| import OAI.Probability.FreeIsing.Stochastic.StoppedSimpleIntegral |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology ENNReal |
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| namespace Problem367.CompensatedSquareLimit |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} |
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| theorem eLpNorm_sq_sub_le {f g : Ω → ℝ} |
| (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) : |
| eLpNorm (fun ω => (f ω) ^ 2 - (g ω) ^ 2) 1 μ ≤ |
| eLpNorm (f - g) 2 μ * |
| (eLpNorm (f - g) 2 μ + (eLpNorm g 2 μ + eLpNorm g 2 μ)) := by |
| have hfactor : (fun ω => (f ω) ^ 2 - (g ω) ^ 2) = |
| fun ω => (f ω - g ω) * (f ω + g ω) := by |
| funext ω |
| ring |
| have hsum : eLpNorm (f + g) 2 μ ≤ |
| eLpNorm (f - g) 2 μ + (eLpNorm g 2 μ + eLpNorm g 2 μ) := by |
| have heq : f + g = (f - g) + (g + g) := by |
| funext ω |
| simp only [Pi.add_apply, Pi.sub_apply] |
| ring |
| rw [heq] |
| exact (eLpNorm_add_le (by norm_num)).trans |
| (add_le_add le_rfl (eLpNorm_add_le (p := 2) (by norm_num))) |
| rw [hfactor] |
| calc |
| _ ≤ eLpNorm (f - g) 2 μ * eLpNorm (f + g) 2 μ := by |
| simpa using |
| (eLpNorm_le_eLpNorm_mul_eLpNorm_of_norm |
| (p := 2) (q := 2) (r := 1) |
| (fun first second : ℝ => first * second) 1 continuous_mul |
| (hf.sub hg) (hf.add hg) |
| (Eventually.of_forall (fun ω => by simp [norm_mul]))) |
| _ ≤ _ := mul_le_mul_right hsum _ |
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| theorem tendsto_eLpNorm_sq_sub {N : Type*} {l : Filter N} |
| {f : N → Ω → ℝ} {g : Ω → ℝ} |
| (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : MemLp g 2 μ) |
| (hlim : Tendsto (fun n => eLpNorm (f n - g) 2 μ) l (𝓝 0)) : |
| Tendsto (fun n => eLpNorm (fun ω => (f n ω) ^ 2 - (g ω) ^ 2) 1 μ) |
| l (𝓝 0) := by |
| have hsum := hlim.add_const (eLpNorm g 2 μ + eLpNorm g 2 μ) |
| have hfinite : (0 : ℝ≥0∞) + (eLpNorm g 2 μ + eLpNorm g 2 μ) ≠ ∞ := by |
| simpa using ENNReal.add_ne_top.mpr ⟨hg.eLpNorm_ne_top, hg.eLpNorm_ne_top⟩ |
| have hb := ENNReal.Tendsto.mul hlim (Or.inr hfinite) hsum |
| (Or.inr ENNReal.zero_ne_top) |
| simp only [zero_mul] at hb |
| exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hb |
| (fun _ => bot_le) (fun n => eLpNorm_sq_sub_le (hf n) hg.aestronglyMeasurable) |
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| theorem martingale_compensatedSquare_of_limits |
| {ι N : Type*} [LinearOrder ι] |
| {P : Measure Ω} [IsFiniteMeasure P] {F : Filtration ι mΩ} |
| {l : Filter N} [l.NeBot] |
| {X A : N → ι → Ω → ℝ} {Y Q : ι → Ω → ℝ} |
| (hX : ∀ n t, MemLp (X n t) 2 P) |
| (_hA : ∀ n t, Integrable (A n t) P) |
| (hcomp : ∀ n, Martingale (fun t ω => (X n t ω) ^ 2 - A n t ω) F P) |
| (hY : StronglyAdapted F Y) (hY2 : ∀ t, MemLp (Y t) 2 P) |
| (hQ : StronglyAdapted F Q) (hQint : ∀ t, Integrable (Q t) P) |
| (hlim : ∀ t, Tendsto (fun n => eLpNorm (X n t - Y t) 2 P) l (𝓝 0)) |
| (hAlim : ∀ t, Tendsto (fun n => eLpNorm (A n t - Q t) 1 P) l (𝓝 0)) : |
| Martingale (fun t ω => (Y t ω) ^ 2 - Q t ω) F P := by |
| apply StoppedSimpleIntegral.martingale_of_pointwise_time_L1_limit (l := l) hcomp |
| · intro t |
| exact ((hY t).pow 2).sub (hQ t) |
| · intro t |
| exact (hY2 t).integrable_sq.sub (hQint t) |
| · intro t |
| have hs := tendsto_eLpNorm_sq_sub |
| (fun n => (hX n t).aestronglyMeasurable) (hY2 t) (hlim t) |
| have hb := hs.add (hAlim t) |
| simp only [add_zero] at hb |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hb (fun _ => bot_le) |
| intro n |
| have heq : (fun ω => (X n t ω) ^ 2 - A n t ω) - |
| (fun ω => (Y t ω) ^ 2 - Q t ω) = |
| (fun ω => (X n t ω) ^ 2 - (Y t ω) ^ 2) - (A n t - Q t) := by |
| funext ω |
| simp only [Pi.sub_apply] |
| ring |
| dsimp only |
| rw [heq] |
| exact eLpNorm_sub_le le_rfl |
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| theorem eLpNorm_two_eq_ofReal_integral_sq {f : Ω → ℝ} (hf : MemLp f 2 μ) : |
| eLpNorm f 2 μ = (ENNReal.ofReal (∫ ω, (f ω) ^ 2 ∂μ)) ^ (1 / 2 : ℝ) := by |
| rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num) (by norm_num) |
| hf.aestronglyMeasurable] |
| simp only [ENNReal.toReal_ofNat] |
| congr 1 |
| rw [ofReal_integral_eq_lintegral_ofReal hf.integrable_sq |
| (Eventually.of_forall (fun ω => sq_nonneg _))] |
| apply lintegral_congr |
| intro ω |
| simp [← ofReal_norm, ← ENNReal.ofReal_pow, Real.norm_eq_abs, sq_abs] |
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| theorem tendsto_eLpNorm_two_of_meanSquare {N : Type*} {l : Filter N} |
| {f : N → Ω → ℝ} {g : Ω → ℝ} (hf : ∀ n, MemLp (f n) 2 μ) (hg : MemLp g 2 μ) |
| (hlim : Tendsto (fun n => ∫ ω, (f n ω - g ω) ^ 2 ∂μ) l (𝓝 0)) : |
| Tendsto (fun n => eLpNorm (f n - g) 2 μ) l (𝓝 0) := by |
| have ho : Tendsto (fun n => ENNReal.ofReal (∫ ω, (f n ω - g ω) ^ 2 ∂μ)) l (𝓝 0) := by |
| simpa only [Function.comp_def, ENNReal.ofReal_zero] using |
| (ENNReal.continuous_ofReal.tendsto 0).comp hlim |
| have h := (ENNReal.continuous_rpow_const (y := 1 / 2).tendsto (0 : ℝ≥0∞)).comp ho |
| have heq (n) := eLpNorm_two_eq_ofReal_integral_sq ((hf n).sub hg) |
| simpa only [Function.comp_def, heq, Pi.sub_apply, ENNReal.ofReal_zero, ENNReal.zero_rpow_of_pos (by norm_num : (0 : ℝ) < 1 / 2)] using h |
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| end Problem367.CompensatedSquareLimit |
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| end |
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| end OAI |
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