Large deviations of Gaussian random functions

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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 useful when analysing

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 deviations 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 is equal to $(\pi /2)^{1/4}C^{1/2}$ (for the sphere of radius $1$ ).

It is assumed that $X$ is twice continuously differentiable (almost surely).

The clue: mean Euler characteristic

The clue to the theory sketched above is, Euler characteristic $\chi _{a}$ of the set $\{X>a\}$ of all points $t$ (of the sphere) such that $X(t)>a$ . Its mean value (in other words, expected value) $E(\chi _{a})$ can be calculated explicitly: $E(\chi _{a})=P(|\xi |>a)+Ca\exp(-a^{2}/2)$ (which is far from being trivial, however).

The set $\{X>a\}$ is empty whenever $M<a$ ; in this case $\chi _{a}=0$ . In the other case, when $M>a$ , the set $\{X>a\}$ is non-empty; its Euler characteristic may take various values, depending on the topology of the set (the number of connected components, and possible holes in these components). However, if $a$ is large and $M>a$ then the set $\{X>a\}$ is usually a small, slightly deformed disk or ellipse (which is easy to guess, but quite difficult to prove). Thus, its Euler characteristic $\chi _{a}$ is usually equal to $1$ (given that $M>a$ ). This is why $E(\chi _{a})$ is close to $P(M>a)$ .

Large deviations of Gaussian random functions

Basic statement

The clue: mean Euler characteristic

Further reading

See also