File size: 4,123 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 | import OAI.Probability.FreeIsing.Martingales.PredictableL2Completion
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped BigOperators Topology
namespace Problem367.PredictableL2
theorem memLp_two_mul_of_indep
{Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
{f g : Ω → ℝ} (hind : IndepFun f g μ)
(hf : MemLp f 2 μ) (hg : MemLp g 2 μ) :
MemLp (fun ω => f ω * g ω) 2 μ := by
apply (memLp_two_iff_integrable_sq (hf.aestronglyMeasurable.mul
hg.aestronglyMeasurable)).2
have hi := (hind.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)).integrable_mul
hf.integrable_sq hg.integrable_sq
simpa only [Function.comp_def, Pi.mul_def, mul_pow, id_eq] using hi
theorem weighted_array_orthogonal
{Ω ι : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
{H D : Ω → ι → ℝ} (hind : IndepFun H D μ)
(hH : ∀ i, MemLp (fun ω => H ω i) 2 μ)
(hD : ∀ i, MemLp (fun ω => D ω i) 2 μ)
(horth : Pairwise fun i j => ∫ ω, D ω i * D ω j ∂μ = 0) :
Pairwise fun i j => ∫ ω, (H ω i * D ω i) * (H ω j * D ω j) ∂μ = 0 := by
intro i j hij
have hp := hind.comp ((measurable_pi_apply i).mul (measurable_pi_apply j))
((measurable_pi_apply i).mul (measurable_pi_apply j))
have hi := hp.integral_fun_mul_eq_mul_integral
((hH i).aestronglyMeasurable.mul (hH j).aestronglyMeasurable)
((hD i).aestronglyMeasurable.mul (hD j).aestronglyMeasurable)
calc
_ = ∫ ω, (H ω i * H ω j) * (D ω i * D ω j) ∂μ := by
congr 1
funext ω
ring
_ = (∫ ω, H ω i * H ω j ∂μ) * ∫ ω, D ω i * D ω j ∂μ := hi
_ = 0 := by rw [horth hij, mul_zero]
theorem independent_weighted_second_moment
{Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
{f g : Ω → ℝ} (hind : IndepFun f g μ)
(hf : MemLp f 2 μ) (hg : MemLp g 2 μ) :
(∫ ω, (f ω * g ω) ^ 2 ∂μ) =
(∫ ω, f ω ^ 2 ∂μ) * ∫ ω, g ω ^ 2 ∂μ := by
have hi := (hind.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)).integral_fun_mul_eq_mul_integral hf.integrable_sq.aestronglyMeasurable
hg.integrable_sq.aestronglyMeasurable
simpa only [Function.comp_def, mul_pow, id_eq] using hi
theorem exists_independent_weighted_sum
{Ω ι : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
{H D : Ω → ι → ℝ} (hind : IndepFun H D μ)
(hH : ∀ i, MemLp (fun ω => H ω i) 2 μ)
(hD : ∀ i, MemLp (fun ω => D ω i) 2 μ)
(hcenter : ∀ i, ∫ ω, D ω i ∂μ = 0)
(horth : Pairwise fun i j => ∫ ω, D ω i * D ω j ∂μ = 0)
(henergy : Summable fun i =>
(∫ ω, H ω i ^ 2 ∂μ) * ∫ ω, D ω i ^ 2 ∂μ) :
∃ Y : Ω → ℝ, Measurable Y ∧ MemLp Y 2 μ ∧
(∫ ω, Y ω ∂μ) = 0 ∧
(∫ ω, Y ω ^ 2 ∂μ) =
∑' i, (∫ ω, H ω i ^ 2 ∂μ) * ∫ ω, D ω i ^ 2 ∂μ ∧
Tendsto (fun s : Finset ι => ∫ ω,
((∑ i ∈ s, H ω i * D ω i) - Y ω) ^ 2 ∂μ) atTop (𝓝 0) := by
have hc (i : ι) : IndepFun (fun ω => H ω i) (fun ω => D ω i) μ :=
hind.comp (measurable_pi_apply i) (measurable_pi_apply i)
have hHD (i : ι) : MemLp (fun ω => H ω i * D ω i) 2 μ :=
memLp_two_mul_of_indep (hc i) (hH i) (hD i)
have hsecond (i : ι) : (∫ ω, (H ω i * D ω i) ^ 2 ∂μ) =
(∫ ω, H ω i ^ 2 ∂μ) * ∫ ω, D ω i ^ 2 ∂μ :=
independent_weighted_second_moment (hc i) (hH i) (hD i)
obtain ⟨Y, hY, hmY, hsY, hiY⟩ := exists_orthogonal_sum
(fun i ω => H ω i * D ω i) hHD (weighted_array_orthogonal hind hH hD horth)
(by simpa only [hsecond] using henergy)
refine ⟨Y, hmY, hY, ?_, by simpa only [hsecond] using hiY, ?_⟩
· apply integral_eq_zero_of_hasSum (fun i ω => H ω i * D ω i) hHD hY hsY
intro i
rw [(hc i).integral_fun_mul_eq_mul_integral
(hH i).aestronglyMeasurable (hD i).aestronglyMeasurable, hcenter i, mul_zero]
· exact tendsto_integral_sq_error_of_hasSum (fun i ω => H ω i * D ω i) hHD hY hsY
end Problem367.PredictableL2
end
end OAI
|