File size: 3,806 Bytes
f205175 | 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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import OAI.Probability.DilutedSpin.CompoundSuperposition
import OAI.Probability.DilutedSpin.InsertionStability
namespace OAI
section
namespace DilutedSpinGlass
open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal BigOperators
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
[MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
omit [NormedSpace ℝ E] in
lemma iid_energy_integrable (μ : Measure E) [IsProbabilityMeasure μ]
(hi : Integrable id μ) (n : ℕ) :
Integrable id (Measure.map (fun x : Fin n → E => ∑ i,x i) (Measure.pi (fun _ : Fin n => μ))) := by
rw [integrable_map_measure aestronglyMeasurable_id (by fun_prop)]
change Integrable (fun x : Fin n → E => ∑ i,x i) (Measure.pi (fun _ : Fin n => μ))
exact integrable_finsetSum _ (fun i _ => integrable_eval (μ := fun _ : Fin n => μ) (i := i) hi)
omit [NormedSpace ℝ E] in
lemma iid_energy_norm_le (μ : Measure E) [IsProbabilityMeasure μ]
(hi : Integrable id μ) (n : ℕ) :
(∫ x, ‖x‖ ∂Measure.map (fun x : Fin n → E => ∑ i,x i) (Measure.pi (fun _ : Fin n => μ))) ≤
n*(∫ x, ‖x‖ ∂μ) := by
rw [integral_map (by fun_prop) (by fun_prop)]
calc
_ ≤ ∫ x : Fin n → E, ∑ i, ‖x i‖ ∂Measure.pi (fun _ : Fin n => μ) :=
integral_mono (integrable_finsetSum _ (fun i _ => integrable_eval hi)).norm
(integrable_finsetSum _ (fun i _ => integrable_comp_eval hi.norm))
(fun x => norm_sum_le _ _)
_ = _ := by
rw [integral_finsetSum _ (fun i _ => integrable_comp_eval hi.norm)]
have he (i : Fin n) : (∫ x : Fin n → E, ‖x i‖ ∂Measure.pi (fun _ : Fin n => μ)) = ∫ x,‖x‖ ∂μ :=
integral_comp_eval (μ := fun _ : Fin n => μ) (i := i) (by fun_prop)
simp only [he,
Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
omit [NormedSpace ℝ E] in
lemma compoundPoisson_integrable_id (r : ℝ≥0) (μ : Measure E) [IsProbabilityMeasure μ]
(hi : Integrable id μ) : Integrable id (compoundPoisson r μ) := by
unfold compoundPoisson
apply integrable_sum_measure
· intro n
exact (iid_energy_integrable μ hi n).smul_measure ENNReal.ofReal_ne_top
· have hs := ((hasSum_integral_poissonMeasure (poisson_integrable_count r)).mul_right
(∫ x, ‖x‖ ∂μ)).summable
apply Summable.of_nonneg_of_le (fun n => integral_nonneg (fun _ => norm_nonneg _)) _ hs
intro n
rw [integral_smul_measure,ENNReal.toReal_ofReal (by positivity)]
change (Real.exp (-(r:ℝ))*(r:ℝ)^n/n.factorial)*_ ≤
((Real.exp (-(r:ℝ))*(r:ℝ)^n/n.factorial)*(n:ℝ))*(∫ x,‖x‖ ∂μ)
rw [mul_assoc]
exact mul_le_mul_of_nonneg_left (iid_energy_norm_le μ hi n) (by positivity)
omit [NormedSpace ℝ E] in
lemma compoundPoisson_norm_le (r : ℝ≥0) (μ : Measure E) [IsProbabilityMeasure μ]
(hi : Integrable id μ) :
(∫ x, ‖x‖ ∂compoundPoisson r μ) ≤ r*(∫ x, ‖x‖ ∂μ) := by
have hc : Integrable (fun x : E => ‖x‖) (compoundPoisson r μ) :=
(compoundPoisson_integrable_id r μ hi).norm
rw [compoundPoisson,integral_sum_measure hc]
have hs := (hasSum_integral_poissonMeasure (poisson_integrable_count r)).mul_right (∫ x,‖x‖ ∂μ)
rw [poisson_mean] at hs
rw [← hs.tsum_eq]
apply Summable.tsum_le_tsum
· intro n
rw [integral_smul_measure,ENNReal.toReal_ofReal (by positivity)]
change (Real.exp (-(r:ℝ))*(r:ℝ)^n/n.factorial)*_ ≤
((Real.exp (-(r:ℝ))*(r:ℝ)^n/n.factorial)*(n:ℝ))*(∫ x,‖x‖ ∂μ)
rw [mul_assoc]
exact mul_le_mul_of_nonneg_left (iid_energy_norm_le μ hi n) (by positivity)
· exact (hasSum_integral_measure hc).summable
· exact hs.summable
end DilutedSpinGlass
end
end OAI
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