# Why is sub-gaussian important?

I encounter a lot of problems and theorems. Many will assume that the distribution is sub-gaussian. However, most cases or problem I deal with is sub-gaussian. Is there any distribution that is not sub-gaussian and plays an important role in analysis or applications?

• a lot of them, like Student's t and exponential, and many others Commented Nov 2, 2019 at 23:48

(In what follows I'm handwaving the typical assumption/condition that sub-Gaussian variables have zero mean. The points being made can be adapted to encompass that -- where necessary we may assume we're dealing with $$X-E(X)$$ when considering whether something is sub-Gaussian)

Is there any distribution that is not sub-gaussian and plays an important role in analysis or applications?

Yes. In some real world application areas almost every variable of interest is modelled using a distribution that is not sub-Gaussian. In others, some variables are reasonable to treat as sub-Gaussian but others are not.

Examples of distributional models that are more-or-less widely used include:

• Exponential waiting times,

• Weibull survival times (at least for some parameter values),

• Lognormal claim sizes,

• Pareto wealth,

• Inverse Gaussian earthquake magnitudes.

There are many others!

It's not just continuous variables; many discrete variables are not sub-Gaussian either.

As for why sub-Gaussian distributions are important (per your title), hopefully some of the study you did will have given you some clues; there are a number of applications for which theorems relating to sub-Gaussian variables have use -- e.g. in forming tail bounds. If you have short-tailed variables (and there's no lack of those), then tail bounds that make use of that information can be handy. Results relating to sub-Gaussian variables have value in the study of random matrices, for example.