| 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 |
| |
