Modern Stochastics: Theory and Applications logo


  • Help
Login Register

  1. Home
  2. To appear
  3. Skorokhod M1 convergence of maxima of mu ...

Modern Stochastics: Theory and Applications

Submit your article Information Become a Peer-reviewer
  • Article info
  • Full article
  • Related articles
  • More
    Article info Full article Related articles

Skorokhod M1 convergence of maxima of multivariate linear processes with heavy-tailed innovations and random coefficients
Danijel Krizmanić ORCID icon link to view author Danijel Krizmanić details  

Authors

 
Placeholder
https://doi.org/10.15559/25-VMSTA271
Pub. online: 28 January 2025      Type: Research Article      Open accessOpen Access

Received
30 August 2024
Revised
11 January 2025
Accepted
11 January 2025
Published
28 January 2025

Abstract

In this paper, functional convergence is derived for the partial maxima stochastic processes of multivariate linear processes with weakly dependent heavy-tailed innovations and random coefficients. The convergence takes place in the space of ${\mathbb{R}^{d}}$-valued càdlàg functions on $[0,1]$ endowed with the weak Skorokhod ${M_{1}}$ topology.

References

[1] 
Avram, F., Taqqu, M.: Weak convergence of sums of moving averages in the α–stable domain of attraction. Ann. Probab. 20, 483–503 (1992) MR1143432
[2] 
Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl. 119, 1055–1080 (2009) MR2508565. https://doi.org/10.1016/j.spa.2008.05.004
[3] 
Basrak, B., Tafro, A.: A complete convergence theorem for stationary regularly varying multivariate time series. Extremes 19, 549–560 (2016) MR3535966. https://doi.org/10.1007/s10687-016-0253-5
[4] 
Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997) MR1464694
[5] 
Krizmanić, D.: Functional limit theorems for weakly dependent regularly varying time series. Ph.D. dissertation, University of Zagreb, Croatia, https://www.math.uniri.hr/~dkrizmanic/DKthesis.pdf. Accessed 28 June 2024.
[6] 
Krizmanić, D.: Skorokhod ${M_{1}}$ convergence of maxima of multivariate linear processes with heavy-tailed innovations and random coefficients. arXiv preprint, https://arxiv.org/abs/2208.04054, 2022. Accessed 28 June 2024.
[7] 
Krizmanić, D.: Functional weak convergence of partial maxima processes. Extremes 19, 7–23 (2016) MR3454028. https://doi.org/10.1007/s10687-015-0236-y
[8] 
Krizmanić, D.: Functional convergence for moving averages with heavy tails and random coefficients. ALEA Lat. Am. J. Probab. Math. Stat. 16, 729–757 (2019) MR3949276. https://doi.org/10.30757/alea.v16-26
[9] 
Krizmanić, D.: Maxima of linear processes with heavy-tailed innovations and random coefficients. J. Time Ser. Anal. 43, 238–262 (2022) MR4400293. https://doi.org/10.1111/jtsa.12610
[10] 
Kulik, R., Soulier, P.: Heavy-Tailed Time Series. Springer, New York (2020) MR4174389. https://doi.org/10.1007/978-1-0716-0737-4
[11] 
Lamperti, J.: On extreme order statistics. Ann. Math. Stat. 35, 1726–1737 (1964) MR0170371. https://doi.org/10.1214/aoms/1177700395
[12] 
Mikosch, T., Wintenberger, O.: Extreme Value Theory for Time Series. Models with Power-Law Tails. Springer, New York (2024) MR4823721. https://doi.org/10.1007/978-3-031-59156-3
[13] 
Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987) MR0900810. https://doi.org/10.1007/978-0-387-75953-1
[14] 
Resnick, S.I.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York (2007) MR2271424
[15] 
Skorohod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl. 1, 261–290 (1956) MR0084897
[16] 
Tyran-Kamińska, M.: Convergence to Lévy stable processes under some weak dependence conditions. Stoch. Process. Appl. 120, 1629–1650 (2010) MR2673968. https://doi.org/10.1016/j.spa.2010.05.010
[17] 
Whitt, W.: Stochastic-Process Limits. Springer, New York (2002) MR1876437

Full article Related articles PDF XML
Full article Related articles PDF XML

Copyright
© 2025 The Author(s). Published by VTeX
by logo by logo
Open access article under the CC BY license.

Keywords
Functional limit theorem multivariate linear process regular variation extremal process M1 topology

MSC2010
60F17 60G70

Funding
This work has been supported by University of Rijeka research grant uniri-iskusni-prirod-23-98.

Metrics
since March 2018
152

Article info
views

211

Full article
views

42

PDF
downloads

14

XML
downloads

Export citation

Copy and paste formatted citation
Placeholder

Download citation in file


Share


RSS

MSTA

MSTA

  • Online ISSN: 2351-6054
  • Print ISSN: 2351-6046
  • Copyright © 2018 VTeX

About

  • About journal
  • Indexed in
  • Editors-in-Chief

For contributors

  • Submit
  • OA Policy
  • Become a Peer-reviewer

Contact us

  • ejournals-vmsta@vtex.lt
  • Mokslininkų 2A
  • LT-08412 Vilnius
  • Lithuania
Powered by PubliMill  •  Privacy policy