| import OAI.Probability.FreeIsing.Martingales.PredictableL2Completion |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators Topology |
|
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| namespace Problem367.PredictableL2 |
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|
| 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 |
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|
| 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 |
|
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| 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 |
| |
