File size: 4,769 Bytes
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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Problem367
theorem tendsto_conditional_covariance_ae
{Ω : Type*} [m : MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(ℱ : Filtration ℕ m) (f g : Ω → ℝ) :
∀ᵐ ω ∂μ, Tendsto
(fun n => (μ[fun x => f x * g x | ℱ n]) ω -
(μ[f | ℱ n]) ω * (μ[g | ℱ n]) ω)
atTop (𝓝 ((μ[fun x => f x * g x | ⨆ n, ℱ n]) ω -
(μ[f | ⨆ n, ℱ n]) ω * (μ[g | ⨆ n, ℱ n]) ω)) := by
filter_upwards [tendsto_ae_condExp (μ := μ) (ℱ := ℱ) (fun x => f x * g x),
tendsto_ae_condExp (μ := μ) (ℱ := ℱ) f,
tendsto_ae_condExp (μ := μ) (ℱ := ℱ) g] with ω hfg hf hg
exact hfg.sub (hf.mul hg)
private theorem artanh_continuous_at_interior {x : ℝ}
(hx : x ∈ Set.Ioo (-1 : ℝ) 1) : ContinuousAt Real.artanh x := by
have hd : 1 - x ≠ 0 := ne_of_gt (sub_pos.mpr hx.2)
have hp : 0 < (1 + x) / (1 - x) := div_pos (by linarith [hx.1]) (sub_pos.mpr hx.2)
unfold Real.artanh
exact ((continuousAt_const.add continuousAt_id).div
(continuousAt_const.sub continuousAt_id) hd).sqrt.log
(ne_of_gt (Real.sqrt_pos.2 hp))
theorem posterior_field_limit_of_tanh_bounds
{u : ℕ → ℝ} {z X K : ℝ} (hu : Tendsto u atTop (𝓝 z))
(hb : ∀ n, Real.tanh (X - K) ≤ u n ∧ u n ≤ Real.tanh (X + K)) :
z ∈ Set.Ioo (-1 : ℝ) 1 ∧ |Real.artanh z - X| ≤ K ∧
Tendsto (fun n => Real.artanh (u n)) atTop (𝓝 (Real.artanh z)) := by
have hlo : Real.tanh (X - K) ≤ z :=
ge_of_tendsto hu (Filter.Eventually.of_forall (fun n => (hb n).1))
have hhi : z ≤ Real.tanh (X + K) :=
le_of_tendsto hu (Filter.Eventually.of_forall (fun n => (hb n).2))
have hz : z ∈ Set.Ioo (-1 : ℝ) 1 :=
⟨lt_of_lt_of_le (Real.neg_one_lt_tanh _) hlo,
lt_of_le_of_lt hhi (Real.tanh_lt_one _)⟩
refine ⟨hz, ?_, (artanh_continuous_at_interior hz).tendsto.comp hu⟩
have hleft := Real.artanh_le_artanh (Real.neg_one_lt_tanh (X - K)) hz.2 hlo
have hright := Real.artanh_le_artanh hz.1 (Real.tanh_lt_one (X + K)) hhi
rw [Real.artanh_tanh] at hleft hright
exact abs_le.mpr ⟨by linarith, by linarith⟩
theorem tendsto_posterior_fields_ae
{Ω : Type*} [m : MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(ℱ : Filtration ℕ m) (f X : Ω → ℝ) (K : ℝ)
(hb : ∀ n, ∀ᵐ ω ∂μ,
Real.tanh (X ω - K) ≤ (μ[f | ℱ n]) ω ∧
(μ[f | ℱ n]) ω ≤ Real.tanh (X ω + K)) :
∀ᵐ ω ∂μ,
(μ[f | ⨆ n, ℱ n]) ω ∈ Set.Ioo (-1 : ℝ) 1 ∧
|Real.artanh ((μ[f | ⨆ n, ℱ n]) ω) - X ω| ≤ K ∧
Tendsto (fun n => Real.artanh ((μ[f | ℱ n]) ω)) atTop
(𝓝 (Real.artanh ((μ[f | ⨆ n, ℱ n]) ω))) := by
filter_upwards [tendsto_ae_condExp (μ := μ) (ℱ := ℱ) f,
ae_all_iff.mpr hb] with ω hlim hbound
exact posterior_field_limit_of_tanh_bounds hlim hbound
theorem tendsto_integral_sq_sub_condExp_of_bounded
{Ω : Type*} [m : MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(ℱ : Filtration ℕ m) (f : Ω → ℝ) (K : ℝ)
(hb : ∀ᵐ ω ∂μ, |f ω| ≤ K) :
Tendsto (fun n => ∫ ω,
((μ[f | ℱ n]) ω - (μ[f | ⨆ n, ℱ n]) ω) ^ 2 ∂μ)
atTop (𝓝 0) := by
have hsup : (⨆ n, ℱ n) ≤ m := iSup_le (fun n => ℱ.le n)
have hmeas (n : ℕ) : AEStronglyMeasurable
(fun ω => ((μ[f | ℱ n]) ω - (μ[f | ⨆ n, ℱ n]) ω) ^ 2) μ :=
(((stronglyMeasurable_condExp.mono (ℱ.le n)).sub
(stronglyMeasurable_condExp.mono hsup)).pow 2).aestronglyMeasurable
have hbound (n : ℕ) : ∀ᵐ ω ∂μ,
‖((μ[f | ℱ n]) ω - (μ[f | ⨆ n, ℱ n]) ω) ^ 2‖ ≤ 4 * K ^ 2 := by
filter_upwards [ae_bdd_abs_condExp_of_ae_bdd_abs (m := ℱ n) hb,
ae_bdd_abs_condExp_of_ae_bdd_abs (m := ⨆ n, ℱ n) hb] with ω hn hlimBound
have hK : 0 ≤ K := (abs_nonneg _).trans hn
have hd : |(μ[f | ℱ n]) ω - (μ[f | ⨆ n, ℱ n]) ω| ≤ 2 * K :=
(abs_sub _ _).trans (by linarith)
have hs := (sq_le_sq₀ (abs_nonneg _) (show 0 ≤ 2 * K by positivity)).2 hd
rw [Real.norm_of_nonneg (sq_nonneg _)]
nlinarith [sq_abs ((μ[f | ℱ n]) ω - (μ[f | ⨆ n, ℱ n]) ω)]
have hlimit : ∀ᵐ ω ∂μ, Tendsto
(fun n => ((μ[f | ℱ n]) ω - (μ[f | ⨆ n, ℱ n]) ω) ^ 2)
atTop (𝓝 (0 : ℝ)) := by
filter_upwards [tendsto_ae_condExp (μ := μ) (ℱ := ℱ) f] with ω hω
simpa using (hω.sub_const ((μ[f | ⨆ n, ℱ n]) ω)).pow 2
simpa using tendsto_integral_of_dominated_convergence (fun _ : Ω => 4 * K ^ 2)
hmeas (integrable_const _) hbound hlimit
end Problem367
end
end OAI
|