We analyze quasiasymptotic boundedness of distributions and their wavelet transforms, in general, as well as for a class of $\alpha-$ exponentially bounded distributions and their wavelet transforms in particular. The main idea of this paper is to use, instead of the quasiasymptotic behaviour, the notion of quasiasymptotic boundedness. In this way we obtain new Abelian type theorems for the wavelet transform of distributions with different growth.