Infinitely divisible laws and regular variation

Infinitely divisible and stable distributions, convolution powers, the Lévy–Khintchin representation, and regularly varying and slowly varying functions (Karamata's theorem, Potter bounds, generalized inverses) with the truncated moments and tail integrals of heavy-tailed laws.

• Infinitely divisible distributions
• Regular variation
• Stable distributions and domains of attraction
• Truncated moments of regularly varying tails

TheoremLévy–Khintchin representation on the real line#✓fully proved

A probability measure \(\mu\) on \(\mathbb{R}\) is infinitely divisible, meaning that for every integer \(n \ge 1\) there is a probability measure \(\rho\) on \(\mathbb{R}\) whose \(n\)-fold convolution power \(\rho^{\ast n}\) equals \(\mu\), if and only if there are real numbers \(\gamma\) and \(\sigma \ge 0\) and a measure \(\nu\) on \(\mathbb{R}\) with \(\nu(\{0\}) = 0\) and \(\int \min(1, x^2)\, d\nu(x) < \infty\) such that the characteristic function \(\hat\mu\) of \(\mu\) satisfies, for every \(t \in \mathbb{R}\),

\[ \hat\mu(t) = \exp\Big(i \gamma t - \frac{\sigma^2 t^2}{2} + \int_{\mathbb{R}} \big(e^{itx} - 1 - itx \, \mathbf{1}_{\{|x| \le 1\}}(x)\big) \, d\nu(x)\Big). \]

Proof: The reverse implication uses compound Poisson approximation plus Lévy continuity; the forward implication uses a distinguished logarithm, canonical measures, and Prokhorov compactness. For the equivalence, see William Feller, An Introduction to Probability Theory and Its Applications, vol. II, Theorem XVII.2.1.

theorem levy_Khintchin :
    ∀ (μ : MeasureTheory.Measure ℝ), MeasureTheory.IsProbabilityMeasure μ →
      (MeasureTheory.Measure.IsInfinitelyDivisible μ ↔
        ∃ γ : ℝ, ∃ σ : ℝ, 0 ≤ σ ∧
          ∃ ν : MeasureTheory.Measure ℝ,
            ν {0} = 0 ∧ (∫⁻ x, ENNReal.ofReal (min 1 (x ^ 2)) ∂ν < ∞) ∧
              ∀ t : ℝ, MeasureTheory.charFun μ t =
                Complex.exp ((Complex.I * (γ : ℂ) * (t : ℂ) -
                  (((σ ^ 2 * t ^ 2 : ℝ) : ℂ) / 2) +
                  ∫ x, (Complex.exp (Complex.I * (t : ℂ) * (x : ℂ)) - 1 -
                    Set.indicator {y : ℝ | |y| ≤ 1}
                      (fun y => Complex.I * (t : ℂ) * (y : ℂ)) x) ∂ν)))

Sources

• Ken-iti Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics 68, Cambridge University Press (1999), Theorem 8.1, pp. 37–38, ISBN 978-0-521-55302-5. https://books.google.com/books?vid=ISBN9780521553025&jscmd=SearchWithinVolume2&q=Theorem+8.1 Scope: restricts Sato Theorem 8.1 (stated for ℝ^d) to ℝ, and formalizes only the existence direction (i), omitting uniqueness of the generating triplet (ii). The jump integrand uses the hard truncation x * 1_{|x| ≤ 1}, which is an equivalent convention to Sato's x / (1 + |x|^2) after reparameterizing the drift γ.

Uses 5

Statement usesInfinitely divisible measures

Proof usesConvolution power of a measure · Convolution powers as iterated left convolution · Infinite divisibility through iterated convolution · Convolution powers of a probability measure are probability measures

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