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9b6e014 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.SharedConditionedPairLawFst
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
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
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma signedHeight_sub {d : ℕ} (e : Direction d) (x y : Lattice d) :
signedHeight e (x-y) = signedHeight e x-signedHeight e y := by
simp only [signedHeight,Pi.sub_apply]
split <;> omega
theorem shared_path_marginal_limits {d : ℕ} (ν : Measure (Row d))
[IsProbabilityMeasure ν] (hue : UniformElliptic ν) (e f : Direction d) (hef : e.1 ≠ f.1)
(htrans : DirectionallyTransient ν (realPosition (step e)))
(r : ℕ → ℝ) (hr : IsGaussianSequence (independentConditionedPairLaw ν (realPosition (step e)))
(commonIncrementProcess (realPosition (step e)) f 0) r) {T : ℝ} (hT : 0 ≤ T) :
let ℓ := realPosition (step e)
let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans)
let F := fun i => recordLinearPath ℓ f (recordMedianSlope ν ℓ hp f (r i)) (r i)
(fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) (r i)) T
∃ W : ProbabilityMeasure C(unitInterval,ℝ),
(∀ I : Finset unitInterval, (W : Measure C(unitInterval,ℝ)).map
(fun g : C(unitInterval,ℝ) => I.restrict g) =
gaussianPathFiniteLaw (T/(2*commonMeanWidth ν ℓ)) I) ∧
∀ x y : ℕ → Lattice d, (∀ i, signedHeight e (x i) = signedHeight e (y i)) →
letI : ∀ i, IsProbabilityMeasure (sharedConditionedPairLaw ν ℓ (x i) (y i)) := fun i =>
sharedConditionedPairLaw_probability ν ℓ (x i) (y i)
(ne_of_gt (sharedNoDropMass_pos ν hue ℓ (signed_direction_unit e) htrans (x i) (y i)))
TendstoInDistribution (fun i Z => F i (fun j => Z.1 j-x i))
atTop id (fun i => sharedConditionedPairLaw ν ℓ (x i) (y i)) W ∧
TendstoInDistribution (fun i Z => F i (fun j => Z.2 j-y i))
atTop id (fun i => sharedConditionedPairLaw ν ℓ (x i) (y i)) W := by
let ℓ := realPosition (step e)
let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans)
let F := fun i => recordLinearPath ℓ f (recordMedianSlope ν ℓ hp f (r i)) (r i)
(fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) (r i)) T
have hF (i : ℕ) : Measurable (F i) := measurable_recordLinearPath _ _ _ _ _ _
obtain ⟨W,hW,hlim⟩ := shared_first_zero_path_limit ν hue e f hef htrans r hr hT
have hfirst (x y : ℕ → Lattice d) (hxy : ∀ i, signedHeight e (x i) = signedHeight e (y i)) :
letI : ∀ i, IsProbabilityMeasure (sharedConditionedPairLaw ν ℓ (x i) (y i)) := fun i =>
sharedConditionedPairLaw_probability ν ℓ (x i) (y i)
(ne_of_gt (sharedNoDropMass_pos ν hue ℓ (signed_direction_unit e) htrans (x i) (y i)))
TendstoInDistribution (fun i Z => F i (fun j => Z.1 j-x i))
atTop id (fun i => sharedConditionedPairLaw ν ℓ (x i) (y i)) W := by
let : ∀ i, IsProbabilityMeasure (sharedConditionedPairLaw ν ℓ (x i) (y i)) := fun i =>
sharedConditionedPairLaw_probability ν ℓ (x i) (y i)
(ne_of_gt (sharedNoDropMass_pos ν hue ℓ (signed_direction_unit e) htrans (x i) (y i)))
let : ∀ i, IsProbabilityMeasure (sharedConditionedPairLaw ν ℓ 0 (y i-x i)) := fun i =>
sharedConditionedPairLaw_probability ν ℓ 0 (y i-x i)
(ne_of_gt (sharedNoDropMass_pos ν hue ℓ (signed_direction_unit e) htrans 0 (y i-x i)))
have hy : ∀ i, signedHeight e (y i-x i) = 0 := fun i => by rw [signedHeight_sub,← hxy i,sub_self]
have hm (i : ℕ) : (sharedConditionedPairLaw ν ℓ (x i) (y i)).map
(fun Z : Path d × Path d => ((fun j => Z.1 j-x i),(fun j => Z.2 j-x i))) =
sharedConditionedPairLaw ν ℓ 0 (y i-x i) := by
simpa only [add_zero,add_sub_cancel] using sharedConditionedPairLaw_translation ν ℓ (x i) 0 (y i-x i)
exact tendstoInDistribution_pullback _ _ _ (fun _ => by fun_prop) hm
(fun i Z => F i Z.1) (fun i => (hF i).comp measurable_fst) id W (hlim _ hy)
refine ⟨W,hW,fun x y hxy => ?_⟩
let : ∀ i, IsProbabilityMeasure (sharedConditionedPairLaw ν ℓ (x i) (y i)) := fun i =>
sharedConditionedPairLaw_probability ν ℓ (x i) (y i)
(ne_of_gt (sharedNoDropMass_pos ν hue ℓ (signed_direction_unit e) htrans (x i) (y i)))
let : ∀ i, IsProbabilityMeasure (sharedConditionedPairLaw ν ℓ (y i) (x i)) := fun i =>
sharedConditionedPairLaw_probability ν ℓ (y i) (x i)
(ne_of_gt (sharedNoDropMass_pos ν hue ℓ (signed_direction_unit e) htrans (y i) (x i)))
refine ⟨hfirst x y hxy,?_⟩
exact tendstoInDistribution_pullback _ _ (fun _ => Prod.swap) (fun _ => measurable_swap)
(fun i => sharedConditionedPairLaw_swap ν ℓ (x i) (y i))
(fun i Z => F i (fun j => Z.1 j-y i)) (fun i => (hF i).comp (by fun_prop)) id W
(hfirst y x (fun i => (hxy i).symm))
lemma joint_tight_of_marginal_limits {Ω Ω' E F : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω']
[MetricSpace E] [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]
[MetricSpace F] [CompleteSpace F] [MeasurableSpace F] [BorelSpace F] [SecondCountableTopology F]
(μ : ℕ → Measure Ω) [∀ i, IsProbabilityMeasure (μ i)]
(X : ℕ → Ω → E) (Y : ℕ → Ω → F) (hX : ∀ i, Measurable (X i)) (hY : ∀ i, Measurable (Y i))
(G : Ω' → E) (H : Ω' → F) (W : Measure Ω') [IsProbabilityMeasure W]
(h₁ : TendstoInDistribution X atTop G μ W) (h₂ : TendstoInDistribution Y atTop H μ W) :
IsTightMeasureSet (Set.range (fun i => (μ i).map (fun ω => (X i ω,Y i ω)))) := by
have hx := distribution_sequence_tight μ X G W h₁
have hy := distribution_sequence_tight μ Y H W h₂
apply IsTightMeasureSet.prodMk
· convert hx using 1
ext η
simp only [Set.mem_image,Set.mem_range,exists_exists_eq_and]
simp only [Measure.fst,Measure.map_map measurable_fst ((hX _).prodMk (hY _)),Function.comp_def]
· convert hy using 1
ext η
simp only [Set.mem_image,Set.mem_range,exists_exists_eq_and]
simp only [Measure.snd,Measure.map_map measurable_snd ((hX _).prodMk (hY _)),Function.comp_def]
end DirectionalTransience
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
end
end
end OAI
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