File size: 7,819 Bytes
180bc03 | 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.WeightedConditioned
namespace OAI
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
|