Theorems · Theorem · probability

ProbabilityTheory.tendsto_charFun_inv_sqrt_mul_pow

∀ {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 →
        ∀ (t : ℝ),
          Filter.Tendsto (fun n => MeasureTheory.charFun (MeasureTheory.Measure.map X P) ((√↑n)⁻¹ * t) ^ n) Filter.atTop
            (nhds (Complex.exp (-↑t ^ 2 / 2)))

Defined in

Mathlib.Probability.CentralLimitTheorem

Cited by

1 results in Mathlib

Foundations

Depth 279 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsProbabilityMeasure