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c4cdbbd | 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 46 47 48 49 | import OAI.Probability.InvariantIsing.Fields.FieldEndpointIntegrability
import OAI.Probability.InvariantIsing.Fields.FieldCanonicalVectorMoments
namespace OAI
/-! The biased Gaussian root followed by the full canonical recursion
has exactly the product endpoint law used in the terminal estimate. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
theorem fieldCanonicalVectorLaw_root (N : ℕ) (h : FieldStep) (b : Fin N → ℝ) :
fieldVectorTailEndpointKernel N (scalarFieldIncrements h) (scalarFieldIncrements_positive h) ∘ₘ
Measure.pi (fun i => gaussianReal (b i) (NNReal.mk (h.height 0) (h.nonneg 0))) =
fieldCanonicalVectorLaw N h b := by
rw [product_kernel_comp (fieldTailEndpointKernel (scalarFieldIncrements h)
(scalarFieldIncrements_positive h)) _
(fieldVectorTailEndpointKernel_product N _ _)]
rfl
lemma fieldCanonicalEndpointLaw_integrable_id (h : FieldStep) (b : ℝ) :
Integrable (fun y : ℝ => y) (fieldCanonicalEndpointLaw h b) := by
have hp := (fieldCanonicalEndpointLaw_mgf h b 1).1
have hn := (fieldCanonicalEndpointLaw_mgf h b (-1)).1
apply (hp.add hn).mono' measurable_id.aestronglyMeasurable
apply ae_of_all
intro y
rw [Real.norm_eq_abs]
simp only [one_mul, neg_one_mul, Pi.add_apply, id_eq]
by_cases hy : 0 ≤ y
· rw [abs_of_nonneg hy]
linarith [Real.add_one_le_exp y, Real.exp_pos (-y)]
· rw [abs_of_nonpos (le_of_not_ge hy)]
linarith [Real.add_one_le_exp (-y), Real.exp_pos y]
lemma fieldCanonicalVectorLaw_linear_integrable (N : ℕ) (h : FieldStep) (b : Fin N → ℝ)
(F : (Fin N → ℝ) → ℝ) (hF : Measurable F) (hG : HasLinearGrowth F) :
Integrable F (fieldCanonicalVectorLaw N h b) :=
integrable_linearGrowth_pi _ (fun i => fieldCanonicalEndpointLaw_integrable_id h (b i)) F hF hG
end InvariantIsing
end
end OAI
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