Monotone invariants and embeddings of statistical manifolds

In this note we prove certain necessary and sufficient conditions for the existence of an embedding of statistical manifolds. In particular, we prove that any compact smooth ($C^1$ resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get an answer to the Lauritzen question on a realization of smooth ($C^1$ resp.) statistical manifolds as statistical models