Large deviations of Gaussian random functions

A random function of one variable (random process) or two and more variables (random field) is called Gaussian if its finite-dimensional distributions are multinormal. For example, Gaussian random fields on the sphere are used for modeling

the anomalies in the cosmic microwave background radiation (see [2], p. 8);

brain images obtained by positron emission tomography (see [2], p. 9).

Sometimes a value of a Gaussian random function deviates from its mean value by several standard deviations. This is a large deviation. Being a rare event in a small domain (of space or/and time), large deviation may be quite usual in a large domain.

The probability of such deviations is an old topic of probability theory. Its recent progress is a good (rather than `pure' or `applied') mathematics: it is more elegant than average `pure' mathematics, and at the same time more useful than average `applied' mathematics.

Basic statement

Let $M$ be the maximal value of a Gaussian random function $X$ on the (two-dimensional) sphere. Assume that the mean value of $X$ is $0$ (at every point of the sphere), and the standard deviation of $X$ is $1$ (at every point of the sphere). Then, for large $a>0$ , $P(M>a)$ is close to $P(|\xi |>a)+Ca\exp(-a^{2}/2)$ , where $\xi$ is distributed $N(0,1)$ , and $C$ is a constant; it does not depend on $a$ , but depends on the correlation function of $X$ (see below). The relative error of the approximation decays exponentially as $a\to \infty$ .

The constant $C$ is easy to determine in the important special case described in terms of the direcrtional derivative of $X$ at a given point (of the sphere) in a given direction (tangential to the sphere). The derivative is random, with zero mean and some standard deviation. The latter may depend on the point and the direction. However, if it does not depend, then it appears to be equal to $(\pi /2)^{1/4}C^{1/2}$ (assuming the the sphere is of radius $1$ ).

Basic statement

Further reading