Stunts with Gaussian distributions.

Let's start here with the basic thing. The (univariate) standard Gaussian pdf

We define

## Left tail of icdf

For small , the quantile function has the useful asymptotic expansion

## What is Erf again?

This erf function is popular, isn't it? Unavoidable if you do computer algebra. But I can never remember what it is. There's these two scaling factors tacked on.

Well…

and

Done.

## Representations

### Rational approximations

TBD.

### ODE representation for the univariate density

TODO: note where I learned this.

### ODE representation for the icdf

From StSh08 via Wikipedia.

### Density PDE representation as a diffusion equation

(see, e.g. BoGK10)

Look, it's the diffusion equation of Wiener process. Surprise.

## Roughness

Univariate -

## Multidimensional marginals

As made famous by Wiener processes in finance and Gaussian processes in Bayesian nonparametrics.

See, e.g. these lectures, or Michael I Jordan's backgrounders.

## Transformed variables

implies

## Refs

- Wich88: (1988) Algorithm AS 241: The Percentage Points of the Normal Distribution.
*Journal of the Royal Statistical Society. Series C (Applied Statistics)*, 37(3), 477–484. DOI - BoGK10: (2010) Kernel density estimation via diffusion.
*The Annals of Statistics*, 38(5), 2916–2957. DOI - StSh08: (2008) Quantile mechanics.
*European Journal of Applied Mathematics*, 19(2), 87–112. DOI - Bote17: (2017) The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 79(1), 125–148. DOI