| import OAI.Probability.FKMaps.MetricLimit |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory ProbabilityTheory |
| open scoped Topology BigOperators ENNReal NNReal |
| namespace FKCRT.Inventory |
| open FKCRT.TreeCoding |
| attribute [local instance] Classical.propDecidable |
| local instance : MeasurableSpace C(unitInterval,ℝ) := borel _ |
| local instance : BorelSpace C(unitInterval,ℝ) := ⟨rfl⟩ |
|
|
| /-- The original map observable converges under its exact marked code law. -/ |
| theorem actual_map_tests (P : Parameters) |
| {Ω : Type} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {B : Fin 3 → ℝ≥0 → Ω → ℝ} (hB : ∀ i,IsBrownianReal (B i) μ) |
| (hI : iIndepFun (fun i ω t => B i t ω) μ) |
| (F : FKCRT.MetricMeasureData → ℝ) (hF : FKCRT.IsGHPTest F) : |
| Tendsto (fun n => ∫ x,FKCRT.markedMapTest (edgeScale P n) n F x ∂actualDrawLaw P n) |
| atTop (𝓝 (FKCRT.brownianExpectation μ B F)) := by |
| have hp := local_probability_transfer (actualHeight_CLT P hB hI) |
| (fun _ => measurable_of_countable _) (fun n => FKCRT.markedMapTest (edgeScale P n) n F) |
| (fun f => F (FKCRT.treeSpace f)) (metricBad P) |
| (fun a ha => metricBad_limit P (hB 0) ha) (metric_test_local P F hF) |
| have hh := (actualHeight_CLT P hB hI).continuous_comp hF.continuous_treeSpace |
| have hy : TendstoInDistribution (fun n => FKCRT.markedMapTest (edgeScale P n) n F) atTop |
| (fun ω => F (FKCRT.treeSpace (excursionPath B ω))) (actualDrawLaw P) μ := by |
| apply tendstoInDistribution_of_close hh (fun _ => (measurable_of_countable _).aemeasurable) |
| intro ε hε |
| simpa only [Function.comp_def,Real.dist_eq,abs_sub_comm] using hp ε hε |
| obtain ⟨C,hC⟩ := hF.1 |
| have hC0 : 0≤C := (abs_nonneg (F (FKCRT.treeSpace (0 : C(unitInterval,ℝ))))).trans (hC _) |
| let clip : ℝ → ℝ := fun x => max (-C) (min C x) |
| have hclip : Continuous clip := continuous_const.max (continuous_const.min continuous_id) |
| have hclipB : ∀ x,‖clip x‖≤C := by |
| intro x |
| rw [Real.norm_eq_abs,abs_le] |
| exact ⟨le_max_left _ _,max_le (by linarith) (min_le_left _ _)⟩ |
| have hclipId : ∀ x,|x|≤C → clip x=x := by |
| intro x hx |
| rw [abs_le] at hx |
| exact max_eq_right (hx.1.trans_eq (min_eq_right hx.2).symm) |>.trans (min_eq_right hx.2) |
| have hY : ∀ n x,clip (FKCRT.markedMapTest (edgeScale P n) n F x)= |
| FKCRT.markedMapTest (edgeScale P n) n F x := by |
| intro n x |
| apply hclipId |
| unfold FKCRT.markedMapTest |
| split |
| · exact hC _ |
| · simpa only [abs_zero] using hC0 |
| have hz : ∀ ω,clip (F (FKCRT.treeSpace (excursionPath B ω)))=F (FKCRT.treeSpace (excursionPath B ω)) := |
| fun _ => hclipId _ (hC _) |
| have hi := integral_tendsto_of_tendstoInDistribution hy hclip hclipB |
| simp only [hY,hz] at hi |
| have he : (fun ω => F (FKCRT.treeSpace (excursionPath B ω)))=ᵐ[μ] |
| (fun ω => F (FKCRT.treeSpace (FKCRT.standardExcursion B ω))) := by |
| filter_upwards [excursionPath_ae_eq hB] with ω hω |
| exact congrArg (fun e => F (FKCRT.treeSpace e)) hω |
| simpa only [integral_congr_ae he,FKCRT.brownianExpectation] using hi |
|
|
| end FKCRT.Inventory |
| namespace FKCRT |
| open Inventory |
|
|
| /-- GHP convergence of the finite FK laws to the |
| standard Brownian-excursion CRT, through every positive edge count. -/ |
| theorem finite_fk_maps_converge_to_brownian_crt : MainStatement := by |
| intro q hq |
| let P := Parameters.ofQ q hq |
| refine ⟨crtScale P,crtScale_pos P,?_⟩ |
| intro Ω mΩ μ hμ B hB hI F hF |
| have hh := actual_map_tests P hB hI F hF |
| convert hh using 1 |
| ext n |
| exact finiteExpectation_marked q hq (edgeScale P n) n F |
|
|
| end FKCRT |
| end |
|
|
| end OAI |
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|
