| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.WeightedConditioned |
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|
| namespace OAI |
|
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| section |
| section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| lemma weak_limit_of_bounded_fresh_map {Ω E : Type*} [MeasurableSpace Ω] |
| [MetricSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (η : ℕ → Measure Ω) |
| [∀ i, IsProbabilityMeasure (η i)] (S : Ω → Ω) (hS : Measurable S) |
| (hmap : ∀ i, (η i).map S = μ) (C : ℝ≥0∞) (hC : C ≠ ⊤) |
| (hdom : ∀ i, η i ≤ C • μ) (F : ℕ → Ω → E) (hF : ∀ i, Measurable (F i)) |
| (W : ProbabilityMeasure E) |
| (hlim : Tendsto (fun i => ProbabilityMeasure.map (⟨μ,inferInstance⟩ : ProbabilityMeasure Ω) |
| (F i)) atTop (𝓝 W)) |
| (hclose : ∀ ε, 0 < ε → Tendsto (fun i => μ.real |
| {ω | ε ≤ dist (F i (S ω)) (F i ω)}) atTop (𝓝 0)) : |
| Tendsto (fun i => ProbabilityMeasure.map (⟨η i,inferInstance⟩ : ProbabilityMeasure Ω) |
| (F i)) atTop (𝓝 W) := by |
| apply weak_map_of_varying_close η (fun i => F i ∘ S) F |
| (fun i => (hF i).comp hS) hF W |
| · have he (i : ℕ) : ProbabilityMeasure.map (⟨η i,inferInstance⟩ : ProbabilityMeasure Ω) |
| (F i ∘ S) = |
| ProbabilityMeasure.map (⟨μ,inferInstance⟩ : ProbabilityMeasure Ω) (F i) := by |
| apply Subtype.ext |
| change (η i).map (F i ∘ S) = μ.map (F i) |
| rw [← Measure.map_map (hF i) hS,hmap] |
| simpa only [he] using hlim |
| · intro ε hε |
| have hb (i : ℕ) : (η i).real {ω | ε ≤ dist (F i (S ω)) (F i ω)} ≤ |
| C.toReal * μ.real {ω | ε ≤ dist (F i (S ω)) (F i ω)} := by |
| have hh := ENNReal.toReal_mono (ENNReal.mul_ne_top hC (measure_ne_top _ _)) (hdom i {ω | ε ≤ dist (F i (S ω)) (F i ω)}) |
| simpa only [Measure.smul_apply,smul_eq_mul,ENNReal.toReal_mul,Measure.real] using hh |
| exact squeeze_zero (fun _ => measureReal_nonneg) hb |
| (by simpa only [mul_zero] using (hclose ε hε).const_mul C.toReal) |
|
|
| lemma weightedConditioned_finite {d : ℕ} (ν : Measure (Row d)) (ℓ : Vector d) |
| [IsProbabilityMeasure (conditionedLaw ν ℓ)] (g : Environment d → ℝ≥0∞) |
| (hg : Measurable g) (hgle : ∀ ω, g ω ≤ 1) : |
| IsFiniteMeasure (weightedConditioned ν ℓ g) := |
| ⟨lt_of_le_of_lt (weightedConditioned_le ν ℓ g hg hgle Set.univ) (measure_lt_top _ _)⟩ |
|
|
| lemma normalized_weightedConditioned_highSuffix {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (htrans : DirectionallyTransient ν ℓ) (J : ℝ) (hJ : 0 ≤ J) |
| (g : Environment d → ℝ≥0∞) |
| (hg : @Measurable _ _ (rowSigma {z | dot (realPosition z) ℓ ≤ J}) _ g) |
| (hgle : ∀ ω, g ω ≤ 1) (hpos : weightedConditioned ν ℓ g Set.univ ≠ 0) : |
| (normalizedMeasure (weightedConditioned ν ℓ g)).map (highSuffix ℓ J) = conditionedLaw ν ℓ := by |
| let : IsProbabilityMeasure (conditionedLaw ν ℓ) := conditionedLaw_probability ν ℓ |
| (ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans)) |
| let : IsFiniteMeasure (weightedConditioned ν ℓ g) := |
| weightedConditioned_finite ν ℓ g (hg.mono (rowSigma_le _) le_rfl) hgle |
| rw [normalizedMeasure,Measure.map_smul _ (measurable_highSuffix ℓ J hJ).aemeasurable, |
| weightedConditioned_highSuffix ν ℓ htrans J hJ g hg hgle, |
| smul_smul,ENNReal.inv_mul_cancel hpos (measure_ne_top _ _),one_smul] |
|
|
| lemma normalized_weightedConditioned_le {d : ℕ} (ν : Measure (Row d)) (ℓ : Vector d) |
| (g : Environment d → ℝ≥0∞) (hg : Measurable g) (hgle : ∀ ω, g ω ≤ 1) |
| (c : ℝ≥0∞) (hc : c ≤ weightedConditioned ν ℓ g Set.univ) : |
| normalizedMeasure (weightedConditioned ν ℓ g) ≤ c⁻¹ • conditionedLaw ν ℓ := by |
| intro A |
| change (weightedConditioned ν ℓ g Set.univ)⁻¹ * weightedConditioned ν ℓ g A ≤ c⁻¹ * conditionedLaw ν ℓ A |
| exact mul_le_mul' (ENNReal.inv_le_inv.mpr hc) (weightedConditioned_le ν ℓ g hg hgle A) |
| |
| end DirectionalTransience |
| |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| |
| end |
| end |
| |
| end OAI |
| |
