Krizmanić, Danijel Joint functional convergence of partial sums and maxima for linear processes. (English) Zbl 1407.60044 Lith. Math. J. 58, No. 4, 457-479 (2018). Summary: For linear processes with independent identically distributed innovations that are regularly varying with tail index \(\alpha\in (0, 2)\), we study the functional convergence of the joint partial-sum and partial-maxima processes. We derive a functional limit theorem under certain assumptions on the coefficients of the linear processes, which enable the functional convergence in the space of \(\mathbb R^2\)-valued càdlàg functions on \([0,1]\) with the Skorokhod weak \(M_2\) topology. We also obtain a joint convergence in the \(M_2\) topology on the first coordinate and in the \(M_1\) topology on the second coordinate. Cited in 4 Documents MSC: 60F17 Functional limit theorems; invariance principles 60G52 Stable stochastic processes Keywords:extremal process; functional limit theorem; linear process; regular variation; Skorokhod \(M_2\) topology; stable Lévy process PDFBibTeX XMLCite \textit{D. Krizmanić}, Lith. Math. J. 58, No. 4, 457--479 (2018; Zbl 1407.60044) Full Text: DOI arXiv References: [1] J. Astrauskas, Limit theorems for sums of linearly generated random variables, Lith.Math. J., 23(2):127-134, 1983. · Zbl 0536.60005 [2] F. Avram and M. Taqqu, Weak convergence of sums of moving averages in the α-stable domain of attraction, Ann. Probab., 20(1):483-503, 1992. · Zbl 0747.60032 [3] R. Balan, A. Jakubowski, and S. Louhichi, Functional convergence of linear processes with heavy-tailed innovations, J. Theor. Probab., 29(2):491-526, 2016. · Zbl 1346.60037 [4] B. Basrak and D. KrizmanićA limit theorem for moving averages in the α-stable domain of attraction, Stochastic Processes Appl., 124(2):1070-1083, 2014. · Zbl 1314.60085 [5] B. Basrak, D. Krizmanić, and J. Segers, A functional limit theorem for partial sums of dependent random variables with infinite variance, Ann. Probab., 40(5):2008-2033, 2012. · Zbl 1295.60041 [6] B. Basrak and A. Tafro, A complete convergence theorem for stationary regularly varying multivariate time series, Extremes, 19(3):549-560, 2016. · Zbl 1357.60034 [7] B. Böttcher, EmbeddedMarkov chain approximations in Skorokhod topologies, arXiv:1409.4656. · Zbl 1434.60010 [8] T.L. Chow and J.L. Teugels, The sum and the maximum of i.i.d. random variables, in P. Mandl and M. Hušková (Eds.), Proceedings of the Second Prague Symposium of Asymptotic Statistics, 21-25, August, 1978, North-Holland, Amsterdam, New York, 1979, pp. 81-92. [9] D. Cline, Infinite series of random variables with regularly varying tails, Tecnical Report No. 83-24, Institute of Applied Mathematics and Statistics, University of British Columbia, Canada, 1983. [10] R. Davis and S.I. Resnick, Limit theorems for moving averages with regularly varying tail probabilities, Ann. Probab., 13(1):179-195, 1985. · Zbl 0562.60026 [11] D. Krizmanić, Weak convergence of partial maxima processes in theM1 topology, Extremes, 17(3):447-465, 2014. · Zbl 1306.60009 [12] D. Krizmanić, A note on joint functional convergence of partial sum and maxima for linear processes, Stat. Probab. Lett., 138:42-46, 2018. · Zbl 1391.60060 [13] S.I. Resnick, Point processes, regular variation and weak convergence, Adv. Appl. Probab., 18(1):66-138, 1986. · Zbl 0597.60048 [14] S.I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987. · Zbl 0633.60001 [15] S.I. Resnick, Heavy-Tail Phenomena: Probabilistic and Statistical Modeling, Springer, New York, 2007. · Zbl 1152.62029 [16] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, Cambridge, 1999. · Zbl 0973.60001 [17] W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer, New York, 2002. · Zbl 0993.60001 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.