In this article we study the behaviour of translation invariant experiments. The main result establishes necessary and sufficient conditions for the local approximation of a sequence of suitably normalized product experiments by Gaussian shifts with respect to the weak convergence. Moreover, it turns out that the local asymptotically normal (LAN) condition with uniform remainders with respect to the local parameter can be characterized in terms of the local behavior of the Hellinger distances. This result depends on the fact that a convergent sequence En is always equicontinous if En is convergent. The result answers a question which has been posed in the author's article about "the convergence of almost regular statistical experiments to Gaussian shifts" (In Proceedings, 4th Pannonian Sympos. Math. Statist., Bad Tatzmannsdorf, 1983). The proofs rely on a theorem showing that binary experiments can be treated by positive definite functions.
