Local invariance principle for independent and identically distributed random variables

It is well known that for a sequence of independent and identically distributed random variables, the corresponding normalized step-processes converge weakly to the Wiener process. A stronger convergence, namely the convergence in variation of the functional distributions of these processes, has been established in [Y. A. Davydov, M. A. Lifshits, and N. V. Smorodina, Local Properties of Distributions of Stochastic Functionals, American Mathematical Society, Providence, RI, 1998] under the finiteness of the Fisher information of the random variables. In this paper we prove such convergences without a Fisher information type condition.