Theorems · Theorem · probability

ProbabilityTheory.hasGaussianLaw_iff_charFun_map_eq

∀ {Ω : Type u_1} {E : Type u_2} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [inst : NormedAddCommGroup E]
  [inst_1 : MeasurableSpace E] [BorelSpace E] {X : Ω → E} [CompleteSpace E] [inst_4 : InnerProductSpace ℝ E]
  [MeasureTheory.IsFiniteMeasure P],
  AEMeasurable X P →
    (ProbabilityTheory.HasGaussianLaw X P ↔
      ∀ (t : E),
        MeasureTheory.charFun (MeasureTheory.Measure.map X P) t =
          Complex.exp
            (↑(∫ (x : Ω), (fun ω => inner ℝ t (X ω)) x ∂P) * Complex.I -
              ↑(ProbabilityTheory.variance (fun ω => inner ℝ t (X ω)) P) / 2))

Defined in

Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic

Cited by

0 results in Mathlib

Foundations

Depth 299 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupMeasurableSpaceBorelSpaceCompleteSpaceInnerProductSpaceMeasureTheory.IsFiniteMeasure