File size: 3,639 Bytes
f61e547 | 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 | 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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