The Lévy–Khintchine formula for stable exponents #
For 0 < α < 2 and |β| ≤ 1, let ν be the stable Lévy measure stableLevyMeasure α β, with
density Γ(1 + α) sin(πα/2) / π * ((1 + β) 1_{x > 0} + (1 - β) 1_{x < 0}) |x| ^ (-1 - α).
This file proves that for every real t,
∫ x, exp (i t x) - 1 - i t sin x ∂ν = stableCharacteristicExponent α β t,
so the sine-compensated Lévy integral is exactly the stable characteristic exponent Ψ_{α,β}
in Nolan's S₀ parameterization, including the logarithmic branch at α = 1 and both
full-skewness boundaries. Combined with the compensated compound-Poisson approximations, this
identifies the limit of their characteristic functions as exp (Ψ_{α,β} t).
Proof #
Reflection reduces the integral to the positive half-line
(TauCeti.Probability.integral_stableLevyMeasure_eq), where the weight x ^ (-(α + 1))
is written as a Gamma mixture of exponentials (TauCeti.integral_Ioi_rpow_neg_smul). The
Laplace transform of the compensated kernel is the rational function
(s - i t)⁻¹ - s⁻¹ - i t / (1 + s ^ 2)
(TauCeti.integral_Ioi_exp_neg_mul_smul_cexp_sub_one_sub_sin), and the remaining
power-weighted integrals reduce to ∫ y in Ioi 0, y ^ (s - 1) / (y ^ 2 + t ^ 2), evaluated by
Euler's reflection formula (TauCeti.integral_rpow_div_sq_add_sq). The imaginary part needs
three cases: below and above shape one the integrand splits into two such integrals, and at
shape one it has an elementary logarithmic primitive
(TauCeti.integral_mul_div_sq_add_sq_sub_div_one_add_sq).
References #
- J. P. Nolan, Univariate Stable Distributions: Models for Heavy Tailed Data, Springer, 2020, chapter 1.
- K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999, section 14.
The Lévy–Khintchine formula for stable exponents. For 0 < α < 2 and |β| ≤ 1,
integrating the sine-compensated characteristic kernel against the stable Lévy measure gives
the stable characteristic exponent in Nolan's S₀ parameterization.