Unimodality of the Lévy spectral function

Infinitely divisible distributions similar to class L distributions

“…A similiar result is proven in [1]. The arguments used there need only be slightly modified to prove the above lemma.…”

“…Because the L6vy representation of ~b is unique, special properties of 4) or its distribution can frequently be expressed in terms of its L6vy spectral function and vice versa. In recent work (Alf-O'Connor [1], O'Connor [5]), we have identified those infinitely divisible characteristic functions whose L6vy spectral function is convex on (-m, 0) and is concave on (0, + oo). If U stands for this class, then q5 e U if and only if a version of the derivative dM(x) exists for which sgnxdM(x) does not increase on (-o%0) and on (0, + oo).…”

“…1) then it is possible to write 0 as 0(u) = lim I] f~k(U) where {f~k: n = 1, 2 .... , n~oo k=l k = 1,..., n} is a u.a.n, array of characteristic functions. Motivated by Theorem 1, we now construct an array {f,k} which has 0 as its limit law.…”

α-selfdecomposable distributions and related Ornstein–Uhlenbeck type processes

“…By virtue of the properties of ℓ ξ (r ), ℓ ξ (∞) := lim r →∞ ℓ ξ (r ) ≥ 0 exists and is measurable in ξ . Defining Lévy measures ν (1) and ν (2) by (1) ,γ ) and µ (2) = µ (2) (0,ν (2) ,0) , we have that µ = µ (1) * µ (2) and that µ (2) is α-stable due to Theorem 14.3 of [22]. Then, in order to prove this proposition, it suffices to show that lim n→∞  µ (1) …”

“…Alf and O'Connor [1] and O'Connor [18] investigated the class of all infinitely divisible distributions on R with unimodal Lévy measures with mode 0, and showed that the class is equal to L ⟨−1⟩ (R). As to this class, Alf and O'Connor [1] studied stochastic integral characterizations with respect to Lévy processes. O'Connor [18] studied the decomposability (1.7) for d = 1 and α = −1, and characterized this class by some limit theorem.…”