Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

We give an overview of the recent asymptotic results on the geometry of excursion sets of stationary random fields. Namely, we cover a number of limit theorems of central type for the volume of excursions of stationary (quasi-, positively or negatively) associated random fields with stochastically continuous realizations for a fixed excursion level. This class includes in particular Gaussian, Poisson shot noise, certain infinitely divisible, α-stable, and max-stable random fields satisfying some extra dependence conditions. Functional limit theorems (with the excursion level being an argument of the limiting Gaussian process) are reviewed as well. For stationary isotropic C 1-smooth Gaussian random fields similar results are available also for the surface area of the excursion set. Statistical tests of Gaussianity of a random field which are of importance to real data analysis as well as results for an increasing excursion level round up the paper.

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Limit Theorems for Excursion Sets of Stationary Random Fields

/pdf/success-epochs-in-bernoulli-trials-with-applications-in-x6mwdfzacg.pdf

Success Epochs in Bernoulli Trials with Applications in Number Theory.

We prove that under certain conditions the excursion sets volumes of stationary positively associated random fields converge after rescaling to the normal distribution as the excursion level and the size of the observation window grow In addition, we provide a number of examples

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Judith Sarah Olszewski

Papers

A Central Limit Theorem for the Volumes of High Excursions of Stationary Associated Random Fields