Large deviations of Gaussian random functions


A random function - of either one variable, or two or more variables
- is called Gaussian if every finite-dimensional distribution is a multivariate normal distribution. Gaussian random fields on the sphere are useful when analysing
Sometimes, a value of a Gaussian random function deviates from its expected value by several standard deviations. This is a large deviation. Though rare in a small domain, large deviations may be quite usual in a large domain.

Basic statement

Let be the maximal value of a Gaussian random function on the
sphere. Assume that the expected value of is , and the standard deviation of is . Then, for large, is close to,
where is distributed , and is a constant; it does not depend on, but depends on the correlation function of . The relative error of the approximation decays exponentially for large.
The constant is easy to determine in the important special case described in terms of the directional derivative of at a given point in a given direction. The derivative is random, with zero expectation and some standard deviation. The latter may depend on the point and the direction. However, if it does not depend, then it is equal to .
The coefficient before is in fact the Euler characteristic of the sphere.
It is assumed that is twice continuously differentiable, and reaches its maximum at a single point.

The clue: mean Euler characteristic

The clue to the theory sketched above is, Euler characteristic of the set of all points such that. Its expected value can be calculated explicitly:
.
The set is the empty set whenever ; in this case. In the other case, when, the set is non-empty; its Euler characteristic may take various values, depending on the topology of the set. However, if is large and then the set is usually a small, slightly deformed disk or ellipse. Thus, its Euler characteristic is usually equal to . This is why is close to.