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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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.Probability.FreeIsing.Martingales.SpatialLimit
import OAI.Probability.FreeIsing.Stochastic.IntegralAssociativityLimit
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped NNReal Topology
namespace Problem367.MartingaleIntegerIdentification
variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
{μ : Measure Ω} {F : Filtration ℝ≥0 mΩ}
theorem ae_eq_of_positive_integer
{X Y : ℝ≥0 → Ω → ℝ}
(hX : Martingale X F μ) (hY : Martingale Y F μ)
(he : ∀ N : ℕ, 0 < N → X N =ᵐ[μ] Y N) (t : ℝ≥0) :
X t =ᵐ[μ] Y t := by
obtain ⟨N, hN⟩ := exists_nat_gt t
have hN0 : 0 < N := by
have hn : (0 : ℝ≥0) < (N : ℝ≥0) := lt_of_le_of_lt (show (0 : ℝ≥0) ≤ t from zero_le) hN
exact_mod_cast hn
exact (hX.condExp_ae_eq hN.le).symm.trans
((condExp_congr_ae (he N hN0)).trans (hY.condExp_ae_eq hN.le))
theorem meanSquare_limit_of_positive_integer [IsFiniteMeasure μ]
{α : Type*} {l : Filter α} {X : α → ℝ≥0 → Ω → ℝ}
{Y : ℝ≥0 → Ω → ℝ}
(hX : ∀ a, Martingale (X a) F μ) (hY : Martingale Y F μ)
(hX2 : ∀ a t, MemLp (X a t) 2 μ) (hY2 : ∀ t, MemLp (Y t) 2 μ)
(hlim : ∀ N : ℕ, 0 < N → Tendsto (fun a =>
∫ ω, (X a N ω - Y N ω) ^ 2 ∂μ) l (𝓝 0)) (t : ℝ≥0) :
Tendsto (fun a => ∫ ω, (X a t ω - Y t ω) ^ 2 ∂μ) l (𝓝 0) := by
obtain ⟨N, hN⟩ := exists_nat_gt t
have hN0 : 0 < N := by
have hn : (0 : ℝ≥0) < (N : ℝ≥0) := lt_of_le_of_lt (show (0 : ℝ≥0) ≤ t from zero_le) hN
exact_mod_cast hn
apply squeeze_zero (fun a => integral_nonneg (fun ω => sq_nonneg _))
(fun a => ?_) (hlim N hN0)
exact MartingaleSquareMaximal.integral_sq_le ((hX a).sub hY)
(fun s => (hX2 a s).sub (hY2 s)) hN.le
theorem finiteSum_meanSquare_of_positive_integer [IsFiniteMeasure μ]
{ι : Type*} {M : ι → ℝ≥0 → Ω → ℝ} {Y : ℝ≥0 → Ω → ℝ}
(hM : ∀ i, Martingale (M i) F μ) (hY : Martingale Y F μ)
(hM2 : ∀ i t, MemLp (M i t) 2 μ) (hY2 : ∀ t, MemLp (Y t) 2 μ)
(hlim : ∀ N : ℕ, 0 < N → Tendsto (fun s : Finset ι =>
∫ ω, ((∑ i ∈ s, M i N ω) - Y N ω) ^ 2 ∂μ) atTop (𝓝 0))
(t : ℝ≥0) :
Tendsto (fun s : Finset ι =>
∫ ω, ((∑ i ∈ s, M i t ω) - Y t ω) ^ 2 ∂μ) atTop (𝓝 0) := by
exact meanSquare_limit_of_positive_integer
(fun s => MartingaleSpatialLimit.finiteSum_martingale hM s) hY
(fun s t => MartingaleSpatialLimit.finiteSum_memLp hM2 s t) hY2 hlim t
theorem hasSum_of_positive_integer [IsFiniteMeasure μ]
{ι : Type*} {M : ι → ℝ≥0 → Ω → ℝ} {Y : ℝ≥0 → Ω → ℝ}
(hM : ∀ i, Martingale (M i) F μ) (hY : Martingale Y F μ)
(hM2 : ∀ i t, MemLp (M i t) 2 μ) (hY2 : ∀ t, MemLp (Y t) 2 μ)
(hlim : ∀ N : ℕ, 0 < N → Tendsto (fun s : Finset ι =>
∫ ω, ((∑ i ∈ s, M i N ω) - Y N ω) ^ 2 ∂μ) atTop (𝓝 0))
(t : ℝ≥0) :
HasSum (fun i => (hM2 i t).toLp (M i t)) ((hY2 t).toLp (Y t)) := by
have hh := (SimpleIntegral.tendsto_toLp_iff_meanSquare
(fun s => MartingaleSpatialLimit.finiteSum_memLp hM2 s t) (hY2 t)).mpr
(finiteSum_meanSquare_of_positive_integer hM hY hM2 hY2 hlim t)
have he (s : Finset ι) :
(MartingaleSpatialLimit.finiteSum_memLp hM2 s t).toLp
(MartingaleSpatialLimit.finiteSum M s t) =
∑ i ∈ s, (hM2 i t).toLp (M i t) :=
PredictableL2.toLp_finsetSum (fun i => M i t) (fun i => hM2 i t) s
change Tendsto (fun s : Finset ι => ∑ i ∈ s, (hM2 i t).toLp (M i t))
atTop (𝓝 ((hY2 t).toLp (Y t)))
simpa only [he] using hh
theorem finiteSum_maximalSquare_of_positive_integer [IsFiniteMeasure μ]
{ι : Type*} {M : ι → ℝ≥0 → Ω → ℝ} {Y : ℝ≥0 → Ω → ℝ}
(hM : ∀ i, Martingale (M i) F μ) (hY : Martingale Y F μ)
(hM2 : ∀ i t, MemLp (M i t) 2 μ) (hY2 : ∀ t, MemLp (Y t) 2 μ)
(hcM : ∀ i, ∀ᵐ ω ∂μ, Continuous (fun t => M i t ω))
(hcY : ∀ᵐ ω ∂μ, Continuous (fun t => Y t ω))
(hlim : ∀ N : ℕ, 0 < N → Tendsto (fun s : Finset ι =>
∫ ω, ((∑ i ∈ s, M i N ω) - Y N ω) ^ 2 ∂μ) atTop (𝓝 0))
(T : ℝ≥0) :
Tendsto (fun s : Finset ι => ∫ ω,
MartingaleStrongMaximal.maximalSquare
(MartingaleSpatialLimit.finiteSum M s - Y) T ω ∂μ) atTop (𝓝 0) := by
have ht := finiteSum_meanSquare_of_positive_integer hM hY hM2 hY2 hlim T
apply squeeze_zero
(fun s => integral_nonneg (fun ω => ENNReal.toReal_nonneg))
(fun s => ?_)
(show Tendsto (fun s : Finset ι => 4 * ∫ ω,
((∑ i ∈ s, M i T ω) - Y T ω) ^ 2 ∂μ) atTop (𝓝 0) from by
simpa only [mul_zero] using ht.const_mul 4)
have hc : ∀ᵐ ω ∂μ,
Continuous (fun t => (MartingaleSpatialLimit.finiteSum M s - Y) t ω) := by
filter_upwards [MartingaleSpatialLimit.finiteSum_ae_continuous hcM s, hcY]
with ω hm hy
exact hm.sub hy
exact MartingaleStrongMaximal.integral_maximalSquare_le
((MartingaleSpatialLimit.finiteSum_martingale hM s).sub hY)
(fun t => (MartingaleSpatialLimit.finiteSum_memLp hM2 s t).sub (hY2 t)) hc T
end Problem367.MartingaleIntegerIdentification
end
end OAI
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