Theorems · Theorem · measure theory

MeasureTheory.taylor_charFun_two

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {X : Ω → ℝ},
  AEMeasurable X P →
    ∫ (x : Ω), X x ∂P = 0 →
      ∫ (x : Ω), (X ^ 2) x ∂P = 1 →
        (fun t => MeasureTheory.charFun (MeasureTheory.Measure.map X P) t - (1 - ↑t ^ 2 / 2)) =o[nhds 0] fun t => t ^ 2

Defined in

Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Cited by

1 results in Mathlib

Foundations

Depth 274 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsProbabilityMeasure