  In this paper we prove that, for asymptotically bounded holomorphic functions
defined in a polysector in ${\mathbb C}^n$, the existence of a strong
asymptotic expansion in Majima's sense following a single multidirection
towards the vertex entails (global) asymptotic expansion in the whole
polysector. Moreover, we specialize this result for Gevrey strong asymptotic
expansions. This is a generalization of a result proved by A. Fruchard and C.
Zhang for asymptotic expansions in one variable, but the proof, mainly in the
Gevrey case, involves different techniques of a functional-analytic nature.
