File size: 4,942 Bytes
f0f05de | 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 | import OAI.Combinatorics.BooleanFunctions.Rounding
import OAI.Combinatorics.BooleanFunctions.FiniteCLT
namespace OAI
open scoped BigOperators Topology NNReal ENNReal
open MeasureTheory ProbabilityTheory Filter Real
noncomputable section
namespace SquareRootDegree
namespace Producer
open MeasureTheory ProbabilityTheory Filter UniformFinite FiniteCLT MomentTransfer GaussianGain
open scoped Topology
attribute [local instance] Classical.propDecidable
variable {α β : Type*} [Fintype α] [Nonempty α] [MeasurableSpace α] [MeasurableSingletonClass α]
theorem normalized_moments_tendsto (T : α → ℝ) (h0 : average T = 0)
(h1 : average (fun a => (T a)^2) = 1) (k : Fin 3) :
Tendsto (fun n => ∫ x, (normalizedSum T n x)^(k:ℕ) ∂uniformMeasure (Fin n → α)) atTop
(𝓝 (∫ x : ℝ, x^(k:ℕ) ∂gaussianReal 0 1)) := by
fin_cases k
· simp only [pow_zero,integral_const,probReal_univ,one_smul]
exact tendsto_const_nhds
· simp only [pow_one,integral_uniform,average_normalizedSum T h0,
integral_id_gaussianReal]
exact tendsto_const_nhds
· change Tendsto (fun n => ∫ x, (normalizedSum T n x)^2 ∂uniformMeasure (Fin n → α)) atTop
(𝓝 (∫ x : ℝ, x^2 ∂gaussianReal 0 1))
rw [gaussian_sq_integral]
have he : (fun n => ∫ x, (normalizedSum T n x)^2 ∂uniformMeasure (Fin n → α)) =ᶠ[atTop]
(fun _ => (1:ℝ)) := by
filter_upwards [eventually_gt_atTop 0] with n hn
rw [integral_uniform,average_normalizedSum_sq T h0 h1 hn]
exact (tendsto_congr' he).mpr tendsto_const_nhds
theorem exists_soft_gain (T : α → ℝ) (h0 : average T = 0)
(h1 : average (fun a => (T a)^2) = 1) :
∃ n : ℕ, 0 < n ∧
(∀ i : Leaves, 0 < average (fun x => leafTest i (normalizedSum T n x))) ∧
eta < SoftEvent.deficit (normalizedSum T n)
((∑ i, means (SoftEvent.rawMoments (normalizedSum T n)
(fun i x => leafTest i (normalizedSum T n x))) i)-1)
(SoftEvent.softWeight (fun i x => selectorTest i (normalizedSum T n x))
(fun i x => leafTest i (normalizedSum T n x))) := by
have hc := uniform_clt T h0 h1
have hp := eventually_positive_gain (fun n => uniformMeasure (Fin n → α))
(normalizedSum T) (fun n => measurable_of_finite _) hc
(fun n k => Integrable.of_finite) (normalized_moments_tendsto T h0 h1)
obtain ⟨n,hn,hp⟩ := ((eventually_gt_atTop 0).and hp).exists
have hl (i : Leaves) : 0 < average (fun x => leafTest i (normalizedSum T n x)) := by
simpa only [moments,Fin.val_zero,pow_zero,mul_one,integral_uniform] using hp.1 i
refine ⟨n,hn,hl,?_⟩
have hs := SoftEvent.soft_deficit_eq_gain
(fun i x => selectorTest i (normalizedSum T n x))
(fun i x => leafTest i (normalizedSum T n x)) (fun i => (hl i).ne')
(normalizedSum T n) (average_normalizedSum T h0 n) (average_normalizedSum_sq T h0 h1 hn)
rw [hs]
have he : (moments (uniformMeasure (Fin n → α)) (normalizedSum T n) leafTest) =
SoftEvent.rawMoments (normalizedSum T n) (fun i x => leafTest i (normalizedSum T n x)) := by
funext k i
exact integral_uniform _
have he' : (moments (uniformMeasure (Fin n → α)) (normalizedSum T n) selectorTest) =
SoftEvent.rawMoments (normalizedSum T n) (fun i x => selectorTest i (normalizedSum T n x)) := by
funext k i
exact integral_uniform _
rw [he,he'] at hp
convert hp.2 using 1
congr 1
theorem exists_table_gain [Fintype β] (F : α → β) (t : β → ℝ)
(h0 : average (fun a => t (F a)) = 0)
(h1 : average (fun a => (t (F a))^2) = 1) :
∃ n : ℕ, 0 < n ∧ ∃ b : ℝ, ∃ S : (Fin n → β) → Leaves,
∃ L : Leaves → (Fin n → β) → Bool,
eta < SoftEvent.deficit (normalizedSum (fun a => t (F a)) n) b
(fun x => if TableRounding.tableEvent S L
((fun j => F (x.1 j)),fun i j => F (x.2 i j)) then 1 else 0) := by
obtain ⟨n,hn,_,hg⟩ := exists_soft_gain (fun a => t (F a)) h0 h1
let b := (∑ i, means (SoftEvent.rawMoments (normalizedSum (fun a => t (F a)) n)
(fun i x => leafTest i (normalizedSum (fun a => t (F a)) n x))) i)-1
have he (x : Fin n → α) : normalizedSum t n (fun j => F (x j)) =
normalizedSum (fun a => t (F a)) n x := rfl
obtain ⟨S,L,hSL⟩ := TableRounding.exists_event_deficit
(fun (i : Leaves) (y : Fin n → β) => selectorTest i (normalizedSum t n y))
(fun (i : Leaves) (y : Fin n → β) => leafTest i (normalizedSum t n y))
(fun y => SmoothSelector.weight_sum gridN step (normalizedSum t n y))
(fun i y => SmoothSelector.weight_nonneg gridN step i (normalizedSum t n y))
(fun i y => SmoothGaussian.tent_nonneg _ _ _)
(fun i y => SmoothGaussian.tent_le_one width_pos _ _)
(fun x : Fin n → α => fun j => F (x j))
(normalizedSum (fun a => t (F a)) n) b eta (by simpa only [he] using hg)
exact ⟨n,hn,b,S,L,hSL⟩
end Producer
end SquareRootDegree
end
end OAI
|