[Paper Review] The converse to Curtiss' theorem for one-sided moment generating functions

Patrick Chareka|ArXiv.org|Jul 22, 2008

Mathematical functions and polynomials8 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for the convergence of one-sided moment generating functions (MGFs) to a limiting MGF on an interval (a,b) not necessarily containing zero. Using weak convergence of distribution functions and uniform integrability, it proves that pointwise convergence of MGFs follows if the MGFs are uniformly bounded and the underlying distributions converge weakly to a distribution with a valid MGF on (a,b). The key contribution is a converse to Curtiss’ theorem for one-sided MGFs, generalizing prior results to intervals excluding zero.

ABSTRACT

A set of necessary and sufficient conditions for a sequence of moment generating functions to converge to a moment generating function on an interval (a,b) not necessarily containing 0, is given. The result is derived using recent results by Mukherjea, et al. (2006) and Chareka (2007).

Motivation & Objective

• To generalize Curtiss’ theorem to one-sided moment generating functions by establishing a converse for intervals (a,b) not necessarily containing zero.
• To address the limitation of prior results that required the interval to include zero, which restricts applicability in cases like the lognormal and Fréchet distributions.
• To provide verifiable, practical conditions—specifically uniform boundedness and weak convergence—for MGF convergence in one-sided settings.
• To simplify proofs of limit theorems (e.g., central limit theorem) by enabling convergence arguments without requiring MGF existence near zero.

Proposed method

• Uses the uniqueness of distributional determination by MGFs on an interval (Chareka, 2007) to ensure that a given MGF corresponds to a unique distribution.
• Applies weak convergence of distribution functions {Fn} to F, ensuring that the limiting distribution has an MGF M(t) on (a,b).
• Imposes uniform boundedness of {Mn(t)} on (a,b) to control the behavior of the sequence of MGFs.
• Employs the dominated convergence theorem and properties of continuous transformations to justify interchange of limit and expectation in E[etXn] → E[etX].
• Leverages uniform convergence of distribution functions to continuous limits (via Prohorov’s theorem) to ensure convergence of integrals involving the survival function.
• Applies the identity E[etX] = ∫₀^∞ (1 − G(t,x)) dx for positive random variables, where G(t,x) = P(etX ≤ x), to express MGFs as integrals of survival functions.

Experimental results

Research questions

• RQ1 Under what conditions does a sequence of one-sided moment generating functions converge to a limiting MGF on an interval (a,b) not containing zero?
• RQ2 Can the converse of Curtiss’ theorem be established without requiring the interval to include zero?
• RQ3 How can uniform boundedness of MGFs and weak convergence of distributions be used to ensure convergence of MGFs in one-sided settings?
• RQ4 Is it possible to avoid verifying uniform boundedness of MGFs under additional regularity conditions, such as continuity of the limiting distribution?
• RQ5 How does the convergence of MGFs relate to the convergence of the underlying distributions in the absence of zero in the interval?

Key findings

• A sequence of MGFs {Mn(t)} converges pointwise to a limiting MGF M(t) on (a,b) if and only if {Mn(t)} is uniformly bounded on (a,b) and the corresponding distribution functions {Fn(x)} converge weakly to a distribution F(x) with MGF M(t) on (a,b).
• The uniform boundedness condition (supn Mn(t) < ∞ for all t ∈ (a,b)) is both necessary and sufficient when combined with weak convergence of distributions.
• The result generalizes Kozakiewicz’s conditions for convergence on intervals containing zero to intervals not necessarily containing zero.
• For continuous distributions, if {Fn} converges weakly to a continuous F and Mn(t) exists for all t ∈ (a,b), then Mn(t) → M(t) for all t ∈ (a,b), making the boundedness condition redundant in this case.
• The theorem applies to the Fréchet and lognormal distributions, where MGFs exist only for t ≤ 0, and provides a rigorous convergence framework for such one-sided cases.
• The central limit theorem for i.i.d. samples can be re-derived via this converse: if Mn(t) = E[etYn] exists for all t ∈ (a,b), then Mn(t) → exp(−t²/2), the standard normal MGF.