Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

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

Convex bodies: the brunn–minkowski theory

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An Introduction To The Theory Of Point Processes

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.

/pdf/critical-gaussian-multiplicative-chaos-convergence-of-the-4y6u4u9v05.pdf

Critical Gaussian multiplicative chaos: Convergence of the derivative martingale

Two integrable random vectors $\xi$ and $\xi^{*}$ in $\mathbb{R} ^{d}$ are said to be zonoid equivalent if, for each $u\in\mathbb{R} ^{d}$, the scalar products $\langle\xi,u\rangle$ and $\langle\xi^{*},u\rangle$ have the same first absolute moments. The paper analyses stochastic processes whose finite-dimensional distributions are zonoid equivalent with respect to time shift (zonoid stationarity) and permutation of its components (swap invariance). While the first concept is weaker than the stationarity, the second one is a weakening of the exchangeability property. It is shown that nonetheless the ergodic theorem holds for swap-invariant sequences and the limits are characterised.

/pdf/invariance-properties-of-random-vectors-and-stochastic-4n9i2e2l9e.pdf

Invariance properties of random vectors and stochastic processes based on the zonoid concept

Stationarity of multivariate particle systems

A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in R^d and in some cases provide a full characterisation of the stationarity property. In particular, a full characterisation of stationary multivariate Brown-Resnick processes is given.

We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.

/pdf/gibbs-point-process-approximation-total-variation-bounds-1v9z8b9sj3.pdf

Gibbs point process approximation: Total variation bounds using Stein’s method

We consider percolation properties of the Boolean model generated  by a Gibbs point process and balls with deterministic radius. We show that for a large class of Gibbs point processes there exists a critical  activity, such that percolation occurs a.s. above criticality. For locally stable Gibbs point processes we show a converse result,  i.e. they do not percolate a.s. at low activity.

/pdf/continuum-percolation-for-gibbs-point-processes-4h7djqzj8b.pdf

Kaspar Stucki

Papers

Invariance properties of random vectors and stochastic processes based on the zonoid concept

Stationarity of multivariate particle systems

Stationarity of multivariate particle systems

Gibbs point process approximation: Total variation bounds using Stein’s method

Continuum percolation for Gibbs point processes