We establish a connection between the structure of a stationary symmetric α-stable random field (0 < α < 2) and ergodic theory of non-singular group actions, elaborating on a previous work by Rosiński (2000). With the help of this connection, we study the extreme values of the field over increasing boxes. Depending on the ergodic theoretical and group theoretical structures of the underlying action, we observe different kinds of asymptotic behavior of this sequence of extreme values.
Abstract. Using the language of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is then used to obtain the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. Because of heavy tails of the step size distribution, we can invoke a one large jump principle at the level of point processes to give an explicit representation of the limiting point process. As a consequence, we extend the main result of Durrett [18] and verify that two related predictions of Brunet & Derrida [14] remain valid for this model.
Abstract. We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main result of Durrett (1983) in our framework.
We establish characterization results for the ergodicity of stationary symmetric α-stable (SαS) and α-Fréchet random fields. We show that the result of Samorodnitsky [Ann. Probab. 33 (2005Probab. 33 ( ) 1782Probab. 33 ( -1803 remains valid in the multiparameter setting, that is, a stationary SαS (0 < α < 2) random field is ergodic (or, equivalently, weakly mixing) if and only if it is generated by a null group action. Similar results are also established for max-stable random fields. The key ingredient is the adaption of a characterization of positive/null recurrence of group actions by Takahashi [Kōdai Math. Sem. Rep. 23 (1971) 131-143], which is dimension-free and different from the one used by Samorodnitsky. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2013, Vol. 41, No. 1, 206-228. This reprint differs from the original in pagination and typographic detail. 1 2 Y. WANG, P. ROY AND S. A. STOEV Kabluchko [9] have developed similar tools to represent and handle general classes of max-stable processes. The ergodic properties of stationary stochastic processes and random fields are of fundamental importance and hence well-studied. See, for example, Maruyama [14], Rosiński andŻak [24, 25] and Roy [26, 27] for results on infinite divisible processes and Cambanis et al. [2], Podgórski [19], Gross and Robertson [8] and Gross [7] for results on stable processes. These culminated in the characterization of Samorodnitsky [34], which shows that the ergodicity of a stationary symmetric stable process is equivalent to the null-recurrence of the underlying nonsingular flow. On the other hand, the ergodic properties of max-stable processes have been recently studied by Stoev [36], Kabluchko [9] and Kabluchko and Schlather [10]. In particular, Kabluchko [9] has shown that as in the sum-stable case, one can associate a nonsingular flow to the stationary max-stable process and that the characterization of Samorodnitsky [34] remains valid. The case of random fields, however, remained open in both sum-and max-stable settings.Our goal in this paper is to establish a Samorodnitsky-type characterization for sum-stable and max-stable random fields. The main obstacle is the unavailability of a higher-dimensional analogue of the work of Krengel [12], which plays a crucial role in Samorodnitsky's approach for processes. We resolve this problem by providing an alternative dimension-free characterization of ergodicity for both classes of sum-and max-stable stationary random fields. For simplicity of exposition as well as mathematical tractability, we work with symmetric α-stable (SαS), (0 < α < 2) sum-stable random fields and α-Fréchet max-stable random fields (α > 0).The key ingredient of our results is the adaptation of the work of Takahashi [39]. Thanks to Takahashi's result, we are able to develop tractable and dimension-free criteria for verifying whether a given spectral representation corresponds to an SαS random field generated by a null ...
Tail estimates are developed for power law probability distributions with exponential tempering, using a conditional maximum likelihood approach based on the upperorder statistics. Tempered power law distributions are intermediate between heavy power-law tails and Laplace or exponential tails, and are sometimes called "semiheavy" tailed distributions. The estimation method is demonstrated on simulateddata from a tempered stable distribution, and for several data sets from geophysics and finance that show a power law probability tail with some tempering.
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