Best,

Jacob

I think there is a flaw in your proof of Lemma 5.

The “\lambda^T x” should be “\lambda^T \phi(x)”

i.e. it is not moment generating function of X, but of the sufficient statistics \phi(x).

Thus, the following exercise works only because there is a component \phi(x) = x in Gaussian case, and the straightforward computation is still exhausting (if by hand), it is the 6-th derivative of exp(f(t)), where f(t) is a 2rd order polynomial about t.

]]>1. Be explicit rather than implicit.

2. All parts of the paper which are high visibility points such as theorem statement, figures, tables, algorithm should be self-sufficient. A reader should be able to look at these things and be able to get what is happening in the paper. ]]>

A corollary of the latter is that any operator monotone function like x^{-1} or x^{1/2] has to be increasing, smooth, and concave.

]]>Best,

Jacob

I was just reading this post by Terrance Tao https://terrytao.wordpress.com/2017/05/22/quantitative-continuity-estimates/

and the last few paragraphs reminded me of the first fact stated here. My gut feeling is the Sylvester equation based approach might be helpful in establishing true/false conclusion of the ‘In general…’ question you asked in the end.