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Let ℱ = {f(x;δ) = exp[xδ+q(δ)], δ ∈ (a,b)}
be a family of exponential-type probability density-functions (exp. p.d.f.’s) with
respect to a σ-finite measure μ. Let M(t;δ),a − δ < t < b − δ, denote the
moment generating function (m.g.f.) corresponding to f(x;δ) ∈ℱ, and let
c(t;δ) = lnM(t;δ) = ∑
k=1∞λk(δ)tk∕k! be the cumulative generating function. The
main results pertain to characterizations of certain exp. p.d.f.’s in terms of the
cumulants λk(δ). First, it is shown that if M(t;δ0) is the m.g.f., respectively, of a
degenerate, Poisson, or normal law for some δ0 ∈ (a,b), then M(t;δ) is the m.g.f. of
the given law for all δ ∈ (a,b), and that infinite divisibility (inf. div) of M(t;δ0)
for some δ0 implies inf. div. for all δ. Further, it is shown that if φ(t) is a
nondegenerate, inf. div. characteristic function (ch. f.) with finite fourth
cumulant λ4, then λ4 = 0 if and only if φ(t) is the ch.f. of a normal law, while if
λ4 = aλ3 = a2λ2≠0, then φ(t) is the ch. f. of a Poisson law. Combining these results,
it follows that if M(t;δ0) is inf. div., and nondegenerate, with λ4(δ0) = 0,
then M(t;δ) is the m.g.f. of a normal law for all δ ∈ (a,b). A similar result
characterizes the Poisson law. Finally, it is proved that the normal law is the unique
exp. p.d.f. which is symmetric.
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