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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.Probability.FreeIsing.Stochastic.StoppedSimpleIntegral
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology ENNReal
namespace Problem367.CompensatedSquareLimit
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
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 _
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)
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
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]
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
end Problem367.CompensatedSquareLimit
end
end OAI
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