Cauchy joins the club

I’ve already mentioned what my co-authors (J. Jacod, A. Nikeghbali) and myself call “mod-Gaussian convergence”, and “mod-Poisson convergence” (and I will hopefully soon write down some summary of the more recent developments of these notions; for the moment I’ll just mention that the first two papers will appear soon, one in Forum Mathematicum and one in International Math. Research Notices). It was obvious how to extend these definitions to other natural families of standard probability distributions, the only condition required being that the characteristic functions must not have zeros (since we need to divide by them). Precisely, consider any set Λ of probability measures on the real line, with

\phi_{\lambda}(t)=\int_{\mathbf{R}}{e^{itx}d\lambda(x)}\not=0

for any

\lambda\in\Lambda,\quad\quad t\in\mathbf{R}.

Then we can say that a sequence Xn of random variables converges mod-Lambda for some parameters

\lambda_n\in\Lambda,

and some limiting function Φ, if we have limits

\lim_{n\rightarrow +\infty} \quad\phi_{\lambda_n}(t)^{-1}\mathbf{E}(e^{itX_n})=\Phi(t),

for all real t, which we assume to be locally uniform to avoid degeneracies.

There are “easy” examples like

X_n=Y_n+Y,

where Yn has law λn and is independent of the fixed random variable Y: in that case the limiting function is simply the characteristic function of Y. But one may wonder about more interesting examples, and part of the works mentioned above was dedicated in finding such examples in number theory when Λ is either the set of Gaussians or the set of Poisson variables.

Now I’d like to mention two cute examples involving Cauchy distributions; they are interesting for a couple of reasons: (i) again, they involve arithmetic, but of quite a different sort (and one could in fact be interpreted as a geometric or topologic phenomenon); (ii) they are not quite of the type above: indeed, for these two examples, the convergence only holds for t in an open interval around t=0, not for the whole real line (or rather, it’s not entirely clear whether they extend; the proofs don’t, and seem to do so for good reasons, but I haven’t yet looked very deeply at this).

I’ve written a short note with the details (short because, like some of the early cases of mod-Gaussian and mod-Poisson convergence, the results are mostly re-interpretations of earlier works; those, however, are by no means trivial), which can be found here. I’ll describe briefly one of them here, which is related to a result of Vardi on the distribution of Dedekind sums (the second is maybe even more fun: it is related to linking numbers of modular knots, a topic much popularized by É. Ghys, and based on a recent preprint of Sarnak; but I’m less familiar with the underlying objects).

First, here is the definition of the Dedekind sums, defined for coprime positive integers c and d with d< c:

s(d,c)=\sum_{h=1}^{d-1}{\Bigl(\Bigl(\frac{hd}{c}\Bigr)\Bigr)\Bigl(\Bigl(\frac{h}{c}\Bigr)\Bigr)},

where ((x)) is the saw-tooth function, periodic of period 1 with

((x))=x-1/2,\quad\quad 0<x<1,\quad\quad ((0))=0.

These look strange or arbitrary when presented so bluntly, but they are quite important and fairly-ubiquitous in certain areas of mathematics (see for instance here for a report on a recent workshop dedicated to them…)

The question solved by Vardi concerned the distribution of these sums. Precisely, he proved that — suitably normalized –, the sums s(d,c), averaged over all c<N and all allowed values of d, have a limiting Cauchy distribution. To state this formally, let first

F_N=\{(c,d)\,\mid\, 1\leq d<c<N,\quad (c,d)=1\},

and write

\mathbf{P}_N(\text{some property})=\frac{1}{|F_N|}|\{(c,d)\in F_N\,\mid\,\text{the property holds}\}|,

and

\mathbf{E}_N(\alpha(d,c))=\frac{1}{|F_N|}\sum_{(c,d)\in F_N}{\alpha(d,c)},

to get some (finite) probability spaces. Then Vardi proves

\lim_{N\rightarrow +\infty}\quad \mathbf{P}_N(a<\frac{s(d,c)}{(\log N)/(2\pi)}<b)=\mu([a,b]),

for any a<b, where μ is a standard Cauchy distribution, namely

d\mu(x)=\frac{1}{\pi}\frac{1}{1+x^2}dx.

The characteristic function of the more general Cauchy distribution with parameter γ>0, namely

d\mu_{\gamma}(x)=\frac{\gamma}{\pi}\ \frac{1}{\gamma^2+x^2}dx

is given by

\int_{\mathbf{R}}{e^{itx}d\mu_{\gamma}(x)}=\exp(-\gamma|t|),

and since those functions do not vanish, one may wonder about the possibility of getting here some example of mod-Cauchy convergence. And indeed, if one looks at Vardi’s proof with such an ulterior motive, one sees that this is derived elementarily from an asymptotic formula for the characteristic function of s(d,c) on FN. Namely, after cleaning up the notation, Vardi proves that

\mathbf{E}_N(e^{its(d,c)})=\exp(-\gamma_N|t|)\Phi(t)+O(N^{-2/3})

uniformly for

|t|<2\pi,

where

\gamma_N=\frac{1}{2\pi}(\log 4N),

and the limiting function is the rather remarkable expression given by

\Phi(t)^{-1}=\Bigl(1-\frac{|t|}{4\pi}\Bigr)\quad\frac{3}{\pi}\quad\int_{SL(2,\mathbf{Z})\backslash \mathbf{H}}\quad\quad (\sqrt{y}|\eta(z)|)^{t/(2\pi)}y^{-2}dxdy,

in which the function η(z) is the Dedekind eta-function!

This is a restricted mod-Cauchy convergence: because of the size of the parameters, the “main term” is in fact not always dominant, and the limit of

\mathbf{E}_N(e^{its(d,c)})\exp(\gamma_N|t|)

is only guaranteed to be Φ(t) in the range

|t|<\frac{4\pi}{3}.

Since the error term in Vardi’s formula depends crucially on the spectrum of some Laplace-like operator acting on modular forms with a multiplier system depending on t, it is however not clear at all that this can be improved to obtain a larger range. (But this may be tested experimentally, since Dedekind sums can be computed very quickly using the reciprocity relation they satisfy).

A puzzling feature of Vardi’s argument is that, although one is not altogether surprised to see the eta function coming up in studies of Dedekind sums (indeed, Dedekind sums were defined from their occurence in the transformation properties of the eta function!), the precise connection leading to this asymptotic formula is quite indirect.

Compared with our examples of mod-Poisson and mod-Gaussian convergence, one very different feature is the shape of the limiting function. At the moment, I do not know anything really about the behavior of Φ(t), in particular, the behavior of the second expression involving the integral of powers of the eta function. I haven’t found anything about them yet in the literature, but it wouldn’t be surprising if some special identities, for instance, were known…

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

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