On orbit dimensions under a simultaneous Lie group action on n copies of a manifold

We show that the maximal orbit dimension of a simultaneous Lie group action on n copies of a manifold does not pseudo-stabilize when n increases. We also show that if a Lie group action is (locally) effective on subsets of a manifold, then the induced Cartesian action is locally free on an open subset of a sufficiently big (but finite) number of copies of the manifold. The latter is the analogue for the Cartesian action to Ovsiannikov's theorem on jet spaces and is an important fact relative to the moving frame method and the computation of joint invariants. Some interesting corollaries are presented.