| import OAI.Probability.FreeIsing.Martingales.SpatialLimit |
| import OAI.Probability.FreeIsing.Stochastic.IntegralAssociativityLimit |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter |
| open scoped NNReal Topology |
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| namespace Problem367.MartingaleIntegerIdentification |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {μ : Measure Ω} {F : Filtration ℝ≥0 mΩ} |
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| 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)) |
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|
| 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 |
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| 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 |
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| 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 |
|
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| 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 |
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|
| end Problem367.MartingaleIntegerIdentification |
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| end |
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| end OAI |
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