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Cumulants have many other names depending on the context (statistics, quantum field theory, statistical mechanics,...): seminvariants, truncated correlation functions, connected correlation functions, …

Maybe the no1 problem of probability is to make rigorous what one finds in just about any textbook in statistical mechanics. In other words it is to put the predictions of Wilson's renormalization gro …

The result is $$ I(k)=\frac{2\pi^4\ \Gamma\left(\frac{k}{2}+1\right)\Gamma\left(\frac{k}{2}+2\right)}{\Gamma(k+4)}\ , $$ which can be simplified a bit more using the Legendre Duplication Formula. More …

The framing of your question is a bit ambiguous and perhaps there are two different questions here depending on the context and interpretation. One could approach your question from the point of view …

Dec 2017 edit: I just put a hopefully useful detailed complement to this post on https://physics.stackexchange.com/questions/372306/wilsonian-definition-of-renormalizability The answer to your ques …

Let me comment on points 4) and 5). The problem with infinities in QFT or traditional equilibrium statistical field theory is related to the one addressed by Martin's theory but there are some differe …

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 …

This is easy to do using a graphical calculus for contractions of old-fashioned tensors. See this recent article for an example of application of such techniques and hopefully useful references. Here …

It's not clear from the post if you are talking about classical or quantum field theories. In QFT, conformal invariance implies scale invariance. If the theory has a mass $m$ then, as John explained, …

I think the result goes back to Polya, see "Some theorems on stable processes" by Blumenthal and Getoor. Another reference is Paul Lévy "Sur une application de la dérivée d’ordre non entier au calcul …

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 …

My answer Wiener measure and Bochner Minlos can help defining a free massless real scalar field as a random element of $S_0'(\mathbb{R}^2)$. Namely, take the Schwartz space of rapidly decaying test fu …

Yes. Wiener measure can be arrived at using the Bochner-Minlos Theorem in at least two ways. One can consider the bilinear form on $S(\mathbb{R})$ $$ C(f,g)=\int_{\mathbb{R}}\ \overline{\widehat{f} …

Yes it is possible. Let $K\in S'(\mathbb{R}^2)$ be the distribution acting on test functions $f(x,y)$ by $$ K(f)=\int_{|x-y|\ge 1} \frac{f(x,y)}{|x-y|}\ dx\ dy \ + \int_{|x-y|< 1} \frac{f(x,y)-f(x,x)} …

The axioms don't tell you what theory you constructed. For that you need to go beyond the construction of correlation functions of the elementary field $\phi$ (the basic chapter on renormalization in …