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1. Tail index estimation for tail adversarial stable time series with an application to high‐dimensional tail clustering

   https://doi.org/10.1111/jtsa.12785  

   Cao, Hanyue; Gao, Jingying; Shao, Yu; Sriram, T_N; Wang, Weiliang; Wen, Fei; Zhang, Ting (October 2024, Journal of Time Series Analysis)

   For stationary time series with regularly varying marginal distributions, an important problem is to estimate the associated tail index which characterizes the power‐law behavior of the tail distribution. For this, various results have been developed for independent data and certain types of dependent data. In this article, we consider the problem of tail index estimation under a recently proposed notion of serial tail dependence called the tail adversarial stability. Using the technique of adversarial innovation coupling and a martingale approximation scheme, we establish the consistency and central limit theorem of the tail index estimator for a general class of tail dependent time series. Based on the asymptotic normal distribution from the obtained central limit theorem, we further consider an application to cluster a large number of regularly varying time series based on their tail indices by using a robust mixture algorithm. The results are illustrated using numerical examples including Monte Carlo simulations and a real data analysis. 

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2. The Elliptic Tail Kernel

   https://doi.org/10.1093/imrn/rnaa038  

   Cuenca, Cesar; Gorin, Vadim; Olshanski, Grigori (March 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We introduce and study a new family of $$q$$-translation-invariant determinantal point processes on the two-sided $$q$$-lattice. We prove that these processes are limits of the $$q$$–$zw$ measures, which arise in the $$q$$-deformation of harmonic analysis on $$U(\infty )$$, and express their correlation kernels in terms of Jacobi theta functions. As an application, we show that the $$q$$–$zw$ measures are diffuse. Our results also hint at a link between the two-sided $$q$$-lattice and rows/columns of Young diagrams. 

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3. Tail-Erasure-Correcting Codes

   https://doi.org/10.1109/TIT.2024.3453935  

   Moav, Boaz; Gabrys, Ryan; Yaakobi, Eitan (December 2024, IEEE Transactions on Information Theory)
4. Asymptotics of sample tail autocorrelations for tail-dependent time series: phase transition and visualization

   https://doi.org/10.1093/biomet/asab038  

   Zhang, Ting (July 2021, Biometrika)

   Summary In this article we develop an asymptotic theory for sample tail autocorrelations of time series data that can exhibit serial dependence in both tail and non-tail regions. Unlike with the traditional autocorrelation function, the study of tail autocorrelations requires a double asymptotic scheme to capture the tail phenomena, and our results do not impose any restrictions on the dependence structure in non-tail regions and allow processes that are not necessarily strongly mixing. The newly developed asymptotic theory reveals a previously undiscovered phase transition phenomenon, where the asymptotic behaviour of sample tail autocorrelations, including their convergence rate, can transition from one phase to another as the lag index moves past the point beyond which serial tail dependence vanishes. The phase transition discovery fills a gap in existing research on tail autocorrelations and can be used to construct the lines of significance, in analogy to the traditional autocorrelation plot, when visualizing sample tail autocorrelations to assess the existence of serial tail dependence or to identify the maximal lag of tail dependence. 

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5. The BIG X-ray tail

   https://doi.org/10.1093/mnrasl/slab108  

   Ge, Chong; Sun, Ming; Yagi, Masafumi; Fossati, Matteo; Forman, William; Jáchym, Pavel; Churazov, Eugene; Zhuravleva, Irina; Boselli, Alessandro; Jones, Christine; et al (October 2021, Monthly Notices of the Royal Astronomical Society: Letters)

   ABSTRACT Galaxy clusters grow primarily through the continuous accretion of group-scale haloes. Group galaxies experience preprocessing during their journey into clusters. A star-bursting compact group, the Blue Infalling Group (BIG), is plunging into the nearby cluster A1367. Previous optical observations reveal rich tidal features in the BIG members, and a long H α trail behind. Here, we report the discovery of a projected ∼250 kpc X-ray tail behind the BIG using Chandra and XMM–Newton observations. The total hot gas mass in the tail is ∼7 × 1010 M⊙ with an X-ray bolometric luminosity of ∼3.8 × 1041 erg s−1. The temperature along the tail is ∼1 keV, but the apparent metallicity is very low, an indication of the multi-T nature of the gas. The X-ray and H α surface brightnesses in the front part of the BIG tail follow the tight correlation established from a sample of stripped tails in nearby clusters, which suggests the multiphase gas originates from the mixing of the stripped interstellar medium (ISM) with the hot intracluster medium (ICM). Because thermal conduction and hydrodynamic instabilities are significantly suppressed, the stripped ISM can be long lived and produce ICM clumps. The BIG provides us a rare laboratory to study galaxy transformation and preprocessing. 

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