Documentation

BrownianMotion.Gaussian.ProjectiveLimit

Pre-Brownian motion as a projective limit #

theorem L2.posSemidef_interMatrix {ι : Type u_1} [Fintype ι] {α : Type u_2} { : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : ιSet α} (hv₁ : ∀ (j : ι), MeasurableSet (v j)) (hv₂ : ∀ (j : ι), μ (v j) := by finiteness) :
(Matrix.of fun (i j : ι) => μ.real (v i v j)).PosSemidef
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      theorem ProbabilityTheory.covariance_eval_gaussianProjectiveFamily {I : Finset NNReal} (s t : { x : NNReal // x I }) :
      covariance (fun (x : { x : NNReal // x I }) => x s) (fun (x : { x : NNReal // x I }) => x t) (gaussianProjectiveFamily I) = (min s t)
      theorem ProbabilityTheory.measurePreserving_gaussianProjectiveFamily₂ {I : Finset NNReal} {s t : { x : NNReal // x I }} :
      MeasureTheory.MeasurePreserving ((fun (x : { x : NNReal // x I }) => x s) - fun (x : { x : NNReal // x I }) => x t) (gaussianProjectiveFamily I) (gaussianReal 0 (max (s - t) (t - s)))
      theorem MeasureTheory.IsProjectiveLimit.measurePreserving_restrict {ι : Type u_2} {X : ιType u_3} {mX : (i : ι) → MeasurableSpace (X i)} {μ : Measure ((i : ι) → X i)} {P : (I : Finset ι) → Measure ((i : { x : ι // x I }) → X i)} (h : IsProjectiveLimit μ P) {I : Finset ι} :