Theorems · Theorem · probability

ProbabilityTheory.IsGaussianProcess.of_isGaussianProcess

∀ {S : Type u_1} {T : Type u_2} {Ω : Type u_3} {E : Type u_4} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  {X : T → Ω → E} [inst : NormedAddCommGroup E] [inst_1 : MeasurableSpace E] [BorelSpace E] [inst_3 : NormedSpace ℝ E]
  [SecondCountableTopology E] {F : Type u_6} [inst_5 : NormedAddCommGroup F] [inst_6 : NormedSpace ℝ F]
  [inst_7 : MeasurableSpace F] [BorelSpace F] [SecondCountableTopology F] {Y : S → Ω → F},
  ProbabilityTheory.IsGaussianProcess X P →
    (∀ (s : S), ∃ I L, ∀ (ω : Ω), Y s ω = L (I.restrict fun x => X x ω)) → ProbabilityTheory.IsGaussianProcess Y P

If a stochastic process Y is such that for each s, Y s can be written as a linear combination of finitely many values of a Gaussian process, then Y is a Gaussian process.

Defined in

Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic

Cited by

4 results in Mathlib

Foundations

Depth 308 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupMeasurableSpaceBorelSpaceNormedSpaceSecondCountableTopologyNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceSecondCountableTopology