  In the previous paper \cite{L-Z}, for a characteristic problem with not
necessarily small initial data given on a complete null cone decaying like that
in the work \cite{Ch-K} of the stability of Minkowski spacetime by
Christodoulou and Klainerman, we proved the local existence in retarded time,
which means the solution to the vacuum Einstein equations exists in a uniform
future neighborhood, while the global existence in retarded time is the weak
cosmic censorship conjecture. In this paper, we prove that the local existence
in retarded time still holds when the data is assumed to decay slower, like
that in Bieri's work \cite{Bie} on the extension to the stability of Minkowski
spacetime. Such decay guarantees the existence of the limit of the Hawking mass
on the initial null cone, when approaching to infinity, in an optimal way.
