File size: 6,649 Bytes
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 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.WeakAddTendsto
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
namespace DirectionalTransience
theorem centered_height_additive_uniform {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (F : ℕ → Ω → ℝ)
(hF : ∀ h, Measurable (F h)) (b : ℕ → ℝ) (hb0 : b 0 = 0) (r n : ℕ → ℝ)
(hn : Tendsto n atTop atTop) (V : ℝ) (hV : 0 ≤ V)
(hlim : ∀ T : ℝ, 0 < T → ∃ W : ProbabilityMeasure C(unitInterval,ℝ),
TendstoInDistribution (fun i x => heightPolygon (fun h => F h x-b h) (r i) (n i) T)
atTop id (fun _ => μ) W ∧
∀ I : Finset unitInterval,
(W : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g) = gaussianPathFiniteLaw (T*V) I)
(p : ℝ) (hp : 0 < p)
(hd : ∀ k j : ℕ, 0 < k → 0 < j →
ENNReal.ofReal p • (μ.prod μ).map (fun P => F k P.1+F j P.2) ≤ μ.map (F (k+j)))
(T : ℝ) {ε : ℝ} (hε : 0 < ε) :
∀ᶠ i in atTop, ∀ k j : ℕ, ((k+j:ℕ):ℝ) ≤ T*n i →
|(b (k+j)-b k-b j)/r i| < ε := by
let S := max T 1
have hS : 0 < S := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
obtain ⟨N,hN⟩ := eventually_atTop.mp (hn.eventually (eventually_gt_atTop (0:ℝ)))
by_contra hf
simp only [eventually_atTop,not_exists,not_forall] at hf
have hex (l : ℕ) : ∃ i ≥ max l N, ∃ k j : ℕ,
((k+j:ℕ):ℝ) ≤ T*n i ∧ ε ≤ |(b (k+j)-b k-b j)/r i| := by
obtain ⟨i,hi,hbreak⟩ := hf (max l N)
push Not at hbreak
obtain ⟨k,j,hkj,hbad⟩ := hbreak
exact ⟨i,hi,k,j,hkj,hbad⟩
choose a ha k j hkj hbad using hex
have halim : Tendsto a atTop atTop := tendsto_atTop_mono (fun l => le_trans (le_max_left _ _) (ha l)) tendsto_id
have hna (l : ℕ) : 0 < n (a l) := hN _ (le_trans (le_max_right _ _) (ha l))
have hbounds (l : ℕ) : ((k l+j l:ℕ):ℝ) ≤ S*n (a l) :=
(hkj l).trans (mul_le_mul_of_nonneg_right (le_max_left _ _) (hna l).le)
have hkpos (l : ℕ) : 0 < k l := by
by_contra hh
have hz : k l = 0 := by omega
have hb := hbad l
simp [hz,hb0] at hb
linarith
have hjpos (l : ℕ) : 0 < j l := by
by_contra hh
have hz : j l = 0 := by omega
have hb := hbad l
simp [hz,hb0] at hb
linarith
have hx (l : ℕ) : (k l:ℝ)/n (a l) ∈ Set.Icc 0 S := by
refine ⟨div_nonneg (Nat.cast_nonneg _) (hna l).le,(div_le_iff₀ (hna l)).mpr ?_⟩
exact le_trans (by exact_mod_cast Nat.le_add_right (k l) (j l)) (hbounds l)
have hy (l : ℕ) : (j l:ℝ)/n (a l) ∈ Set.Icc 0 S := by
refine ⟨div_nonneg (Nat.cast_nonneg _) (hna l).le,(div_le_iff₀ (hna l)).mpr ?_⟩
exact le_trans (by exact_mod_cast Nat.le_add_left (j l) (k l)) (hbounds l)
obtain ⟨s,_,q,hq,hqs⟩ := isCompact_Icc.tendsto_subseq hx
obtain ⟨t,_,u,hu,hut⟩ := isCompact_Icc.tendsto_subseq (fun i => hy (q i))
let v := fun i => a (q (u i))
have hv : Tendsto v atTop atTop := halim.comp (hq.tendsto_atTop.comp hu.tendsto_atTop)
obtain ⟨W,hW,hWf⟩ := hlim S hS
have hsub : TendstoInDistribution
(fun i x => heightPolygon (fun h => F h x-b h) (r (v i)) (n (v i)) S)
atTop id (fun _ => μ) W :=
⟨fun i => hW.forall_aemeasurable (v i),hW.aemeasurable_limit,hW.tendsto.comp hv⟩
have hadd := centered_height_additive_sequence μ F hF b (r ∘ v) (n ∘ v)
(fun i => hna (q (u i))) S V hS hV W hsub hWf p hp hd
(fun i => k (q (u i))) (fun i => j (q (u i)))
(fun i => hkpos _) (fun i => hjpos _) (fun i => hbounds _)
(hqs.comp hu.tendsto_atTop) hut
have he := hadd.abs.eventually (gt_mem_nhds (show |(0:ℝ)| < ε by simpa using hε))
obtain ⟨i,hi⟩ := he.exists
exact not_lt_of_ge (hbad (q (u i))) hi
end DirectionalTransience
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
end
end
end OAI
|