Level crossings and other level functionals of stationary Gaussian processes

This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some extensions. The main results [(factorial) moments, representation into the Wiener Chaos, asymptotic results, rate of convergence, local time and number of crossings] are described, as well as the different approaches [normal comparison method, Rice method, Stein-Chen method, a general $m$-dependent method] used to obtain them; these methods are also very useful in the general context of Gaussian fields. Finally some extensions [time occupation functionals, number of maxima in an interval, process indexed by a bidimensional set] are proposed, illustrating the generality of the methods. A large inventory of papers and books on the subject ends the survey.Comment: Published at http://dx.doi.org/10.1214/154957806000000087 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Continuous and discrete properties of stochastic processes

This thesis considers the interplay between the continuous and discrete properties of random stochastic processes. It is shown that the special cases of the one-sided Lévy-stable distributions can be connected to the class of discrete-stable distributions through a doubly-stochastic Poisson transform. This facilitates the creation of a one-sided stable process for which the N-fold statistics can be factorised explicitly. The evolution of the probability density functions is found through a Fokker-Planck style equation which is of the integro-differential type and contains non-local effects which are different for those postulated for a symmetric-stable process, or indeed the Gaussian process. Using the same Poisson transform interrelationship, an exact method for generating discrete-stable variates is found. It has already been shown that discrete-stable distributions occur in the crossing statistics of continuous processes whose autocorrelation exhibits fractal properties. The statistical properties of a nonlinear filter analogue of a phase-screen model are calculated, and the level crossings of the intensity analysed. It is found that rather than being Poisson, the distribution of the number of crossings over a long integration time is either binomial or negative binomial, depending solely on the Fano factor. The asymptotic properties of the inter-event density of the process are found to be accurately approximated by a function of the Fano factor and the mean of the crossings alone

A New Approach To Estimate The Collision Probability For Automotive Applications

We revisit the computation of probability of collision in the context of automotive collision avoidance (the estimation of a potential collision is also referred to as conflict detection in other contexts). After reviewing existing approaches to the definition and computation of a collision probability we argue that the question "What is the probability of collision within the next three seconds?" can be answered on the basis of a collision probability rate. Using results on level crossings for vector stochastic processes we derive a general expression for the upper bound of the distribution of the collision probability rate. This expression is valid for arbitrary prediction models including process noise. We demonstrate in several examples that distributions obtained by large-scale Monte-Carlo simulations obey this bound and in many cases approximately saturate the bound. We derive an approximation for the distribution of the collision probability rate that can be computed on an embedded platform. In order to efficiently sample this probability rate distribution for determination of its characteristic shape an adaptive method to obtain the sampling points is proposed. An upper bound of the probability of collision is then obtained by one-dimensional numerical integration over the time period of interest. A straightforward application of this method applies to the collision of an extended object with a second point-like object. Using an abstraction of the second object by salient points of its boundary we propose an application of this method to two extended objects with arbitrary orientation. Finally, the distribution of the collision probability rate is identified as the distribution of the time-to-collision.Comment: Revised and restructured version, discussion of extended vehicles expanded, section on TTC expanded, references added, other minor changes, 17 pages, 18 figure