| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.AlignedRecordPrefix |
|
|
| 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 |
| namespace DirectionalTransience |
|
|
| lemma block_transfer_bound {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] |
| (F B : ℕ → Ω → ℝ) {q : ℝ} (hq : 0 ≤ q) {H b M : ℕ} (hM : H/b ≤ M) |
| {a : ℝ} |
| (htransfer : q*μ.real (HeightOscillation F (HeightBlockRelation H b) (2*a)) ≤ |
| μ.real {ω | ∃ s h, (s,h) ∈ HeightBlockRelation H b ∧ |
| 2*a < |B h ω-B s ω|+|F s ω-B s ω|}) : |
| q*μ.real (HeightGapEvent F B H (6*a)) ≤ |
| (q+1)*(μ.real (HeightOscillation B (HeightBlockRelation H b) a)+ |
| μ.real (BlockDecoration F B M b a)) := by |
| have hs : {ω | ∃ s h, (s,h) ∈ HeightBlockRelation H b ∧ |
| 2*a < |B h ω-B s ω|+|F s ω-B s ω|} ⊆ |
| HeightOscillation B (HeightBlockRelation H b) a ∪ BlockDecoration F B M b a := by |
| rintro ω ⟨s,h,hsj,hbig⟩ |
| by_cases hb : a < |B h ω-B s ω| |
| · exact Or.inl ⟨s,h,hsj,hb⟩ |
| · apply Or.inr |
| refine ⟨h/b,(Nat.div_le_div_right hsj.1).trans hM,?_⟩ |
| rw [← hsj.2] |
| have := le_of_not_gt hb |
| linarith |
| have ht := htransfer.trans ((measureReal_mono hs (measure_ne_top _ _)).trans |
| (measureReal_union_le (μ := μ) _ _)) |
| have hg := (measureReal_mono (heightGap_subset F B hM (a := a)) (measure_ne_top _ _)).trans |
| ((measureReal_union_le (μ := μ) _ _).trans |
| (add_le_add (measureReal_union_le (μ := μ) _ _) le_rfl)) |
| have hmul := mul_le_mul_of_nonneg_left hg hq |
| nlinarith |
|
|
| lemma blockDecoration_finite_union {Ω : Type*} (F B : ℕ → Ω → ℝ) (M b : ℕ) (z : ℝ) : |
| BlockDecoration F B M b z = ⋃ j ∈ Finset.range (M+1), {ω | z < |F (j*b) ω-B (j*b) ω|} := by |
| ext ω |
| simp only [BlockDecoration,Set.mem_ofPred_eq,Set.mem_iUnion,Finset.mem_range,Nat.lt_succ_iff,exists_prop] |
|
|
| lemma blockDecoration_small {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] |
| (F B : ℕ → Ω → ℝ) (r : ℕ → ℝ) (hr : Tendsto r atTop atTop) |
| (hdec : ∀ h : ℕ → ℕ, TendstoInMeasure μ (fun i ω => (F (h i) ω-B (h i) ω)/r i) atTop 0) |
| (M : ℕ) (b : ℕ → ℕ) {a : ℝ} (ha : 0 < a) : |
| Tendsto (fun i => μ.real (BlockDecoration F B M (b i) (a*r i))) atTop (𝓝 0) := by |
| have hj (j : ℕ) : Tendsto (fun i => μ.real {ω | a*r i < |F (j*b i) ω-B (j*b i) ω|}) atTop (𝓝 0) := by |
| have ht := tendstoInMeasure_iff_measureReal_norm.mp (hdec (fun i => j*b i)) a ha |
| simp only [Pi.zero_apply,sub_zero,Real.norm_eq_abs] at ht |
| apply squeeze_zero' (Eventually.of_forall fun _ => measureReal_nonneg) _ ht |
| filter_upwards [hr.eventually (eventually_gt_atTop (0:ℝ))] with i hri |
| apply measureReal_mono _ (measure_ne_top _ _) |
| intro ω hω |
| simp only [Set.mem_ofPred_eq,abs_div,abs_of_pos hri] |
| exact ((lt_div_iff₀ hri).2 hω).le |
| have ht := tendsto_finsetSum (Finset.range (M+1)) (fun j _ => hj j) |
| simp only [Finset.sum_const_zero] at ht |
| apply squeeze_zero (fun _ => measureReal_nonneg) _ ht |
| intro i |
| rw [blockDecoration_finite_union] |
| exact measureReal_biUnion_finset_le _ _ |
| |
| lemma renewal_height_block_subset {Ω : Type*} (X : ℕ → Ω → ℝ) (L : ℕ → Ω → ℕ) |
| {n T c δ z : ℝ} (hn : 1 ≤ n) (hT : 0 ≤ T) (hc : 1 ≤ c) (hδ : 0 ≤ δ) (hδn : 4 ≤ δ*n) : |
| HeightOscillation (fun h ω => realPartialSum (fun k => X k ω) |
| (renewalCount (fun k => L k ω) h)) (HeightBlockRelation (⌊T*n⌋₊+1) ⌊(δ/4)*n⌋₊) z ⊆ |
| {ω | ∃ h : ℕ, (h:ℝ) ≤ (T+1)*n ∧ |
| δ/8 < |(renewalCount (fun j => L j ω) h : ℝ)/n-(h:ℝ)/(c*n)|} ∪ |
| GridOscillation X ⌊(T+1)*n⌋₊ ⌊δ*n⌋₊ z := by |
| rintro ω ⟨s,h,hsj,hbad⟩ |
| by_cases hclock : ω ∈ {ω | ∃ h : ℕ, (h:ℝ) ≤ (T+1)*n ∧ |
| δ/8 < |(renewalCount (fun j => L j ω) h : ℝ)/n-(h:ℝ)/(c*n)|} |
| · exact Or.inl hclock |
| apply Or.inr |
| let C := renewalCount (fun k => L k ω) |
| have hn0 : 0 < n := by linarith |
| have hc0 : 0 < c := by linarith |
| have hs : s ≤ h := heightBlockRelation_order _ _ _ hsj |
| have hH : (h:ℝ) ≤ (T+1)*n := by |
| have := (show (h:ℝ) ≤ (⌊T*n⌋₊:ℝ)+1 by exact_mod_cast hsj.1) |
| have hf := Nat.floor_le (mul_nonneg hT hn0.le) |
| nlinarith |
| have hsH := (Nat.cast_le.mpr hs).trans hH |
| have hb : (h:ℝ)-(s:ℝ) ≤ (δ/4)*n := by |
| have hrem : h-s = h % ⌊(δ/4)*n⌋₊ := by |
| have he : s = h / ⌊(δ/4)*n⌋₊ * ⌊(δ/4)*n⌋₊ := hsj.2 |
| have heq : h % ⌊(δ/4)*n⌋₊ + h / ⌊(δ/4)*n⌋₊ * ⌊(δ/4)*n⌋₊ = h := by |
| simpa only [Nat.mul_comm] using Nat.mod_add_div h ⌊(δ/4)*n⌋₊ |
| omega |
| have hb0 : 0 < ⌊(δ/4)*n⌋₊ := by |
| apply Nat.succ_le_iff.mp |
| apply Nat.le_floor |
| norm_num |
| linarith |
| have hh : h-s ≤ ⌊(δ/4)*n⌋₊ := by rw [hrem]; exact (Nat.mod_lt _ hb0).le |
| have hh' : (h:ℝ)-(s:ℝ) ≤ (⌊(δ/4)*n⌋₊:ℝ) := by |
| rw [← Nat.cast_sub hs] |
| exact_mod_cast hh |
| exact hh'.trans (Nat.floor_le (mul_nonneg (by positivity) hn0.le)) |
| have hclock' (j : ℕ) (hj : (j:ℝ) ≤ (T+1)*n) : |(C j:ℝ)-(j:ℝ)/c| ≤ δ*n/8 := by |
| have he : |(C j:ℝ)/n-(j:ℝ)/(c*n)| ≤ δ/8 := |
| le_of_not_gt (fun hh => hclock ⟨j,hj,hh⟩) |
| have heq : ((C j:ℝ)/n-(j:ℝ)/(c*n))*n = (C j:ℝ)-(j:ℝ)/c := by field_simp |
| have he' := mul_le_mul_of_nonneg_right he hn0.le |
| have he'' : |((C j:ℝ)/n-(j:ℝ)/(c*n))*n| ≤ δ/8*n := by |
| simpa only [abs_mul,abs_of_pos hn0] using he' |
| rw [heq] at he'' |
| convert he'' using 1; ring |
| have hch := (abs_le.mp (hclock' h hH)).2 |
| have hcs := (abs_le.mp (hclock' s hsH)).1 |
| have hdiv : ((h:ℝ)-(s:ℝ))/c ≤ (h:ℝ)-(s:ℝ) := div_le_self (sub_nonneg.mpr (Nat.cast_le.mpr hs)) hc |
| have hd : (C h:ℝ)-(C s:ℝ) ≤ δ*n := by |
| have he : (h:ℝ)/c-(s:ℝ)/c = ((h:ℝ)-(s:ℝ))/c := (sub_div _ _ _).symm |
| have hδn := mul_nonneg hδ hn0.le |
| linarith |
| have hC := renewalCount_monotone (fun k => L k ω) hs |
| have hdiff : C h-C s ≤ ⌊δ*n⌋₊ := by |
| apply Nat.le_floor |
| rw [Nat.cast_sub hC] |
| exact hd |
| refine ⟨C s,?_,C h,?_,hC,by omega,hbad⟩ |
| · exact (renewalCount_le _ _).trans (Nat.le_floor hsH) |
| · exact (renewalCount_le _ _).trans (Nat.le_floor hH) |
|
|
| lemma gaussian_renewal_block_modulus {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → Ω → ℝ) |
| (hX : ∀ k, Measurable (X k)) (hind : iIndepFun X μ) |
| (hident : ∀ k, IdentDistrib (X k) (X 0) μ μ) |
| (hsym : IdentDistrib (X 0) (fun ω => -X 0 ω) μ μ) |
| (hI : Integrable (X 0) μ) (hne : 0 < μ {ω | X 0 ω ≠ 0}) |
| (L : ℕ → Ω → ℕ) (hL : ∀ k, Measurable (L k)) |
| (hpos : ∀ᵐ ω ∂μ, ∀ k, 0 < L k ω) {c : ℝ} (hc : 1 ≤ c) |
| (hmean : ∀ᵐ ω ∂μ, Tendsto (fun k : ℕ => (renewalSum (fun j => L j ω) k : ℝ)/k) |
| atTop (𝓝 c)) (r : ℕ → ℝ) (hr : IsGaussianSequence μ (X 0) r) |
| {T a β : ℝ} (hT : 0 ≤ T) (ha : 0 < a) (hβ : 0 < β) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ᶠ i in atTop, |
| μ.real (HeightOscillation (fun h ω => realPartialSum (fun k => X k ω) |
| (renewalCount (fun k => L k ω) h)) |
| (HeightBlockRelation (⌊T*fluctuationScale μ (X 0) (r i)⌋₊+1) |
| ⌊δ*fluctuationScale μ (X 0) (r i)⌋₊) (a*r i)) ≤ β := by |
| let n := fun i => fluctuationScale μ (X 0) (r i) |
| have hn : Tendsto n atTop atTop := (fluctuationScale_tendsto μ (X 0) (hX 0) hI hne).comp hr.1 |
| obtain ⟨δ,hδ,hgrid⟩ := gaussian_grid_modulus μ X hX hind hident hsym hI hne r hr |
| (by linarith : 0 ≤ T+1) ha (by linarith : 0 < β/2) |
| let A := fun i => {ω | ∃ h : ℕ, (h:ℝ) ≤ (T+1)*n i ∧ |
| δ/8 < |(renewalCount (fun j => L j ω) h : ℝ)/n i-(h:ℝ)/(c*n i)|} |
| have hclock : Tendsto (fun i => μ (A i)) atTop (𝓝 0) := |
| renewalCount_uniform_in_probability μ L hL hpos (by linarith) hmean n hn |
| (by linarith) (by positivity) |
| have hclockR : Tendsto (fun i => μ.real (A i)) atTop (𝓝 0) := by |
| simpa only [ENNReal.toReal_zero,Function.comp_def,Measure.real] using (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp hclock |
| have hevent := (tendsto_order.mp hclockR).2 (β/2) (by linarith) |
| refine ⟨δ/4,by positivity,?_⟩ |
| filter_upwards [hgrid,hevent,hn.eventually (eventually_ge_atTop (1:ℝ)), |
| (hn.const_mul_atTop hδ).eventually (eventually_ge_atTop (4:ℝ))] with i hg ha hni hdni |
| have hs := renewal_height_block_subset X L hni hT hc hδ.le hdni (z := a*r i) |
| have hh := (measureReal_mono hs (measure_ne_top _ _)).trans (measureReal_union_le (μ := μ) _ _) |
| change μ.real (A i) < β/2 at ha |
| change μ.real (GridOscillation X ⌊(T+1)*n i⌋₊ ⌊δ*n i⌋₊ (a*r i)) ≤ β/2 at hg |
| exact hh.trans (by linarith) |
|
|
| theorem heightGap_small_of_transfer {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (F B : ℕ → Ω → ℝ) |
| (r n : ℕ → ℝ) (hr : Tendsto r atTop atTop) (hn : Tendsto n atTop atTop) |
| {q T : ℝ} (hq : 0 < q) (hT : 0 ≤ T) |
| (htransfer : ∀ J : Set (ℕ × ℕ), (∀ sr ∈ J, sr.1 ≤ sr.2) → ∀ z, 0 ≤ z → |
| q*μ.real (HeightOscillation F J z) ≤ |
| μ.real {ω | ∃ s h, (s,h) ∈ J ∧ z < |B h ω-B s ω|+|F s ω-B s ω|}) |
| (hdec : ∀ h : ℕ → ℕ, TendstoInMeasure μ (fun i ω => (F (h i) ω-B (h i) ω)/r i) atTop 0) |
| (hmod : ∀ a β : ℝ, 0 < a → 0 < β → ∃ δ : ℝ, 0 < δ ∧ ∀ᶠ i in atTop, |
| μ.real (HeightOscillation B (HeightBlockRelation (⌊T*n i⌋₊+1) ⌊δ*n i⌋₊) (a*r i)) ≤ β) |
| {ε : ℝ} (hε : 0 < ε) : |
| Tendsto (fun i => μ.real (HeightGapEvent F B (⌊T*n i⌋₊+1) (ε*r i))) atTop (𝓝 0) := by |
| apply tendsto_order.2 |
| constructor |
| · intro a ha |
| exact Eventually.of_forall fun _ => ha.trans_le measureReal_nonneg |
| · intro β hβ |
| let a := ε/6 |
| let b := β*q/(4*(q+1)) |
| have ha : 0 < a := by dsimp [a]; positivity |
| have hb : 0 < b := by dsimp [b]; positivity |
| obtain ⟨δ,hδ,hm⟩ := hmod a b ha hb |
| obtain ⟨M,hM⟩ := exists_nat_gt ((T+1)/δ) |
| have hM' : T+1 < (M:ℝ)*δ := (div_lt_iff₀ hδ).mp hM |
| have hcovered := eventually_floor_horizon_covered n hn (by linarith : 0 ≤ T+1) hδ M hM' |
| have hd := blockDecoration_small μ F B r hr hdec M (fun i => ⌊δ*n i⌋₊) ha |
| have hdevent := (tendsto_order.mp hd).2 b hb |
| filter_upwards [hm,hcovered,hdevent,hr.eventually (eventually_gt_atTop (0:ℝ)), |
| hn.eventually (eventually_ge_atTop (1:ℝ))] with i hmi hci hdi hri hni |
| have hH : ⌊T*n i⌋₊+1 ≤ ⌊(T+1)*n i⌋₊ := by |
| apply Nat.le_floor |
| push_cast |
| have hf := Nat.floor_le (mul_nonneg hT (by linarith : 0 ≤ n i)) |
| nlinarith |
| have hdiv : (⌊T*n i⌋₊+1)/⌊δ*n i⌋₊ ≤ M := by |
| exact ((Nat.div_lt_iff_lt_mul hci.1).mpr (hH.trans_lt hci.2)).le |
| have hz : 0 ≤ 2*(a*r i) := by positivity |
| have ht := block_transfer_bound μ F B hq.le hdiv |
| (htransfer _ (heightBlockRelation_order _ _) (2*(a*r i)) hz) |
| have he : 6*(a*r i) = ε*r i := by dsimp [a]; ring |
| rw [he] at ht |
| have hfin : q*μ.real (HeightGapEvent F B (⌊T*n i⌋₊+1) (ε*r i)) < q*β := by |
| have hbq : (q+1)*(2*b) = q*β/2 := by dsimp [b]; field_simp; ring |
| have := mul_lt_mul_of_pos_left (add_lt_add_of_le_of_lt hmi hdi) (by linarith : 0 < q+1) |
| rw [← two_mul,hbq] at this |
| nlinarith |
| exact (mul_lt_mul_iff_of_pos_left hq).mp hfin |
|
|
| theorem transverse_firstHit_uniform_error {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 ε : ℝ} (hT : 0 ≤ T) (hε : 0 < ε) : |
| Tendsto (fun i => (independentConditionedPairLaw ν (realPosition (step e))).real |
| (HeightGapEvent (firstHitPairGap (realPosition (step e)) f) |
| (commonBoundaryGap (realPosition (step e)) f) |
| (⌊T*fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) (r i)⌋₊+1) (ε*r i))) atTop (𝓝 0) := by |
| let ℓ := realPosition (step e) |
| let μ := independentConditionedPairLaw ν ℓ |
| have hp := noDrop_positive_of_directionallyTransient ν ℓ htrans |
| let : IsProbabilityMeasure μ := independentConditionedPairLaw_probability ν ℓ (ne_of_gt hp) |
| have hi := independent_commonWordIncrement_integrable ν hue ℓ (signed_direction_unit e) |
| htrans (signedHeight e) (signedHeight_projection e) (signedHeight_step_le e) f |
| have hne := independent_commonWordIncrement_nonzero ν hue e f hef htrans |
| have hn := (fluctuationScale_tendsto μ (commonIncrementProcess ℓ f 0) |
| (measurable_commonIncrementProcess ℓ f 0) hi hne).comp hr.1 |
| apply heightGap_small_of_transfer μ (firstHitPairGap ℓ f) (commonBoundaryGap ℓ f) |
| r _ hr.1 hn (q := (annealedLaw ν (NoDrop ℓ 0)).toReal^2) |
| (sq_pos_of_pos (ENNReal.toReal_pos (ne_of_gt hp) (measure_ne_top _ _))) hT _ _ _ hε |
| · intro J hJ z hz |
| have hh := independent_firstHit_gap_transfer ν ℓ f htrans (signedHeight e) |
| (signedHeight_projection e) (signedHeight_step_le e) J hJ hz |
| have hh' := ENNReal.toReal_mono (measure_ne_top _ _) hh |
| simpa only [HeightOscillation,Measure.real,ENNReal.toReal_mul,ENNReal.toReal_pow,μ] using hh' |
| · intro h |
| exact independent_firstHit_decoration_small ν ℓ f htrans (signedHeight e) |
| (signedHeight_projection e) (signedHeight_step_le e) r hr.1 h |
| · intro a β ha hβ |
| apply gaussian_renewal_block_modulus μ (commonIncrementProcess ℓ f) |
| (measurable_commonIncrementProcess ℓ f) |
| (commonIncrements_independent ν ℓ htrans (signedHeight e) |
| (signedHeight_projection e) (signedHeight_step_le e) f) |
| (commonIncrements_identDistrib ν ℓ htrans (signedHeight e) |
| (signedHeight_projection e) (signedHeight_step_le e) f) |
| (independent_commonWordIncrement_symmetric ν ℓ htrans f) hi hne |
| (commonWidthProcess ℓ) (measurable_commonWidthProcess ℓ) _ |
| (commonMeanWidth_ge_one ν ℓ htrans (signedHeight e) (signedHeight_projection e) (signedHeight_step_le e)) |
| (commonWidth_strongLaw ν ℓ htrans (signedHeight e) (signedHeight_projection e) (signedHeight_step_le e)) |
| r hr hT ha hβ |
| filter_upwards [independent_conditioned_all_firstPairWords ν ℓ htrans |
| (signedHeight e) (signedHeight_projection e) (signedHeight_step_le e)] with P hP k |
| exact commonWordWidth_positive ℓ _ _ (hP k) |
|
|
| def heightIncrement (F : ℕ → ℝ) (k : ℕ) : ℝ := F (k+1)-F k |
|
|
| lemma realPartialSum_heightIncrement (F : ℕ → ℝ) (h : ℕ) : |
| realPartialSum (heightIncrement F) h = F h-F 0 := by |
| induction h with |
| | zero => simp [realPartialSum] |
| | succ h ih => rw [realPartialSum_succ,ih,heightIncrement]; ring |
|
|
| lemma measurable_heightIncrement {Ω : Type*} [MeasurableSpace Ω] |
| (F : ℕ → Ω → ℝ) (hF : ∀ k, Measurable (F k)) (j : ℕ) : |
| Measurable (fun ω => heightIncrement (fun k => F k ω) j) := (hF (j+1)).sub (hF j) |
|
|
| lemma height_polygon_difference_le (F B : ℕ → ℝ) (h0 : F 0 = B 0) |
| {r n T a : ℝ} (hr : 0 < r) (hn : 0 ≤ n) (hT : 0 ≤ T) |
| (hgap : ∀ h ≤ ⌊T*n⌋₊+1, |F h-B h| ≤ a*r) : |
| ‖scaledPolygon (heightIncrement F) r n T-scaledPolygon (heightIncrement B) r n T‖ ≤ a := by |
| apply (ContinuousMap.norm_le_of_nonempty _).mpr |
| intro t |
| let u := T*n*(t:ℝ) |
| let j := ⌊u⌋₊ |
| have hu : 0 ≤ u := mul_nonneg (mul_nonneg hT hn) t.2.1 |
| have huT : u ≤ T*n := mul_le_of_le_one_right (mul_nonneg hT hn) t.2.2 |
| have hj : j < ⌊T*n⌋₊+1 := Nat.lt_succ_of_le (Nat.floor_mono huT) |
| have hj0 : (j:ℝ) ≤ u := Nat.floor_le hu |
| have hj1 : u < (j:ℝ)+1 := Nat.lt_floor_add_one u |
| have hdet := abs_convex_sub_le (q := u-j) (by linarith) (by linarith) |
| (z := (0:ℝ)) (by simpa only [sub_zero] using hgap j hj.le) |
| (by simpa only [sub_zero] using hgap (j+1) hj) |
| have he : linearPolygon (heightIncrement F) (⌊T*n⌋₊+1) u- |
| linearPolygon (heightIncrement B) (⌊T*n⌋₊+1) u = |
| (1-(u-j))*(F j-B j)+(u-j)*(F (j+1)-B (j+1)) := by |
| rw [linearPolygon_formula _ _ hu hj,linearPolygon_formula _ _ hu hj, |
| realPartialSum_heightIncrement,realPartialSum_heightIncrement] |
| simp only [heightIncrement,h0] |
| dsimp only [j] |
| ring |
| rw [sub_zero,← he] at hdet |
| simp only [ContinuousMap.sub_apply,Real.norm_eq_abs,scaledPolygon_apply] |
| rw [← sub_div,abs_div,abs_of_pos hr] |
| exact (div_le_iff₀ hr).2 hdet |
|
|
| lemma height_polygon_difference_small {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (F B : ℕ → Ω → ℝ) |
| (h0 : ∀ᵐ ω ∂μ, F 0 ω = B 0 ω) (r n : ℕ → ℝ) |
| (hr : Tendsto r atTop atTop) (hn : Tendsto n atTop atTop) {T : ℝ} (hT : 0 ≤ T) |
| (hgap : ∀ ε, 0 < ε → Tendsto (fun i => μ.real |
| (HeightGapEvent F B (⌊T*n i⌋₊+1) (ε*r i))) atTop (𝓝 0)) : |
| TendstoInMeasure μ (fun i ω => scaledPolygon (heightIncrement (fun h => F h ω)) (r i) (n i) T- |
| scaledPolygon (heightIncrement (fun h => B h ω)) (r i) (n i) T) atTop 0 := by |
| apply tendstoInMeasure_iff_measureReal_norm.2 |
| intro ε hε |
| simp only [Pi.zero_apply,sub_zero] |
| apply squeeze_zero' (Eventually.of_forall fun _ => measureReal_nonneg) _ (hgap (ε/2) (by linarith)) |
| filter_upwards [hr.eventually (eventually_gt_atTop (0:ℝ)), |
| hn.eventually (eventually_ge_atTop (0:ℝ))] with i hri hni |
| apply ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae _) |
| filter_upwards [h0] with ω h0 hbig |
| by_contra hnot |
| have hb (h : ℕ) (hh : h ≤ ⌊T*n i⌋₊+1) : |F h ω-B h ω| ≤ ε/2*r i := |
| le_of_not_gt (fun hbad => hnot ⟨h,hh,hbad⟩) |
| have hh := height_polygon_difference_le (fun h => F h ω) (fun h => B h ω) h0 hri hni hT hb |
| change ε ≤ ‖scaledPolygon (heightIncrement (fun h => F h ω)) (r i) (n i) T- |
| scaledPolygon (heightIncrement (fun h => B h ω)) (r i) (n i) T‖ at hbig |
| linarith |
| |
| lemma commonBoundary_heightIncrement {d : ℕ} (ℓ : Vector d) (e : Direction d) (P : Path d × Path d) : |
| heightIncrement (fun h => commonBoundaryGap ℓ e h P) = |
| renewalRewardIncrement (fun k => commonIncrementProcess ℓ e k P) |
| (fun k => commonWidthProcess ℓ k P) := rfl |
| |
| theorem transverse_firstHit_path_error {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) : |
| TendstoInMeasure (independentConditionedPairLaw ν (realPosition (step e))) (fun i P => |
| scaledPolygon (heightIncrement (fun h => firstHitPairGap (realPosition (step e)) f h P)) |
| (r i) (fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) (r i)) T- |
| scaledPolygon (renewalRewardIncrement (fun k => commonIncrementProcess (realPosition (step e)) f k P) |
| (fun k => commonWidthProcess (realPosition (step e)) k P)) |
| (r i) (fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) (r i)) T) atTop 0 := by |
| let ℓ := realPosition (step e) |
| let μ := independentConditionedPairLaw ν ℓ |
| have hp := noDrop_positive_of_directionallyTransient ν ℓ htrans |
| let : IsProbabilityMeasure μ := independentConditionedPairLaw_probability ν ℓ (ne_of_gt hp) |
| have hi := independent_commonWordIncrement_integrable ν hue ℓ (signed_direction_unit e) |
| htrans (signedHeight e) (signedHeight_projection e) (signedHeight_step_le e) f |
| have hne := independent_commonWordIncrement_nonzero ν hue e f hef htrans |
| have hn := (fluctuationScale_tendsto μ (commonIncrementProcess ℓ f 0) |
| (measurable_commonIncrementProcess ℓ f 0) hi hne).comp hr.1 |
| apply height_polygon_difference_small μ (firstHitPairGap ℓ f) (commonBoundaryGap ℓ f) _ r _ hr.1 hn hT |
| · exact fun ε hε => transverse_firstHit_uniform_error ν hue e f hef htrans r hr hT hε |
| |
| · let : IsProbabilityMeasure (conditionedLaw ν ℓ) := conditionedLaw_probability ν ℓ (ne_of_gt hp) |
| have hac := conditionedLaw_absolutelyContinuous ν ℓ |
| have h0 := (Measure.quasiMeasurePreserving_fst.ae (hac.ae_le (annealed_initial ν))).and |
| (Measure.quasiMeasurePreserving_snd.ae (hac.ae_le (annealed_initial ν))) |
| filter_upwards [h0] with P hP |
| simp only [firstHitPairGap,recordIndexPosition,recordIndexTime_zero,hP.1,hP.2,sub_self, |
| commonBoundaryGap,renewalCount_zero,realPartialSum_zero] |
| |
| theorem transverse_firstHit_path_limit {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) : |
| letI : IsProbabilityMeasure (independentConditionedPairLaw ν (realPosition (step e))) := |
| independentConditionedPairLaw_probability ν _ |
| (ne_of_gt (noDrop_positive_of_directionallyTransient ν _ htrans)) |
| ∃ W : ProbabilityMeasure C(unitInterval,ℝ), |
| TendstoInDistribution (fun i P => scaledPolygon |
| (heightIncrement (fun h => firstHitPairGap (realPosition (step e)) f h P)) (r i) |
| (fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) (r i)) T) |
| atTop id (fun _ => independentConditionedPairLaw ν (realPosition (step e))) W ∧ |
| ∀ I : Finset unitInterval, |
| (W : Measure C(unitInterval,ℝ)).map (fun f : C(unitInterval,ℝ) => I.restrict f) = |
| gaussianPathFiniteLaw (T/commonMeanWidth ν (realPosition (step e))) I := by |
| let ℓ := realPosition (step e) |
| let μ := independentConditionedPairLaw ν ℓ |
| let : IsProbabilityMeasure μ := independentConditionedPairLaw_probability ν ℓ |
| (ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans)) |
| obtain ⟨W,hW,hf⟩ := transverse_common_boundary_path_limit ν hue e f hef htrans r hr hT |
| refine ⟨W,?_,hf⟩ |
| apply tendstoInDistribution_of_tendstoInMeasure_sub _ id hW |
| · exact transverse_firstHit_path_error ν hue e f hef htrans r hr hT |
| · intro i |
| exact (measurable_scaledPolygon _ |
| (measurable_heightIncrement (firstHitPairGap ℓ f) (measurable_firstHitPairGap ℓ f)) _ _ _).aemeasurable |
| |
| end DirectionalTransience |
| |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| |
| end |
| end |
| |
| end OAI |
| |
