Theorems · Theorem · probability

ProbabilityTheory.isGaussian_iff_gaussian_charFunDual

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [SecondCountableTopology E] [CompleteSpace E]
  [inst_3 : MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [inst_5 : NormedSpace ℝ E]
  [MeasureTheory.IsFiniteMeasure μ],
  ProbabilityTheory.IsGaussian μ ↔
    ∃ m f,
      f.toBilinForm.IsPosSemidef ∧
        ∀ (L : StrongDual ℝ E), MeasureTheory.charFunDual μ L = Complex.exp (↑(L m) * Complex.I - ↑((f L) L) / 2)

The measure μ is Gaussian if and only if there exist m : E and f : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ satisfying f.toBilinForm.IsPosSemidef and charFunDual μ L = exp (L m * I - f L L / 2).

Defined in

Mathlib.Probability.Distributions.Gaussian.CharFun

Cited by

4 results in Mathlib

Foundations

Depth 313 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupSecondCountableTopologyCompleteSpaceMeasurableSpaceBorelSpaceNormedSpaceMeasureTheory.IsFiniteMeasure