Existence of moments and an asymptotic result based on a mixture of exponential distributions

On the small-time behavior of subordinators

“…Remark 1. The implication "(S6) for some γ > 0 ⇒ (S1)" was already shown in [1]. as t tends to zero is the somewhat uninteresting limit 1, which is excluded from Theorem 3.1.…”

“…Let F t (y) = P(Y t ≤ y) and ψ t (u) = E(e −uYt ) be the distribution function and the Laplace-Stieltjes transform (LST) of Y t and let d → denote convergence in distribution. We start with the following observation from [1], which is not difficult to prove. It states that the convergence of Y −t → Y * as t → 0 if and only if ψ t (u 1/t ) → 1 − F * (u) as t → 0 at all continuity points u of F * .…”

Weak convergence of subordinators to extremal processes

“…It was shown in [1] that if (X t ) t>0 is a family of positive random variables and if X is a non-constant random variable with distribution function F , then X −t t converges weakly to X as t → 0 if and only if ψ t (u 1/t ) → 1 − F (u) as t → 0 at all continuity points u of F , where ψ t is the Laplace transform of X t . In [2] it was found that for the convolution family ψ t (u) = ϕ(u) t , where ϕ is the Laplace transform of an infinitely divisible random variable, i.e.…”

The limiting behavior of some infinitely divisible exponential dispersion models

“…Accordingly, their Theorem 1 suggests a consideration of g t (Y t ) = Y −t t (or, equivalently, of −t ln Y t ) whose limiting distribution is non-degenerate (provided that (1) is satisfied). Bar-Lev and Enis presented several examples which satisfy (1). However, these examples heavily depend on the explicit (and relatively 'nice') form of L t .…”