  In recent years, the theory of superstatistics, which aims to describe
non-equilibrium steady state systems, has gained attention due to its different
real world applications, highlighting its versatility and concise mathematical
formulation in terms of a probability density for the inverse temperature
$\beta=1/k_{B}T$. When exploring the domain of application of the
superstatistical theory, recent works have shown some necessary conditions for
a superstatistical description of a given steady state, in terms of the
fundamental and microcanonical inverse temperature. In this work, a new theorem
that establishes a sufficient condition for the existence of a superstatistical
description of a particular steady state is presented, using the language of
moment-generating functions and connecting them with properties of the
derivatives of the fundamental inverse temperature.
