A closely related perspective on fluctuations of sequences of random variables has been studied recently in pure probability theory under the name of “mod-Gaussian convergence” (and more generally “mod-phi convergence”). Mod-Gaussian convergence of a sequence of RVs or random vectors is just the right amount of control over the characteristic functions—or in a useful variant, the whole complex Laplace transforms—to imply a clean description of the fluctuations at various scales (CLT, Edgeworth expansion, “normality zone”, local CLT, moderate deviations, sharp large deviations,...). Unsurprisingly, the theory is full of cumulants.
Here is a nice introduction with applications to statistical mechanics models:
This leads for instance to a clean approach of some “anomalous” CLTs with non-Gaussian limit laws (not for the mod-Gaussian convergent sequences themselves but for modified versions thereof) for some stat mech models at continuous phase transitions, see Theorems 8 and 11 in the first reference above. As far as I know, those theorems are the simplest “SLT-like” phenomenon in probability theory!
This is very nice! So the way I understand what you linked is this: the class of perturbative expansions in the “Edgeworth expansion” picture I was distilling is that the order-d approximation for the probability distribution PSn(x) associated to the sum variable S_n above is pnGauss(x)⋅Fn(x) where pnGauss(x) is the probability distribution associated with a Gaussian N(0,const/n) and Fn(x) is a polynomial in t and the perturbative parameter 1/√n. The paper you linked says that a related natural thing to do is to take the Fourier transform, which will be the product of the Gaussian pdf N(0,n/const) and a different polynomial F′n in the fourier parameter t and the inverse perturbation parameter √n. You can then look at the leading terms, which will be (maybe up to some fixed scaling) a polynomial in t⋅√n, and this gives some kind of “leading” Edgeworth contribution.
Here this can be interpreted as a stationary phase formula, but you can only get “perturbative” theories, i.e. the relevant critical set will be nonsingular (and everything is expressed as a Feynman diagram with edges decorated by the inverse Hessian). But you’re saying that if you take this idea and apply it to different interesting sequences of random variables (not sum variables, but other natural asymptotic limits of other random processes), you can get singular stationary phase (i.e. the Watanabe expansion). Is there an easy way to describe the simplest case that gives an interesting Watanabe expansion?
Here is the simplest case I know, which is a sum of dependent identically distributed variables. In physical terms, it is about the magnetisation of the 1d Curie-Weiss (=mean-field Ising) model. I follow the notation of the paper https://arxiv.org/abs/1409.2849 for ease of reference, this is roughly Theorem 8 + Theorem 10:
Let $M_n=\sum_{i=1}^n \sigma(i)$ be the sum of n dependent Bernouilli random variables $\sigma(i)\in\{\pm 1}$, where the joint distribution is given by
When $\beta=1$, the fluctuations of $M_n$ are very large and we have an anomalous CLT: $\frac{M_n}{n^{3/4}}$ converges in law to the probability distribution $\sim \exp(-frac{x^4}{12})$.
When $\beta<1$, $M_n$ satisfies a normal CLT: $\frac{M_n}{n^{1/2}}$ converges to a Gaussian.
When $\beta>1$, $M_n$ does not satisfy a limit theorem (there are two lower energy configurations)
In statistical mechanics, this is an old result of Ellis-Newman from 1978; the paper above puts it into a more systematic probabilistic framework, and proves finer results about the fluctuations (Theorems 16 and 17).
The physical intuition is that $\beta=1$ is the critical inverse temperature at which the 1d Curie-Weiss model goes through a continuous phase transition. In general, one should expect such anomalous CLTs in the thermodynamic limit of continuous phase transitions in statistical mechanics, with the shape of the CLT controlled by the Taylor expansion of the microcanonical entropy around the critical parameters. Indeed Ellis and his collaborators have worked out a number of such cases for various mean-field models (which according to Meliot-Nikeghbali also fit in their mod-Gaussian framework). It is of course very difficult to prove such results rigorously outside of mean-field models, since even proving that there is a phase transition is often out of reach.
A limitation of the Curie-Weiss result is that it is 1d and so the “singularity” is pretty limited. The Meliot-Nikeghbali paper has 2d and 3d generalisations where the singularities are a bit more interesting: see Theorem 11 and Equations (10) and (11). And here is another recent example from the stat mech literature
You were actually asking about Edgeworth expansions rather than just the CLT. It may be that with this method of producing anomalous CLTs, starting with a nice mod-Gaussian convergent sequence and doing a change of measure, one could write down further terms in the expansion? I haven’t thought about this.
Since the main result of SLT is roughly speaking an “anomalous CLT for the Bayesian posterior”, I would love to use the results above to think of singular Bayesian statistical models as “at a continuous phase transition” (probably with quenched disorder to be more physically accurate), with the tuning to criticality coming from a combination of structure in data and hyperparameter tuning, but I don’t really know what to do with this analogy!
A closely related perspective on fluctuations of sequences of random variables has been studied recently in pure probability theory under the name of “mod-Gaussian convergence” (and more generally “mod-phi convergence”). Mod-Gaussian convergence of a sequence of RVs or random vectors is just the right amount of control over the characteristic functions—or in a useful variant, the whole complex Laplace transforms—to imply a clean description of the fluctuations at various scales (CLT, Edgeworth expansion, “normality zone”, local CLT, moderate deviations, sharp large deviations,...). Unsurprisingly, the theory is full of cumulants.
Here is a nice introduction with applications to statistical mechanics models:
https://arxiv.org/abs/1409.2849
and the book with the general theory (which I still have to read!)
https://link.springer.com/book/10.1007/978-3-319-46822-8
This leads for instance to a clean approach of some “anomalous” CLTs with non-Gaussian limit laws (not for the mod-Gaussian convergent sequences themselves but for modified versions thereof) for some stat mech models at continuous phase transitions, see Theorems 8 and 11 in the first reference above. As far as I know, those theorems are the simplest “SLT-like” phenomenon in probability theory!
This is very nice! So the way I understand what you linked is this: the class of perturbative expansions in the “Edgeworth expansion” picture I was distilling is that the order-d approximation for the probability distribution PSn(x) associated to the sum variable S_n above is pnGauss(x)⋅Fn(x) where pnGauss(x) is the probability distribution associated with a Gaussian N(0,const/n) and Fn(x) is a polynomial in t and the perturbative parameter 1/√n. The paper you linked says that a related natural thing to do is to take the Fourier transform, which will be the product of the Gaussian pdf N(0,n/const) and a different polynomial F′n in the fourier parameter t and the inverse perturbation parameter √n. You can then look at the leading terms, which will be (maybe up to some fixed scaling) a polynomial in t⋅√n, and this gives some kind of “leading” Edgeworth contribution.
Here this can be interpreted as a stationary phase formula, but you can only get “perturbative” theories, i.e. the relevant critical set will be nonsingular (and everything is expressed as a Feynman diagram with edges decorated by the inverse Hessian). But you’re saying that if you take this idea and apply it to different interesting sequences of random variables (not sum variables, but other natural asymptotic limits of other random processes), you can get singular stationary phase (i.e. the Watanabe expansion). Is there an easy way to describe the simplest case that gives an interesting Watanabe expansion?
Q: How can I use LaTeX in these comments? I tried to follow https://www.lesswrong.com/tag/guide-to-the-lesswrong-editor#LaTeX but it does not seem to render.
Here is the simplest case I know, which is a sum of dependent identically distributed variables. In physical terms, it is about the magnetisation of the 1d Curie-Weiss (=mean-field Ising) model. I follow the notation of the paper https://arxiv.org/abs/1409.2849 for ease of reference, this is roughly Theorem 8 + Theorem 10:
Let $M_n=\sum_{i=1}^n \sigma(i)$ be the sum of n dependent Bernouilli random variables $\sigma(i)\in\{\pm 1}$, where the joint distribution is given by
$$
\mathbb{P}(\sigma)\sim \exp(\frac{\beta}{n}M_n^2))
$$
Then
When $\beta=1$, the fluctuations of $M_n$ are very large and we have an anomalous CLT: $\frac{M_n}{n^{3/4}}$ converges in law to the probability distribution $\sim \exp(-frac{x^4}{12})$.
When $\beta<1$, $M_n$ satisfies a normal CLT: $\frac{M_n}{n^{1/2}}$ converges to a Gaussian.
When $\beta>1$, $M_n$ does not satisfy a limit theorem (there are two lower energy configurations)
In statistical mechanics, this is an old result of Ellis-Newman from 1978; the paper above puts it into a more systematic probabilistic framework, and proves finer results about the fluctuations (Theorems 16 and 17).
The physical intuition is that $\beta=1$ is the critical inverse temperature at which the 1d Curie-Weiss model goes through a continuous phase transition. In general, one should expect such anomalous CLTs in the thermodynamic limit of continuous phase transitions in statistical mechanics, with the shape of the CLT controlled by the Taylor expansion of the microcanonical entropy around the critical parameters. Indeed Ellis and his collaborators have worked out a number of such cases for various mean-field models (which according to Meliot-Nikeghbali also fit in their mod-Gaussian framework). It is of course very difficult to prove such results rigorously outside of mean-field models, since even proving that there is a phase transition is often out of reach.
A limitation of the Curie-Weiss result is that it is 1d and so the “singularity” is pretty limited. The Meliot-Nikeghbali paper has 2d and 3d generalisations where the singularities are a bit more interesting: see Theorem 11 and Equations (10) and (11). And here is another recent example from the stat mech literature
https://link.springer.com/article/10.1007/s10955-016-1667-9
You were actually asking about Edgeworth expansions rather than just the CLT. It may be that with this method of producing anomalous CLTs, starting with a nice mod-Gaussian convergent sequence and doing a change of measure, one could write down further terms in the expansion? I haven’t thought about this.
Since the main result of SLT is roughly speaking an “anomalous CLT for the Bayesian posterior”, I would love to use the results above to think of singular Bayesian statistical models as “at a continuous phase transition” (probably with quenched disorder to be more physically accurate), with the tuning to criticality coming from a combination of structure in data and hyperparameter tuning, but I don’t really know what to do with this analogy!