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We introduce the concept of asymptotic stability for a set of complex functions analytic around the origin, implicitly contained in an earlier paper of the first mentioned author (“Finite group actions and asymptotic expansion of $e^{P\left(z\right)}$", Combinatorica 17 (1997), 523 – 554). As a consequence of our main result we find that the collection of entire functions $exp\left(𝔓\right)$ with $𝔓$ the set of all real polynomials $P\left(z\right)$ satisfying Hayman’s condition $\left[z^n\right]exp\left(P\left(z\right)\right)>0(n\ge n_0)$ is asymptotically stable. This answers a question raised in loc. cit.