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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 46 47 48 49 50 51 | import OAI.Probability.InvariantIsing.Gaussian.GaussianColumnResolventContinuity
import OAI.Probability.InvariantIsing.Spectral.PositiveResolventFluctuation
namespace OAI
/-! Integrability of the actual leave-one-column quadratic deviation. -/
noncomputable section
open MeasureTheory ProbabilityTheory Matrix
namespace InvariantIsing
def gaussianGramColumnDeviation {N m : ℕ} (t : ℝ) (z : EuclideanSpace ℝ (Fin N × Fin m))
(j : Fin m) : ℝ := |gaussianGramColumnQuadratic t z j-(gaussianGramLeaveResolvent t z j).trace/N|
lemma continuous_gaussianGramColumnDeviation {N m : ℕ} {t : ℝ} (ht : 0 < t) (j : Fin m) :
Continuous (gaussianGramColumnDeviation (N := N) t · j) :=
((continuous_gaussianGramColumnQuadratic ht j).sub
((continuous_gaussianGramLeaveResolvent ht j).matrix_trace.div_const _)).abs
lemma gaussianGramColumnDeviation_bound {N m : ℕ} (hN : 0 < N) {t : ℝ} (ht : 0 < t)
(g : Fin m → Fin N → ℝ) (j : Fin m) :
gaussianGramColumnDeviation t (gaussianColumnArray g) j ≤
(1/((N : ℝ)*t))*‖WithLp.toLp 2 (g j)‖^2+1/t := by
unfold gaussianGramColumnDeviation
rw [gaussianGramColumnQuadratic_eq]
exact positiveResolvent_normalized_abs_bound hN ht
(gaussianGramLeaveOneOut_posSemidef (gaussianColumnArray g) j) _
lemma gaussianGramColumnDeviation_integrable {N m : ℕ} (hN : 0 < N) {t : ℝ} (ht : 0 < t)
(j : Fin m) : Integrable (fun g : Fin m → Fin N → ℝ =>
gaussianGramColumnDeviation t (gaussianColumnArray g) j)
(Measure.pi (fun _ : Fin m => Measure.pi (fun _ : Fin N => gaussianReal 0 1))) := by
have hi := (measurePreserving_eval
(fun _ : Fin m => Measure.pi (fun _ : Fin N => gaussianReal 0 1)) j).hasLaw.integrable_comp
(gaussianCoordinateVector_sq_integrable (id : Fin N → Fin N))
have hm : AEStronglyMeasurable (fun g : Fin m → Fin N → ℝ =>
gaussianGramColumnDeviation t (gaussianColumnArray g) j)
(Measure.pi (fun _ : Fin m => Measure.pi (fun _ : Fin N => gaussianReal 0 1))) :=
((continuous_gaussianGramColumnDeviation (N := N) ht j).comp
(continuous_gaussianColumnArray (N := N) (m := m))).aestronglyMeasurable
apply ((hi.const_mul (1/((N : ℝ)*t))).add (integrable_const (1/t))).mono' hm
apply ae_of_all
intro g
rw [Real.norm_eq_abs,abs_of_nonneg (show 0 ≤ gaussianGramColumnDeviation t (gaussianColumnArray g) j from abs_nonneg _)]
exact gaussianGramColumnDeviation_bound hN ht g j
end InvariantIsing
end
end OAI
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