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1 The Lévy-Khinchine Formula Given the characterization of linear functionals that satisfy the minimum principle and are quasi-local, it is quite easy to derive the Lévy-Khinchine formula. Let µ ∈ M1 (RN ) be infinitely devisable in the sence that, for each n ≥ 1, there is a µ n1 ∈ M1 (RN ) such that µ = µ∗n 1 . To prove that there is a Lévy system n (m, C, M ) such that µ̂ = e` where Z `(ξ) = i(m, ξ)RN − 12 (ξ, Cξ)RN + ei(ξ,y)RN − 1 − i1B(0,1) (y)(ξ, y)RN M (dy), RN it suffices to show that (i) µ̂ never vanishes (ii) If ` ∈ C(RN ; C) is determined by `(0) = 1 and µ̂ = e` , then |`(ξ)| ≤ C 1 + |ξ|2 for some C < ∞. Indeed, (i) guarantees that the ` described in (ii) exists and is unique. Further, because µ̂ = (µcn1 )n , (i) guarantees that µcn1 never vanishes, and clearly n1 ` is the `1 unique choice of ` n1 ∈ C(RN ; C) such that ` n1 (0) = 0 and µcn1 = e n . Hence, using (ii) and Parseval’s Identity, one sees, that Z −N Aϕ ≡ lim n hϕ, µ n1 i − ϕ(0) = (2π) `(−ξ)ϕ̂(ξ) dξ n→∞ RN for S (RN ; C). Obviously A satisfies the minimum principle. In addition, if ϕ ∈ S (RN ; C), then Z Z N N (2π) AϕR = R `(−ξ)ϕ̂(Rξ) dξ = `(−R−1 ξ)ϕ̂(ξ) dξ −→ 0 as R → ∞. RN RN Thus A is quasi-local. As a consequence, there exists a (m, C, M ) such that Z −N Aϕ = (2π) i(m, ξ)RN − 12 (ξ, Cξ)RN N R Z i(ξ,y)RN N + − 1 − i1B(0,1) (y)(ξ, y)R M (y) ϕ̂(ξ) dξ. e RN To verify that (i) and (ii) hold, begin by observing that Z |1 − µ̂(ξ)| ≥ Re 1 − µ̂(ξ) = 1 − cos(ξ, y) µ(dy), RN and therefore that, for any e ∈ SN −1 , Z Z r Z 1 sin r(e, y)RN 1 − cos t(ξ, y) dt µ(dy) = 1− µ(dy). sup |1−µ̂(ξ)| ≥ r RN r(e, y)RN |ξ|≤r 0 RN 2 Hence, if s(T ) ≡ inf |t|≥T 1 − R > 0, sin r(e,y)RN r(e,y)RN for T > 0, then, s > 0 and, for any sup |1 − µ̂(ξ)| ≥ s(rR)µ {y : |(e, y)RN | ≥ R} . (*) |ξ|≤r Now choose r > 0 so that sup|ξ|≤r |1 − µ̂(ξ)| ≤ 12 . Then, since |1 − z| ≤ | log z| ≤ 2|1 − z|, |`(ξ)| ≤ 1, and so 1 2 =⇒ `(ξ) 1 |1 − µcn1 (ξ)| = 1 − e n ≤ n for |ξ| ≤ r. Applying (*) with µ replaced by µ n1 and using this estimate, we have that, for any R > 0, 1 ≥ s(rR)µ n1 {y : |(e, y)RN | ≥ R} , n which, because |1 − µcn1 (ρe)| ≤ ρR + 2µ n1 {y : |(e, y)RN | ≥ R} , means that sup |1 − µcn1 (ξ)| ≤ ρR + |ξ|≤ρ In particularly, by taking R = 1 4ρ , 2 ns(rR) for all (ρ, R) ∈ (0, ∞). we get sup |1 − µcn1 (ξ)| ≤ |ξ|≤ρ 1 2 + r , 4 ns 4ρ and so, for all ρ > 0, (**) sup |1 − µcn1 (ξ)| ≤ |ξ|≤ρ 1 2 if n ≥ 8 s r 4ρ . From (**), it is clear that, for each ρ > 0, there is an n such that |µcn1 (ξ)| ≥ 12 and therefore |µ̂(ξ)| ≥ 2−n for |ξ| ≤ ρ, and this completes the proof of (i). Finally, from (**), |`(ξ)| ≤ n if |ξ| ≤ ρ and n ≥ 8r . Since there is an such that s 4ρ s(T ) ≥ T 2 for T ∈ (0, 1], it follows that |`(ξ)| ≤ 1 + 128|ξ|2 . r2