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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 | import OAI.Probability.FreeIsing.Conditioning.CovarianceLimit
import OAI.Probability.FreeIsing.Estimates.ContinuousModification
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Problem367.ObservationStepApproximation
open ContinuousModification
variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
theorem tendsto_integral_sq_dyadic_sample
(ν : Measure ℝ) (μ : Measure Ω) [IsFiniteMeasure ν] [IsFiniteMeasure μ]
{F : ℝ → Ω → ℝ}
(hF : Measurable (Function.uncurry F))
(hcont : ∀ᵐ ω ∂μ, Continuous (fun t => F t ω))
{R : ℝ} (hR : 0 ≤ R) (hbound : ∀ t, ∀ᵐ ω ∂μ, ‖F t ω‖ ≤ R) :
Tendsto (fun n => ∫ p : ℝ × Ω,
(F (dyadicLeft p.1 n) p.2 - F p.1 p.2) ^ 2 ∂ν.prod μ) atTop (𝓝 0) := by
have hFt (t : ℝ) : Measurable (F t) :=
hF.comp (measurable_const.prodMk measurable_id)
apply tendsto_integral_prod_sq_sub_of_bounded_ae ν μ
(measurable_dyadic_sample hFt) hF hR
· exact fun n => Filter.Eventually.of_forall (fun t => hbound (dyadicLeft t n))
· exact Filter.Eventually.of_forall hbound
· apply Filter.Eventually.of_forall
intro t
filter_upwards [hcont] with ω hω
exact (hω.tendsto t).comp (tendsto_dyadicLeft t)
theorem tendsto_integral_sq_dyadic_sample_of_continuous
(ν : Measure ℝ) (μ : Measure Ω) [IsFiniteMeasure ν] [IsFiniteMeasure μ]
{F : ℝ → Ω → ℝ}
(hF : ∀ t, Measurable (F t)) (hcont : ∀ ω, Continuous (fun t => F t ω))
{R : ℝ} (hR : 0 ≤ R) (hbound : ∀ t, ∀ᵐ ω ∂μ, ‖F t ω‖ ≤ R) :
Tendsto (fun n => ∫ p : ℝ × Ω,
(F (dyadicLeft p.1 n) p.2 - F p.1 p.2) ^ 2 ∂ν.prod μ) atTop (𝓝 0) :=
tendsto_integral_sq_dyadic_sample ν μ
(measurable_uncurry_of_continuous_of_measurable hcont hF)
(Filter.Eventually.of_forall hcont) hR hbound
theorem stronglyProgressive_dyadic_sample (Flt : Filtration ℝ mΩ)
{F : ℝ → Ω → ℝ} (hF : Adapted Flt F) (n : ℕ) :
IsStronglyProgressive Flt (fun t ω => F (dyadicLeft t n) ω) :=
(isProgressive_dyadic_sample Flt hF n).isStronglyProgressive
theorem tendsto_integral_sq_dyadic_observable
{E : Type*} [PseudoMetricSpace E]
[MeasurableSpace E] [BorelSpace E]
(ν : Measure ℝ) (μ : Measure Ω) [IsFiniteMeasure ν] [IsFiniteMeasure μ]
(X : ℝ → Ω → E) (M : E → ℝ)
(hX : ∀ t, Measurable (X t)) (hXcont : ∀ ω, Continuous (fun t => X t ω))
(hM : Continuous M) {R : ℝ} (hR : 0 ≤ R) (hbound : ∀ x, ‖M x‖ ≤ R) :
Tendsto (fun n => ∫ p : ℝ × Ω,
(M (X (dyadicLeft p.1 n) p.2) - M (X p.1 p.2)) ^ 2 ∂ν.prod μ)
atTop (𝓝 0) := by
apply tendsto_integral_sq_dyadic_sample_of_continuous ν μ
(fun t => hM.measurable.comp (hX t)) (fun ω => hM.comp (hXcont ω)) hR
exact fun t => Filter.Eventually.of_forall (fun ω => hbound (X t ω))
theorem finite_dyadicLeft_image_Icc (a b : ℝ) (n : ℕ) :
((fun t => dyadicLeft t n) '' Set.Icc a b).Finite := by
let I : Set ℤ := Set.Icc ⌊(2 : ℝ)^n * a⌋ ⌊(2 : ℝ)^n * b⌋
apply ((Set.finite_Icc _ _).image (fun k : ℤ => (k : ℝ) / (2 : ℝ)^n)).subset
rintro y ⟨t, ht, rfl⟩
refine ⟨⌊(2 : ℝ)^n * t⌋, ?_, rfl⟩
exact ⟨Int.floor_mono (mul_le_mul_of_nonneg_left ht.1 (by positivity)),
Int.floor_mono (mul_le_mul_of_nonneg_left ht.2 (by positivity))⟩
theorem dyadicLeft_eq_on_cell (n : ℕ) (k : ℤ) {t : ℝ}
(ht : t ∈ Set.Ico ((k : ℝ) / (2 : ℝ)^n) (((k : ℝ) + 1) / (2 : ℝ)^n)) :
dyadicLeft t n = (k : ℝ) / (2 : ℝ)^n := by
have hpos : 0 < (2 : ℝ)^n := by positivity
have hlo := (div_le_iff₀ hpos).mp ht.1
have hhi := (lt_div_iff₀ hpos).mp ht.2
have hfloor : ⌊(2 : ℝ)^n * t⌋ = k :=
Int.floor_eq_iff.mpr ⟨by nlinarith, by nlinarith⟩
simp only [dyadicLeft, hfloor]
theorem dyadic_coefficient_stronglyMeasurable
(Flt : Filtration ℝ mΩ) {F : ℝ → Ω → ℝ} (hF : StronglyAdapted Flt F)
(n : ℕ) (k : ℤ) :
StronglyMeasurable[Flt ((k : ℝ) / (2 : ℝ)^n)]
(F ((k : ℝ) / (2 : ℝ)^n)) := hF _
theorem exists_diagonal_meanSquare
(μ : Measure Ω) {X : ℕ → ℕ → Ω → ℝ} {Y : ℕ → Ω → ℝ} {Z : Ω → ℝ}
(hX : ∀ n k, MemLp (X n k) 2 μ) (hY : ∀ n, MemLp (Y n) 2 μ)
(hZ : MemLp Z 2 μ)
(hXY : ∀ n, Tendsto (fun k => ∫ ω, (X n k ω - Y n ω) ^ 2 ∂μ) atTop (𝓝 0))
(hYZ : Tendsto (fun n => ∫ ω, (Y n ω - Z ω) ^ 2 ∂μ) atTop (𝓝 0)) :
∃ k : ℕ → ℕ, (∀ n, n ≤ k n) ∧
Tendsto (fun n => ∫ ω, (X n (k n) ω - Z ω) ^ 2 ∂μ) atTop (𝓝 0) := by
classical
have hex (n : ℕ) : ∃ k, n ≤ k ∧
(∫ ω, (X n k ω - Y n ω) ^ 2 ∂μ) < 1 / ((n : ℝ) + 1) := by
have hpos : (0 : ℝ) < 1 / ((n : ℝ) + 1) := by positivity
exact ((eventually_ge_atTop n).and ((hXY n).eventually (gt_mem_nhds hpos))).exists
choose k hk herror using hex
refine ⟨k, hk, ?_⟩
have hupper (n : ℕ) :
(∫ ω, (X n (k n) ω - Z ω) ^ 2 ∂μ) ≤
2 * (1 / ((n : ℝ) + 1)) + 2 * ∫ ω, (Y n ω - Z ω) ^ 2 ∂μ := by
have hA : Integrable (fun ω => (X n (k n) ω - Y n ω) ^ 2) μ :=
((hX n (k n)).sub (hY n)).integrable_sq
have hB : Integrable (fun ω => (Y n ω - Z ω) ^ 2) μ :=
((hY n).sub hZ).integrable_sq
have hC : Integrable (fun ω => (X n (k n) ω - Z ω) ^ 2) μ :=
((hX n (k n)).sub hZ).integrable_sq
calc
(∫ ω, (X n (k n) ω - Z ω) ^ 2 ∂μ) ≤
∫ ω, 2 * (X n (k n) ω - Y n ω) ^ 2 + 2 * (Y n ω - Z ω) ^ 2 ∂μ := by
apply integral_mono hC ((hA.const_mul 2).add (hB.const_mul 2))
intro ω
change (X n (k n) ω - Z ω) ^ 2 ≤
2 * (X n (k n) ω - Y n ω) ^ 2 + 2 * (Y n ω - Z ω) ^ 2
nlinarith [sq_nonneg (X n (k n) ω - 2 * Y n ω + Z ω)]
_ = 2 * (∫ ω, (X n (k n) ω - Y n ω) ^ 2 ∂μ) +
2 * ∫ ω, (Y n ω - Z ω) ^ 2 ∂μ := by
rw [integral_add (hA.const_mul 2) (hB.const_mul 2), integral_const_mul,
integral_const_mul]
_ ≤ _ := by linarith [herror n]
have hlim : Tendsto (fun n : ℕ => 2 * (1 / ((n : ℝ) + 1)) +
2 * ∫ ω, (Y n ω - Z ω) ^ 2 ∂μ) atTop (𝓝 0) := by
simpa using (tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (2 : ℝ)).add
(hYZ.const_mul 2)
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hlim
(fun n => integral_nonneg (fun ω => sq_nonneg _)) hupper
theorem exists_dyadic_diagonal_meanSquare
(ν : Measure ℝ) (μ : Measure Ω) [IsFiniteMeasure ν] [IsFiniteMeasure μ]
{F : ℕ → ℝ → Ω → ℝ} {Z : ℝ × Ω → ℝ}
(hF : ∀ n, Measurable (Function.uncurry (F n))) (hZ : Measurable Z)
(hcont : ∀ n, ∀ᵐ ω ∂μ, Continuous (fun t => F n t ω))
{R : ℝ} (hR : 0 ≤ R) (hbound : ∀ n t ω, ‖F n t ω‖ ≤ R)
(hZbound : ∀ᵐ p ∂ν.prod μ, ‖Z p‖ ≤ R)
(hlimit : Tendsto (fun n => ∫ p : ℝ × Ω,
(F n p.1 p.2 - Z p) ^ 2 ∂ν.prod μ) atTop (𝓝 0)) :
∃ k : ℕ → ℕ, (∀ n, n ≤ k n) ∧
Tendsto (fun n => ∫ p : ℝ × Ω,
(F n (dyadicLeft p.1 (k n)) p.2 - Z p) ^ 2 ∂ν.prod μ) atTop (𝓝 0) := by
have hFt (n : ℕ) (t : ℝ) : Measurable (F n t) :=
(hF n).comp (measurable_const.prodMk measurable_id)
apply exists_diagonal_meanSquare (ν.prod μ)
(X := fun n k p => F n (dyadicLeft p.1 k) p.2)
(Y := fun n p => F n p.1 p.2) (Z := Z)
· intro n k
apply (memLp_const R).mono'
((measurable_dyadic_sample (hFt n) k).aestronglyMeasurable)
exact Filter.Eventually.of_forall (fun p => hbound n (dyadicLeft p.1 k) p.2)
· intro n
exact (memLp_const R).mono' (hF n).aestronglyMeasurable
(Filter.Eventually.of_forall (fun p => hbound n p.1 p.2))
· exact (memLp_const R).mono' hZ.aestronglyMeasurable hZbound
· intro n
exact tendsto_integral_sq_dyadic_sample ν μ (hF n) (hcont n) hR
(fun t => Filter.Eventually.of_forall (fun ω => hbound n t ω))
· exact hlimit
end Problem367.ObservationStepApproximation
end
end OAI
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