The Distribution of Cauchy
Back into the swing of teaching after a short break, I have been doing some lectures this week about complex analysis to theoretical physics students. The name of a brilliant French mathematician called Augustin Louis Cauchy (1789-1857) crops up very regularly in this branch of mathematics, e.g. in the Cauchy integral formula and the Cauchy-Riemann conditions, which reminded me of some old jottings aI made about the Cauchy distribution, which I never used in the publication to which they related, so I thought I’d just quickly pop the main idea on here in the hope that some amongst you might find it interesting and/or amusing.
What sparked this off is that the simplest cosmological models (including the particular one we now call the standard model) assume that the primordial density fluctuations we see imprinted in the pattern of temperature fluctuations in the cosmic microwave background and which we think gave rise to the large-scale structure of the Universe through the action of gravitational instability, were distributed according to Gaussian statistics (as predicted by the simplest versions of the inflationary universe theory). Departures from Gaussianity would therefore, if found, yield important clues about physics beyond the standard model.
Cosmology isn’t the only place where Gaussian (normal) statistics apply. In fact they arise fairly generically, in circumstances where variation results from the linear superposition of independent influences, by virtue of the Central Limit Theorem. Thermal noise in experimental detectors is often treated as following Gaussian statistics, for example.
The Gaussian distribution has some nice properties that make it possible to place meaningful bounds on the statistical accuracy of measurements made in the presence of Gaussian fluctuations. For example, we all know that the margin of error of the determination of the mean value of a quantity from a sample of size independent Gaussian-dsitributed varies as ; the larger the sample, the more accurately the global mean can be known. In the cosmological context this is basically why mapping a larger volume of space can lead, for instance, to a more accurate determination of the overall mean density of matter in the Universe.
However, although the Gaussian assumption often applies it doesn’t always apply, so if we want to think about non-Gaussian effects we have to think also about how well we can do statistical inference if we don’t have Gaussianity to rely on.
That’s why I was playing around with the peculiarities of the Cauchy distribution. This distribution comes up in a variety of real physics problems so it isn’t an artificially pathological case. Imagine you have two independent variables and each of which has a Gaussian distribution with zero mean and unit variance. The ratio has a probability density function of the form
which is a Cauchy distribution. There’s nothing at all wrong with this as a distribution – it’s not singular anywhere and integrates to unity as a pdf should. However, it does have a peculiar property that none of its moments is finite, not even the mean value!
Following on from this property is the fact that Cauchy-distributed quantities violate the Central Limit Theorem. If we take independent Gaussian variables then the distribution of sum has the normal form, but this is also true (for large enough ) for the sum of independent variables having any distribution as long as it has finite variance.
The Cauchy distribution has infinite variance so the distribution of the sum of independent Cauchy-distributed quantities doesn’t tend to a Gaussian. In fact the distribution of the sum of any number of independent Cauchy variates is itself a Cauchy distribution. Moreover the distribution of the mean of a sample of size does not depend on for Cauchy variates. This means that making a larger sample doesn’t reduce the margin of error on the mean value!
This was essentially the point I made in a previous post about the dangers of using standard statistical techniques – which usually involve the Gaussian assumption – to distributions of quantities formed as ratios.
We cosmologists should be grateful that we don’t seem to live in a Universe whose fluctuations are governed by Cauchy, rather than (nearly) Gaussian, statistics. Measuring more of the Universe wouldn’t be any use in determining its global properties as we’d always be dominated by cosmic variance…