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Jensen's inequality gives a lower bound, but it might be too trivial for your needs. For comparing your expectation with the independent case, one can use the method of cluster expansions in statistic …

The answer for vectors uses the theory of multi-symmetric polynomials and in particular power sums. You can learn about that in Emmanuel Briand's thesis mentioned in myprevious MO answer Generalizing …

As far as references for these things, have a look at this webpage for lecture notes by Martin Hairer. In particular, the lectures on "Ergodic properties of Markov processes" are I think exactly what …

I think your reformulation of the conjecture in the Rota-Shen paper is correct. Also your counterexample is correct. So I guess the conjecture was not stated properly in that article. Perhaps one shou …

In general, you need the covariance to be smooth, especially near the diagonal. For some particular covariances there are very quick ways to prove what you want: for example if you can rewrite the fie …

The result is true if the pair of random variables $(X,Y)$ is an FKG system: see Lecture 8 from the course given by David Brydges at the 2009 PIMS Probability Summer School. For example if the joint l …

In addition to the "canonical" references mentioned above, the book "A Basic Course in Probability Theory" by Bhattacharya and Waymire has a very nice treatment of this topic in Chapter 5.

This is easy to compute using techniques from perturbative quantum field theory, i.e., Wick's Theorem (due to Isserlis) for moments of Gaussian measures. Consider $$ (2\pi)^{-\frac{p}{2}}\int_{\mathbb …

Indeed, as J.C. said this has to do with the renormalization group (RG) which in the present context is a transformation $\mu\rightarrow \mu\ast\mu$ followed by rescaling by $\sqrt{2}$ to keep the var …

There is a book on the subject: "Information Theory and The Central Limit Theorem" by Oliver Johnson. The article by Anshelevich mentioned by Yemon considers the operator $T$ acting on probability den …

Just a small bibliographical complement to the very nice answers already given. The identity of the two $\sigma$-algebras is proven in Proposition 2.1 of the article An investigation of the propertie …

It seems you may want to turn the large $k$ property to your advantage and use a Laplace-type asymptotic method with controlled bounds. For fixed $d$ you may find some useful tools in the book "Analyt …

The following (between the horizontal lines) is a statement of what one may call the abstract change of variable theorem. It is taken from some exercise I gave my students a while ago: Let $(X,\mat …

The source of the confusion is not saying explicitly what are the sets and $\sigma$-algebras the measures are supposed to be on. For example, a sentence like ''By Kolmogorov's Extension Theorem, there …

I don't have time to go back and read exactly the definitions, but I suspect the issue here is that Brydges explicitly writes the dependence on the supporting sets, but not the dependence on the field …