Miao, Yu; Xue, Tianyu; Wang, Ke; Zhao, Fangfang Large deviations for dependent heavy tailed random variables. (English) Zbl 1296.60070 J. Korean Stat. Soc. 41, No. 2, 235-245 (2012). Summary: Let \(\{X_{n},n\geq1\}\) be a stationary sequence of random variables with heavy tails. In this paper, we study the logarithmic asymptotic behaviors for the distributions of the partial sums \(S_n=\sum_{i=1}^{n}X_i\) under the assumption that \(\{X_{n},n\geq1\}\) is a sequence of dependent random variables. Our main interest is in the crude estimates \(P(|S_{n}|>n^{x})\approx n^{-\alpha x+1}\) for appropriate values \(x\) where \(\alpha\) is a specific parameter. Some results in this paper extend the works of Y. Hu and H. Nyrhinen [J. Theor. Probab. 17, No. 3, 761–768 (2004; Zbl 1068.60038)]. Cited in 5 Documents MSC: 60F10 Large deviations Keywords:logarithmic asymptotic behaviors; large deviations; heavy tails; \(m\)-dependent; negatively associated; stationary sequence Citations:Zbl 1068.60038 PDFBibTeX XMLCite \textit{Y. Miao} et al., J. Korean Stat. Soc. 41, No. 2, 235--245 (2012; Zbl 1296.60070) Full Text: DOI References: [1] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling extremal events (1997), Springer-Verlag: Springer-Verlag Berlin · Zbl 0873.62116 [2] Gantert, N., A note on logarithmic tail asymptotics and mixing, Statistics & Probability Letters, 49, 2, 113-118 (2000) · Zbl 0963.60024 [3] Goldie, C. M.; Klüppelberg, C., Subexponential distributions, (A practical guide to heavy tails (Santa Barbara, CA, 1995) (1998), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 435-459 · Zbl 0923.62021 [4] Hu, Y.; Nyrhinen, H., Large deviations view points for heavy-tailed random walks, Journal of Theoretical Probability, 17, 3, 761-768 (2004) · Zbl 1068.60038 [5] Joag-Dev, K.; Proschan, F., Negative association of random variables with applications, Annals of Statistics, 11, 286-295 (1983) · Zbl 0508.62041 [6] Ledoux, M.; Talagrand, M., Probability in Banach spaces. Isoperimetry and processes (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0748.60004 [7] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic processes for insurance and finance (1999), John Wiley & Sons, Ltd.: John Wiley & Sons, Ltd. Chichester · Zbl 0940.60005 [8] Roussas, G. G., Positive and negative dependence with some statistical application, (Ghosh, S., Asymptotics nonparametrics and time series (1999), Marcel Dekker: Marcel Dekker New York), 757-788 · Zbl 1069.62518 [9] Shao, Q. M., A comparison theorem on moment inequalities between negatively associated and independent random variables, Journal of Theoretical Probability, 13, 2, 343-356 (2000) · Zbl 0971.60015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.