Theorems · Theorem · probability

ProbabilityTheory.isGaussian_iff_gaussian_charFun

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

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

Defined in

Mathlib.Probability.Distributions.Gaussian.CharFun

Cited by

0 results in Mathlib

Foundations

Depth 314 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupSecondCountableTopologyCompleteSpaceMeasurableSpaceBorelSpaceInnerProductSpaceMeasureTheory.IsFiniteMeasure