Abstract
The tail process \(\varvec{Y}=(Y_{\varvec{i}})_{\varvec{i}\in \mathbb {Z}^d}\) of a stationary regularly varying random field \(\varvec{X}=(X_{\varvec{i}})_{\varvec{i}\in \mathbb {Z}^d}\) represents the asymptotic local distribution of \(\varvec{X}\) as seen from its typical exceedance over a threshold u as \(u\rightarrow \infty\). Motivated by the standard Palm theory, we show that every tail process satisfies an invariance property called exceedance-stationarity and that this property, together with the spectral decomposition of the tail process, characterizes the class of all tail processes. We then restrict to the case when \(Y_{\varvec{i}}\rightarrow 0\) as \(|\varvec{i}|\rightarrow \infty\) and establish a couple of Palm-like dualities between the tail process and the so-called anchored tail process which, under suitable conditions, represents the asymptotic distribution of a typical cluster of extremes of \(\varvec{X}\). The main message is that the distribution of the tail process is biased towards clusters with more exceedances. Finally, we use these results to determine the distribution of a typical cluster of extremes for moving average processes with random coefficients and heavy-tailed innovations.
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The dataset from 1 is available from the author on reasonable request.
Notes
In the rest of the paper we will use the term “tail process” instead of “tail field”. This is mainly due to personal preference, but also since in almost all of the results the geometric structure of the index set is not important, i.e. almost all proofs for the case \(d\ge 2\) are essentially the same as for the case \(d=1\).
The term “typical” used here (and throughout the paper) has the strong motivation of being “chosen uniformly at random” (or simply “randomly chosen”) only under certain dependence assumptions on \(\varvec{X}\), see Sects. 4 and 5.5 below, and also Last and Thorisson (2011); Thorisson (2018) for a general discussion.
It is not clear how one could state a corresponding Campbell theorem since the tail process, as a single cluster, always contains precisely one anchor (that is, point). One would need to consider an infinite number of clusters and maybe use or modify the framework of Sigman and Whitt (2019), but we will not pursue this approach here.
The topology is the weak topology, while \(\delta _x\) denotes the Dirac measure concentrated at \(x\in l_0\).
Observe that \(\varvec{X}_n^{T_n}\) is not well-defined if \(T_n\in \{k_n r_n +1,\dots ,n \}\). However, since
$$\begin{aligned} \mathbb {P}(T_n\in \{k_n r_n +1,\dots ,n \}) \le r_n\mathbb {P}(|X_0|>c_n)\rightarrow 0\, , \; \; \text {as } n\rightarrow \infty \, \end{aligned}$$we can and will neglect this edge effect.
“\({\mathop {\longrightarrow }\limits ^{\mathbb P}}\)” denotes convergence in probability.
References
Basrak, B., Planinić, H.: Compound poisson approximation for random fields with application to sequence alignment. Bernoulli (2021+)
Basrak, B., Segers, J.: Regularly varying multivariate time series. Stochastic Process. Appl. 119(4), 1055–1080 (2009)
Billingsley, P.: Convergence of probability measures. Wiley, New York (1968)
Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic geometry and its applications, third edn. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester (2013). https://doi.org/10.1002/9781118658222
Davis, R.A., Hsing, T.: Point Process and Partial Sum Convergence for Weakly Dependent Random Variables with Infinite Variance. Ann. Probab. 23(2), 879–917 (1995). https://doi.org/10.1214/aop/1176988294
Dombry, C., Hashorva, E., Soulier, P.: Tail measure and spectral tail process of regularly varying time series. Ann. Appl. Probab. 28(6), 3884–3921 (2018). https://doi.org/10.1214/18-AAP1410
Drees, H., Janßen, A., Neblung, S.: Cluster based inference for extremes of time series. (2021). ar**v preprint ar**v:2103.08512
Hashorva, E.: On extremal index of max-stable random fields. (2020). ar**v preprint ar**v:2003.00727
Heveling, M.: Bijective point maps, point-stationarity and characterization of palm measures. Ph.D. thesis (2006). https://doi.org/10.5445/KSP/1000004284
Heveling, M., Last, G.: Characterization of Palm measures via bijective point-shifts. Ann. Probab. 33(5), 1698–1715 (2005). https://doi.org/10.1214/009117905000000224.
Hult, H., Samorodnitsky, G.: Tail probabilities for infinite series of regularly varying random vectors. Bernoulli 14(3), 838–864 (2008). https://doi.org/10.3150/08-BEJ125.
Janßen, A.: Spectral tail processes and max-stable approximations of multivariate regularly varying time series. Stochastic Processes and their Applications 129(6), 1993–2009 (2019). https://doi.org/10.1016/j.spa.2018.06.010, http://www.sciencedirect.com/science/article/pii/S0304414918303016
Kulik, R., Soulier, P.: Heavy-tailed time series. Springer (2020)
Last, G., Penrose, M.: Lectures on the Poisson Process. Institute of Mathematical Statistics Textbooks. Cambridge University Press (2017). https://doi.org/10.1017/9781316104477
Last, G., Thorisson, H.: What is typical? J. Appl. Probab. 48(A), 379–389 (2011). https://doi.org/10.1239/jap/1318940478
Planinić, H., Soulier, P.: The tail process revisited. Extremes (2018). https://doi.org/10.1007/s10687-018-0312-1.
Sigman, K., Whitt, W.: Marked point processes in discrete time. Queueing Systems 92(1), 47–81 (2019)
Soulier, P.: The tail process and tail measure of continuous time regularly varying stochastic processes. Extremes (2021+)
Thorisson, H.: Coupling, stationarity, and regeneration. Probability and its Applications (New York). Springer-Verlag, New York (2000). https://doi.org/10.1007/978-1-4612-1236-2
Thorisson, H.: The Palm-duality for random subsets of \(d\)-dimensional grids. Adv. in Appl. Probab. 39(2), 318–325 (2007). https://doi.org/10.1239/aap/1183667612
Thorisson, H.: On the modified palm version. Adv. Appl. Probab. 50, 271–280 (2018). https://doi.org/10.1017/apr.2018.85
Wu, L., Samorodnitsky, G.: Regularly varying random fields. Stochastic Process. Appl. 130(7), 4470–4492 (2020). https://doi.org/10.1016/j.spa.2020.01.005
Acknowledgements
The work of the author was supported by the grant IZHRZ0_180549 from the Swiss National Science Foundation and Croatian Science Foundation, project “Probabilistic and analytical aspects of generalised regular variation”.
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Planinić, H. Palm theory for extremes of stationary regularly varying time series and random fields. Extremes 26, 45–82 (2023). https://doi.org/10.1007/s10687-022-00447-5
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DOI: https://doi.org/10.1007/s10687-022-00447-5