Theorem 10.2.1. in the book 'The methods of distances in the theory of probability and statistics' by Svetlozar T.Rachev et al. states that, for given non-degenerate distribution function on the real line with existing moments of each order and 'a suitable moment condition', the weak convergence of a chain of distribution functions is equivalent to the uniform convergence of the corresponding characteristic functions in some interval around zero. The prove of this statement is omitted.
With the motivation of the multi-dimensional extension of this theorem in mind, I have two questions:
Is it true that the uniform convergence in an interval around zero of a chain of analytic characteristic function implies point wise convergence of the chain in the whole domain?
Is the condition on the random vectors to be compactly supported enough to extend the theorem to the multi-dimensional setting?
I have tried to prove 1 in order to apply Levy's convergence theorem in the multi-dimensional setting but with minor success. Any publicly available reference would be helpful.