Documentation

Lean 4 Proof

theorem cumulant3_is_derivative_of_variance [NeZero J]
    (x : Fin J → ℝ) (hx : ∀ j, 0 < x j) (ρ : ℝ) :
    HasDerivAt (fun p => escortCumulant2 x p)
      (escortCumulant3 x ρ) ρ := by
  unfold escortCumulant2 escortCumulant3
  -- Get the moment recursion for M_2 and M_1
  have hM2 := escortRawMoment_hasDerivAt x hx ρ 2
  have hM1 := escortRawMoment_hasDerivAt x hx ρ 1
  -- Derivative of M_1^2 via chain rule
  have hM1sq := hM1.pow 2
  -- Combine: d/drho (M_2 - M_1^2) = dM_2 - d(M_1^2)
  have h := hM2.sub hM1sq
  refine h.congr_deriv ?_
  simp only [Nat.cast_ofNat]
  ring

Dependency Graph

Used by

Location

thesis/CESProofs/Foundations/CumulantTower.lean:231

Module Section

The Cumulant Tower: Higher-Order Bridges Between CES and Escort Statistics

In the same file