A large sample test for the length of memory of stationary symmetric stable random fields via nonsingular ℤd-actions

2018 ◽  
Vol 55 (1) ◽  
pp. 179-195
Author(s):  
Ayan Bhattacharya ◽  
Parthanil Roy
Keyword(s):  
Random Field ◽  
Sample Size ◽  
Power Function ◽  
Ergodic Theory ◽  
Random Fields ◽  
Large Sample ◽  
Discrete Parameter ◽  
Sample Test ◽  
Block Maxima ◽  
Stable Random Fields

AbstractBased on the ratio of two block maxima, we propose a large sample test for the length of memory of a stationary symmetric α-stable discrete parameter random field. We show that the power function converges to 1 as the sample-size increases to ∞ under various classes of alternatives having longer memory in the sense of Samorodnitsky (2004). Ergodic theory of nonsingular ℤd-actions plays a very important role in the design and analysis of our large sample test.

scholarly journals On generalized max-linear models in max-stable random fields

2017 ◽  
Vol 54 (3) ◽  
pp. 797-810
Author(s):  
Michael Falk ◽  
Maximilian Zott
Keyword(s):  
Random Field ◽  
Linear Model ◽  
Random Fields ◽  
Linear Models ◽  
Higher Dimensions ◽  
Natural Order ◽  
Stable Random Fields ◽  
Index Space

Abstract In practice, it is not possible to observe a whole max-stable random field. Therefore, we propose a method to reconstruct a max-stable random field in C([0, 1]k) by interpolating its realizations at finitely many points. The resulting interpolating process is again a max-stable random field. This approach uses a generalized max-linear model. Promising results have been established in the k = 1 case of Falk et al. (2015). However, the extension to higher dimensions is not straightforward since we lose the natural order of the index space.

scholarly journals A Large Sample Test for the Independence of Two Renewal Processes

1967 ◽  
Vol 38 (4) ◽  
pp. 1037-1041
Author(s):  
Sidney C. Port ◽  
Charles J. Stone
Keyword(s):  
Renewal Processes ◽  
Large Sample ◽  
Sample Test

scholarly journals Identification and isotropy characterization of deformed random fields through excursion sets

2018 ◽  
Vol 50 (3) ◽  
pp. 706-725
Author(s):  
Julie Fournier
Keyword(s):  
Random Field ◽  
Euler Characteristic ◽  
Random Fields ◽  
Weak Form ◽  
Invariance Property ◽  
Basic Domain ◽  
Excursion Sets ◽  
The Mean ◽  
Isotropic Random

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

scholarly journals Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure

2017 ◽  
Vol 54 (3) ◽  
pp. 833-851 ◽  
Author(s):  
Anders Rønn-Nielsen ◽  
Eva B. Vedel Jensen
Keyword(s):  
Random Field ◽  
Large Class ◽  
Random Fields ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Excursion Sets ◽  
Excursion Set ◽  
Fixed Radius ◽  
Asymptotic Probability ◽  
The Right

Abstract We consider a continuous, infinitely divisible random field in ℝd, d = 1, 2, 3, given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields, we compute the asymptotic probability that the excursion set at level x contains some rotation of an object with fixed radius as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

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