Develop an exponential-type Tauberian equivalence

Determine whether asymptotic information of the form ∫_0^∞ λ^k e^{(l/λ)}e^{-λx}dG(x)→1 as λ→0^+ implies concrete asymptotic information about a monotone increasing function G, for arbitrary real parameters k and l, and establish corresponding converse conditions if possible.

Background

The domain-of-attraction theorem in the supercritical regime involves a Laplace-transform asymptotic with exponential rather than polynomial scaling. The authors recast the relevant condition into an exponential-type Tauberian form for a random monotone function G.

Unlike the Hardy–Littlewood–Karamata theorem used in the critical regime, standard regular-variation methods do not generally convert this exponential Laplace asymptotic into an asymptotic formula for G. The paper gives a counterexample showing that no unrestricted equivalence of the simplest expected form can hold, while noting that specialized Tauberian theorems may apply under additional assumptions. A general characterization remains unresolved and is identified as a problem relevant both to this BBM setting and to Abelian–Tauberian theory more broadly.

References

More precisely, the right question is: Let k, l ∈ ℝ. Is it possible to conclude some concrete information about the asymptotics of G if ∫_0∞ λk e{l/λ} e{-λ x} dG(x) → 1 as λ → 0+? To the best of our knowledge, such a question has not been resolved (even partially, that is, for specific values of k and l) in the literature, although there are some works in similar directions . Thus, we would like to formally pose this as an open question, and an answer to this will not only enhance our understanding in this specific context but also contribute significantly to the broader development of Abelian-Tauberian theory.

— Locally finite fixed points of branching Brownian motion  (2608.23202 - Chen et al., 24 Aug 2026) in Section 5, subsection “A discussion on Tauberian theorems,” and subsection “A counterexample to exponential type Tauberian equivalence”