Numerical bounds for the distributions of the maxima of some one- and two-parameter Gaussian processes

Abstract

We consider the class of real-valued stochastic processes indexed on a compact subset of R or R 2 with almost surely absolutely continuous sample paths. We obtain an implicit formula for the distributions of their maxima. The main result is the derivation of numerical bounds that turn out to be very accurate, in the Gaussian case, for levels that are not large. We also present the first explicit upper bound for the distribution tail of the maximum in the two-dimensional Gaussian framework. Numerical comparisons are performed with known tools such as the Rice upper bound and expansions based on the Euler characteristic. We deal numerically with the determination of the persistence exponent.

Probability Distribution of the Maximum of a Smooth Temporal Signal

“…For a Gaussian process, the same expression was obtained from a heuristic argument in [3]. In this limit of large M , the interval distributions are found to become Poissonian.…”

“…The problem of evaluating the distribution of the maximum of a time-correlated random variable X(t) has elicited a large body of work by mathematicians [1,2,3], and physicists, both theorists [4,5,6,7,8,9,10,11,12] and experimentalists [13,14,15,16,17]. In the physics literature, this is related to the persistence problem, the probability that a temporal signal X (and hence its maximum) remains below a given level M up to time t. The mathematical literature has mainly focused on evaluating…”

“…Recently [3], and for Gaussian processes only, a numerical method to obtain valuable bounds has been extended to all values of M , although the required numerical effort can become quite considerable for large t.…”

Distribution of the maximum in air pollution fields

“…For example, one may be interested in the expected number of upcrossings such that the derivative is larger than some specific value or, as is our case, the expected number of points such that (W (x), W 01 (x)) = (u, 0) and a statement A concerning derivatives and process values is fulfilled. The following generalized Rice's formula, valid for Gaussian random fields, can be found in Mercadier (2005). …”

“…The methodology we propose can be used for a fairly large class of air pollution models, although it relies on assumptions of Gaussianity. The theoretical foundations of our methodology can be found in Azaïs and Delmas (2002) and in a recent paper by Mercadier(2005). Other approaches to approximate the distribution of the maximum are taken by Adler (1981), Sun (1993), Takemura and Kuriki (2002) and Piterbarg (1996).…”