Theorems · Theorem · probability

ProbabilityTheory.IsGaussian.charFunDual_eq_of_forall_strongDual_eq_zero

∀ {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), ∫ (x : E), L x ∂μ = 0) →
    ∀ (L : StrongDual ℝ E), MeasureTheory.charFunDual μ L = Complex.exp (-↑(ProbabilityTheory.variance (⇑L) μ) / 2)

Characteristic function of a centered Gaussian measure. For a Gaussian measure, the hypothesis ∀ L : StrongDual ℝ E, μ[L] = 0 is equivalent to the simpler μ[id] = 0, but at this point we don't know yet that μ has a first moment so we can't use it. See charFunDual_eq_of_integral_eq_zero

Defined in

Mathlib.Probability.Distributions.Gaussian.Fernique

Cited by

2 results in Mathlib

Foundations

Depth 296 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceProbabilityTheory.IsGaussian