| import OAI.Probability.FreeIsing.Estimates.CompensatedSquareLimit |
| import OAI.Probability.FreeIsing.Estimates.NonnegativeSeriesL1 |
| import OAI.Probability.FreeIsing.Martingales.PredictableL2Completion |
| import OAI.Probability.FreeIsing.Stochastic.StoppedBrownianArraySquare |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
|
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| namespace Problem367.MartingaleSpatialCompensator |
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| variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] |
| {μ : Measure Ω} [IsFiniteMeasure μ] {F : Filtration ℝ≥0 mΩ} |
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| theorem spatial_sum_compensatedSquare_martingale [Countable ι] |
| {X A : ι → ℝ≥0 → Ω → ℝ} {Y : ℝ≥0 → Ω → ℝ} |
| (hX2 : ∀ i t, MemLp (X i t) 2 μ) |
| (hdiag : ∀ i, Martingale (fun t ω => X i t ω ^ 2 - A i t ω) F μ) |
| (hcross : Pairwise fun i j => Martingale (fun t ω => X i t ω * X j t ω) F μ) |
| (hAad : ∀ i, StronglyAdapted F (A i)) |
| (hAint : ∀ i t, Integrable (A i t) μ) |
| (hAnonneg : ∀ i t ω, 0 ≤ A i t ω) |
| (hAsum : ∀ t, Summable (fun i => ∫ ω, A i t ω ∂μ)) |
| (hYad : StronglyAdapted F Y) (hY2 : ∀ t, MemLp (Y t) 2 μ) |
| (hYsum : ∀ t, HasSum (fun i => (hX2 i t).toLp (X i t)) ((hY2 t).toLp (Y t))) : |
| Martingale (fun t ω => Y t ω ^ 2 - ∑' i, A i t ω) F μ := by |
| classical |
| apply CompensatedSquareLimit.martingale_compensatedSquare_of_limits |
| (l := (atTop : Filter (Finset ι))) |
| (X := fun s t ω => ∑ i ∈ s, X i t ω) |
| (A := fun s t ω => ∑ i ∈ s, A i t ω) |
| · intro s t |
| exact memLp_finsetSum s (fun i _ => hX2 i t) |
| · intro s t |
| exact integrable_finsetSum s (fun i _ => hAint i t) |
| · exact StoppedBrownianArraySquare.finite_sum_compensated_square_martingale X A hdiag hcross |
| · exact hYad |
| · exact hY2 |
| · intro t |
| let : MeasurableSpace Ω := F t |
| exact StronglyMeasurable.tsum (fun i => hAad i t) |
| · intro t |
| exact (NonnegativeSeriesL1.integrable_tsum_of_nonnegative |
| (fun i => hAint i t) (fun i ω => hAnonneg i t ω) (hAsum t)).2.1 |
| · intro t |
| apply CompensatedSquareLimit.tendsto_eLpNorm_two_of_meanSquare |
| (fun s => memLp_finsetSum s (fun i _ => hX2 i t)) (hY2 t) |
| exact PredictableL2.tendsto_integral_sq_error_of_hasSum |
| (fun i => X i t) (fun i => hX2 i t) (hY2 t) (hYsum t) |
| · intro t |
| exact NonnegativeSeriesL1.tendsto_finsetSum_L1_of_nonnegative |
| (fun i => hAint i t) (fun i ω => hAnonneg i t ω) (hAsum t) |
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| end Problem367.MartingaleSpatialCompensator |
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| end |
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| end OAI |
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