File size: 3,781 Bytes
ddc84c5 | 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 | import OAI.Analysis.FiniteFactor.Convolution
import OAI.Analysis.FiniteFactor.TailEstimate
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal
noncomputable section
/-- Cauchy--Schwarz for genuinely infinite nonnegative coefficient series. -/
theorem nonnegative_series_cauchy_schwarz {ι : Type*} (f g : ι → ℝ)
(hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hfs : Summable (fun i => f i^2)) (hgs : Summable (fun i => g i^2)) :
Summable (fun i => f i*g i) ∧
(∑' i, f i*g i) ≤ Real.sqrt (∑' i, f i^2)*Real.sqrt (∑' i, g i^2) := by
have hp : Real.HolderConjugate 2 2 := by norm_num [Real.holderConjugate_iff]
have hF : 0 ≤ ∑' i, f i^2 := tsum_nonneg (fun i => sq_nonneg _)
have hG : 0 ≤ ∑' i, g i^2 := tsum_nonneg (fun i => sq_nonneg _)
have hsf : HasSum (fun i => f i^(2 : ℝ)) ((Real.sqrt (∑' i, f i^2))^(2 : ℝ)) := by
simpa only [Real.rpow_two,Real.sq_sqrt hF] using hfs.hasSum
have hsg : HasSum (fun i => g i^(2 : ℝ)) ((Real.sqrt (∑' i, g i^2))^(2 : ℝ)) := by
simpa only [Real.rpow_two,Real.sq_sqrt hG] using hgs.hasSum
obtain ⟨C,_,hC,hs⟩ := Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg hp
(Real.sqrt_nonneg _) (Real.sqrt_nonneg _) hf hg hsf hsg
exact ⟨hs.summable,hs.tsum_eq ▸ hC⟩
/-- Probability-space Cauchy--Schwarz after the pointwise series estimate.
Equivalent to the product-with-counting-measure argument; no independence or
finite truncation is assumed. -/
theorem countable_error_integral_le {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι]
{μ : Measure Ω} [IsProbabilityMeasure μ] (f g : ι → Ω → ℝ) (δ : ℝ)
(hf : ∀ i, Measurable (f i)) (hg : ∀ i, Measurable (g i))
(hf0 : ∀ i ω, 0 ≤ f i ω) (hg0 : ∀ i ω, 0 ≤ g i ω)
(hfs : ∀ᵐ ω ∂μ, Summable (fun i => f i ω^2))
(hgs : ∀ᵐ ω ∂μ, Summable (fun i => g i ω^2))
(hfi : Integrable (fun ω => ∑' i, f i ω^2) μ)
(hfm : (∫ ω, (∑' i, f i ω^2) ∂μ) ≤ δ)
(hg1 : ∀ᵐ ω ∂μ, (∑' i, g i ω^2) ≤ 1) :
Integrable (fun ω => ∑' i, f i ω*g i ω) μ ∧
(∫ ω, (∑' i, f i ω*g i ω) ∂μ) ≤ Real.sqrt δ := by
let F := fun ω => ∑' i, f i ω^2
let S := fun ω => ∑' i, f i ω*g i ω
have hF0 (ω) : 0 ≤ F ω := tsum_nonneg (fun i => sq_nonneg _)
have hFm : Measurable F := Measurable.tsum (fun i => (hf i).pow_const 2)
have hSm : Measurable S := Measurable.tsum (fun i => (hf i).mul (hg i))
have hS0 (ω) : 0 ≤ S ω := tsum_nonneg (fun i => mul_nonneg (hf0 i ω) (hg0 i ω))
have hFroot : MemLp (fun ω => Real.sqrt (F ω)) 2 μ := by
apply (memLp_two_iff_integrable_sq (hFm.sqrt.aestronglyMeasurable)).mpr
simpa only [Real.sq_sqrt (hF0 _)] using hfi
have hS : ∀ᵐ ω ∂μ, S ω ≤ Real.sqrt (F ω) := by
filter_upwards [hfs,hgs,hg1] with ω hs ht h1
have hh := (nonnegative_series_cauchy_schwarz (fun i => f i ω) (fun i => g i ω)
(fun i => hf0 i ω) (fun i => hg0 i ω) hs ht).2
apply hh.trans
simpa only [Real.sqrt_one,mul_one] using
mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt h1) (Real.sqrt_nonneg (F ω))
have hSi : Integrable S μ := (hFroot.integrable (by norm_num)).mono'
hSm.aestronglyMeasurable (hS.mono fun ω h => by rwa [Real.norm_of_nonneg (hS0 ω)])
refine ⟨hSi,?_⟩
have hh := integral_mul_le_sqrt_of_square_bounds hFroot (memLp_const (1 : ℝ))
(Eventually.of_forall fun ω => Real.sqrt_nonneg (F ω)) (Eventually.of_forall fun _ => zero_le_one)
(by simpa only [Real.sq_sqrt (hF0 _)] using hfm)
(by simp : (∫ _ω : Ω, (1 : ℝ)^2 ∂μ) ≤ 1)
apply (integral_mono_ae hSi (hFroot.integrable (by norm_num)) hS).trans
simpa only [mul_one] using hh
end
end FiniteFactor
end OAI
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