Theorems · Theorem · probability

ProbabilityTheory.gaussian_charFun_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 : InnerProductSpace ℝ E]
  [MeasureTheory.IsFiniteMeasure μ] (m : E) (f : E →L[ℝ] E →L[ℝ] ℝ),
  f.toBilinForm.IsPosSemidef →
    (∀ (t : E), MeasureTheory.charFun μ t = Complex.exp (↑(inner ℝ t m) * Complex.I - ↑((f t) t) / 2)) →
      m = ∫ (x : E), x ∂μ ∧ f = ProbabilityTheory.covarianceBilin μ

If the characteristic function of μ takes the form of a gaussian characteristic function, then the parameters have to be the expectation and the covariance bilinear form.

Defined in

Mathlib.Probability.Distributions.Gaussian.CharFun

Cited by

1 results in Mathlib

Foundations

Depth 315 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupSecondCountableTopologyCompleteSpaceMeasurableSpaceBorelSpaceInnerProductSpaceMeasureTheory.IsFiniteMeasure