Theorems · Theorem · probability

ProbabilityTheory.gaussian_charFunDual_congr

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [SecondCountableTopology E] [CompleteSpace E]
  [inst_3 : MeasurableSpace E] [inst_4 : BorelSpace E] {μ : MeasureTheory.Measure E} [inst_5 : NormedSpace ℝ E]
  [MeasureTheory.IsFiniteMeasure μ] {m : E} {f : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ},
  f.toBilinForm.IsPosSemidef →
    (∀ (L : StrongDual ℝ E), MeasureTheory.charFunDual μ L = Complex.exp (↑(L m) * Complex.I - ↑((f L) L) / 2)) →
      m = ∫ (x : E), x ∂μ ∧ f = ProbabilityTheory.covarianceBilinDual μ

Defined in

Mathlib.Probability.Distributions.Gaussian.CharFun

Cited by

1 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupSecondCountableTopologyCompleteSpaceMeasurableSpaceBorelSpaceNormedSpaceMeasureTheory.IsFiniteMeasure