Extensions to Basu's theorem, factorizations, and infinite divisibility

We define a notion of approximate sufficiency and approximate ancillarity and show that such statistics are approximately independent pointwise under each value of the parameter. We do so without mentioning the somewhat nonintuitive concept of completeness, thus providing a more transparent version of Basu's theorem. Two total variation inequalities are given, which we call approximate Basu theorems.We also show some new types of applications of Basu's theorem in the theory of probability. The applications are to showing that large classes of random variables are infinitely divisible (id), and that others admit a decomposition in the form YZ, where Y is infinitely divisible, Z is not, both are nondegenerate, and Y and Z are independent.These applications indicate that the possible spectrum of applications of Basu's theorem is much broader than has been realized.