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Testing randomness of spatial point patterns with the Ripley statistic

Gabriel Lang, Eric Marcon (2013)

ESAIM: Probability and Statistics

Aggregation patterns are often visually detected in sets of location data. These clusters may be the result of interesting dynamics or the effect of pure randomness. We build an asymptotically Gaussian test for the hypothesis of randomness corresponding to a homogeneous Poisson point process. We first compute the exact first and second moment of the Ripley K-statistic under the homogeneous Poisson point process model. Then we prove the asymptotic normality of a vector of such statistics for different...

The distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous planar anisotropic STIT Tessellations

Christoph Thäle (2010)

Commentationes Mathematicae Universitatis Carolinae

A result about the distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous and isotropic random tessellations stable under iteration (STIT tessellations) is extended to the anisotropic case using recent findings from Schreiber/Thäle, Typical geometry, second-order properties and central limit theory for iteration stable tessellations, arXiv:1001.0990 [math.PR] (2010). Moreover a new expression for the values of this probability distribution is presented...

The Dyson Brownian Minor Process

Mark Adler, Eric Nordenstam, Pierre Van Moerbeke (2014)

Annales de l’institut Fourier

Consider an n × n Hermitean matrix valued stochastic process { H t } t ≥ 0 where the elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian motion, that is they behave as Ornstein-Uhlenbeck processes conditioned never to intersect.In this paper we study not only the eigenvalues of the full matrix, but also the eigenvalues of all the principal minors. That is, the eigenvalues of the k × k minors in the upper left corner of H t . Projecting this...

The weak convergence of regenerative processes using some excursion path decompositions

Amaury Lambert, Florian Simatos (2014)

Annales de l'I.H.P. Probabilités et statistiques

We consider regenerative processes with values in some general Polish space. We define their ε -big excursions as excursions e such that ϕ ( e ) g t ; ε , where ϕ is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of e . We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of ε -big excursions and of their endpoints, for all ε in a set whose closure contains 0 . Finally, we provide...

Transience/recurrence and the speed of a one-dimensional random walk in a “have your cookie and eat it” environment

Ross G. Pinsky (2010)

Annales de l'I.H.P. Probabilités et statistiques

Consider a variant of the simple random walk on the integers, with the following transition mechanism. At each site x, the probability of jumping to the right is ω(x)∈[½, 1), until the first time the process jumps to the left from site x, from which time onward the probability of jumping to the right is ½. We investigate the transience/recurrence properties of this process in both deterministic and stationary, ergodic environments {ω(x)}x∈Z. In deterministic environments, we also study the speed...

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