro f
simp only [empiricalSpectralLaw_integral]
exact continuous_const.mul (continuous_finsetSum _ fun i _ => f.continuous.comp (continuous_apply i))
theorem continuous_scaled_gaussianEmpirical {N m : ℕ} (hN : 0 < N) (c : ℝ) :
Continuous (fun z : EuclideanSpace ℝ (Fin N × Fin m) => LevyProkhorov.ofMeasure
(empiricalSpectralLaw hN (fun i => c*gaussianPatternEigenvalues z i))) := by
apply LevyProkhorov.continuous_ofMeasure_probabilityMeasure.comp
apply (continuous_empiricalSpectralLaw hN).comp
apply continuous_pi
intro i
exact continuous_const.mul ((continuous_apply i).comp (continuous_gaussianPatternEigenvalues N m))
end InvariantIsing
end
end OAI
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