Abstract
We consider a class of stationary processes exhibiting both long-range dependence and heavy tails. Separate limit theorems for sums and for extremes have been established recently in the literature with novel objects appearing in the limits. In this article, we establish the joint sum-and-max limit theorems for this class of processes. In the finite-variance case, the limit consists of two independent components: a fractional Brownian motion arising from the sum and a long-range dependent random sup measure arising from the maximum. In the infinite-variance case, we obtain in the limit two dependent components: a stable process and a random sup measure whose dependence structure is described through the local time and range of a stable subordinator. For establishing the limit theorem in the latter case, we also develop a joint convergence result for the local time and range of subordinators, which may be of independent interest.
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References
Anderson, C.W., Turkman, K.F.: Sums and maxima of stationary sequences with heavy tailed distributions. Sankhya Indian J. Stat. Ser. A. 57, 1–10 (1995)
Anderson, C.W., Turkman, K.F.: The joint limiting distribution of sums and maxima of stationary sequences. J. Appl. Probab. 28(1), 33–44 (1991)
Bai, S., Owada, T., Wang, Y.: A functional non-central limit theorem for multiple-stable processes with long-range dependence. Stoch. Process. Appl. 130(9), 5768–5801 (2020)
Bai, S., Wang, Y.: Phase transition for extremes of a family of stationary multiple-stable processes. To appear in Annales de l’finstitut Henri Poincaré (2023). arXiv:2110.07497
Bai, S., Wang, Y.: Tail processes for stable-regenerative model. To appear in Bernoulli (2022). arXiv:2110.07499
Bertoin, J.: Subordinators: Examples and Applications. Lectures on Probability Theory and Statistics, pp. 1–91. Springer, Berlin, Heidelberg (1999)
Billingsley, P.: Convergence of Probability Measures.Wiley series in probability and statistics. Wiley, New York (1999)
Bingham, N.H., et al.: Regular Variation, vol. 27. Cambridge University Press, Cambridge (1989)
Chen, Z., Samorodnitsky, G.: A new shape of extremal clusters for certain stationary semi-exponential processes with moderate long range dependence (2021). arXiv preprint arXiv:2107.01517
Chow, T.L., Teugels, J.L.: The sum and the maximum of iid random variables. In: Proceedings of the 2nd Prague symposium on asymptotic statistics. 45, 394–403 (1978)
Ho, H.-C., Hsing, T.: On the asymptotic joint distribution of the sum and maximum of stationary normal random variables. J. Appl. Probab. 33(1), 138–145 (1996)
Ho, H.-C., McCormick, W.P.: Asymptotic distribution of sum and maximum for Gaussian processes. J. Appl. Probab. 36(4), 1031–1044 (1999)
Hsing, T.: A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables. Ann. Probab. 23, 938–947 (1995)
Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002)
Krizmanić, D.: On joint weak convergence of partial sum and maxima processes. Stochastics 92(6), 876–899 (2020)
Lacaux, C., Samorodnitsky, G.: Time-changed extremal process as a random sup measure. Bernoulli 22(4), 1979–2000 (2016)
McCormick, W.P., Qi, Y.: Asymptotic distribution for the sum and maximum of Gaussian processes. J. Appl. Probab. 37(4), 958–971 (2000)
Molchanov, I.: Theory of random sets. Springer, London (2005)
O’Brien, G.L., Torfs, P.J.J.F., Vervaat, W.: Stationary self-similar extremal processes. Probab. Theory Relat. Fields 87(1), 97–119 (1990)
Owada, T., Samorodnitsky, G.: Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 43(1), 240–285 (2015)
Owada, T., Samorodnitsky, G.: Maxima of long memory stationary symmetric alpha-stable processes, and self-similar processes with stationary max-increments. Bernoulli 21(3), 1575–1599 (2015)
Rosiński, J., Samorodnitsky, G.: Classes of mixing stable processes. Bernoulli. 365–377 (1996)
Salinetti, G., Wets, R.J.-B.: On the convergence of closed-valued measurable multifunctions. Trans. Am. Math. Soc. 266(1), 275–289 (1981)
Samorodnitsky, G.: Stochastic Processes and Long Range Dependence, vol. 26. Springer, Berlin (2016)
Samorodnitsky, G., Szulga, J.: An asymptotic evaluation of the tail of a multiple symmetric \(\alpha \)-stable integral. Ann. Probab. 17, 1503–1520 (1989)
Samorodnitsky, G., Wang, Y.: Extremal theory for long range dependent infinitely divisible processes. Ann. Probab. 47(4), 2529–2562 (2019)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Taqqu, M.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Adv. Appl. Probab. 7(2), 249–249 (1975)
Vervaat, W.: Random upper semicontinuous functions and extremal processes. Department of Mathematical Statistics R 8801 (1988)
Whitt, W.: Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, Berlin (2002)
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Bai, S., Tang, H. Joint Sum-and-Max Limit for a Class of Long-Range Dependent Processes with Heavy Tails. J Theor Probab (2023). https://doi.org/10.1007/s10959-023-01289-y
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DOI: https://doi.org/10.1007/s10959-023-01289-y
Keywords
- Asymptotic dependence
- Infinitely divisible process
- Long-range dependence
- Stable subordinator
- Weak convergence