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Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays

Zakhar Kabluchko, Axel Munk (2009)

ESAIM: Probability and Statistics

We generalize a theorem of Shao [Proc. Amer. Math. Soc.123 (1995) 575–582] on the almost-sure limiting behavior of the maximum of standardized random walk increments to multidimensional arrays of i.i.d. random variables. The main difficulty is the absence of an appropriate strong approximation result in the multidimensional setting. The multiscale statistic under consideration was used recently for the selection of the regularization parameter in a number of statistical algorithms as well as...

Stable random fields and geometry

Shigeo Takenaka (2010)

Banach Center Publications

Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(O) = 0. 1. X(a) - X(b) is subject to the Gaussian law of mean 0 and variance d(a,b). He gave an example for M = S m , the m-dimensional sphere. Let Y ( B ) ; B ∈ ℬ ( S m ) be the Gaussian random measure on S m , that is, 1. Y(B) is a centered Gaussian system, 2. the variance of Y(B) is equal of μ(B), where μ is the uniform measure on S m , 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for B i , i = 1,2,..., B i ∩ B j = ∅ , i ≠ j, we...

Stationary gaussian random fields on hyperbolic spaces and on euclidean spheres

S. Cohen, M. A. Lifshits (2012)

ESAIM: Probability and Statistics

We recall necessary notions about the geometry and harmonic analysis on a hyperbolic space and provide lecture notes about homogeneous random functions parameterized by this space. The general principles are illustrated by construction of numerous examples analogous to Euclidean case. We also give a brief survey of the fields parameterized by Euclidean spheres. At the end we give a list of important open questions in hyperbolic case.

Stationary Gaussian random fields on hyperbolic spaces and on Euclidean spheres∗∗∗

S. Cohen, M. A. Lifshits (2012)

ESAIM: Probability and Statistics

We recall necessary notions about the geometry and harmonic analysis on a hyperbolic space and provide lecture notes about homogeneous random functions parameterized by this space. The general principles are illustrated by construction of numerous examples analogous to Euclidean case. We also give a brief survey of the fields parameterized by Euclidean spheres. At the end we give a list of important open questions in hyperbolic case.

Stochastic Poisson-Sigma model

Rémi Léandre (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We produce a stochastic regularization of the Poisson-Sigma model of Cattaneo-Felder, which is an analogue regularization of Klauder’s stochastic regularization of the hamiltonian path integral [23] in field theory. We perform also semi-classical limits.

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