Theorems · Theorem · probability

ProbabilityTheory.IsGaussian.charFunDual_eq

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : MeasurableSpace E] [BorelSpace E]
  {μ : MeasureTheory.Measure E} [ProbabilityTheory.IsGaussian μ] (L : StrongDual ℝ E),
  MeasureTheory.charFunDual μ L =
    Complex.exp ((∫ (x : E), ↑(L x) ∂μ) * Complex.I - ↑(ProbabilityTheory.variance (⇑L) μ) / 2)

The characteristic function of a Gaussian measure μ has value exp (μ[L] * I - Var[L; μ] / 2) at L : Dual ℝ E.

Defined in

Mathlib.Probability.Distributions.Gaussian.Basic

Cited by

9 results in Mathlib

Foundations

Depth 295 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceProbabilityTheory.IsGaussian