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Asymptotic results for sums and extremes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  13 March 2024

Rita Giuliano ,
Claudio Macci  and
Barbara Pacchiarotti
Rita Giuliano*
Affiliation:
University of Pisa
Claudio Macci*
Affiliation:
University of Roma Tor Vergata
Barbara Pacchiarotti*
Affiliation:
University of Roma Tor Vergata
*
*Postal address: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa, Italy. Email: rita.giuliano@unipi.it
**Postal address: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy.
**Postal address: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy.
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Abstract

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant, and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of independent and identically distributed random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of independent and identically distributed exponential random variables.

Keywords

Central limit theoremFisher–Tippett–Gnedenko theoremjoint distribution of sum and maximalarge deviationsmoderate deviationssums of partial minima

MSC classification

Primary: 60F10: Large deviations
Secondary: 60F05: Central limit and other weak theorems 60G70: Extreme value theory; extremal processes
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 19
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Anderson, C. W. and Turkman, K. F. (1991). The joint limiting distribution of sums and maxima of stationary sequences. J. Appl. Prob. 28, 33–44.CrossRefGoogle Scholar
Arendarczyk, M., Kozubowski, T. J. and Panorska, A. K. (2018). The joint distribution of the sum and maximum of dependent Pareto risks. J. Multivariate Anal. 167, 136–156.CrossRefGoogle Scholar
Chaganty, N. R. (1997). Large deviations for joint distributions and statistical applications. Sankhyā A 59, 147–166.Google Scholar
Chow, T. L. and Teugels, J. L. (1979). The sum and the maximum of i.i.d. random variables. In Proceedings of the Second Prague Symposium on Asymptotic Statistics, eds P. Mandl and M. Hušková, North-Holland, Amsterdam, pp. 81–92.Google Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, T. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.CrossRefGoogle Scholar
Ganesh, A. and Torrisi, G. L. (2008). Large deviations of the interference in a wireless communication model. IEEE Trans. Inform. Theory 54, 3505–3517.CrossRefGoogle Scholar
Ghosh, M., Babu, G. J. and Mukhopadhyay, N. (1975). Almost sure convergence of sums of maxima and minima of positive random variables. Z. Wahrscheinlichkeitsth. 33, 49–54.CrossRefGoogle Scholar
Giuliano, R. and Macci, C. (2014). Large deviation principles for sequences of maxima and minima. Commun. Statist. Theory Meth. 43, 1077–1098.CrossRefGoogle Scholar
Giuliano, R. and Macci, C. (2015). Asymptotic results for weighted means of random variables which converge to a Dickman distribution, and some number theoretical applications. ESAIM Prob. Statist. 19, 395–413.CrossRefGoogle Scholar
Giuliano, R. and Macci, C. (2023). Some examples of noncentral moderate deviations for sequences of real random variables. Mod. Stoch. Theory Appl. 10, 111–144.CrossRefGoogle Scholar
Höglund, T. (1972). Asymptotic normality of sums of minima of random variables. Ann. Math. Statist. 43, 351–353.CrossRefGoogle Scholar
Hsing, T. (1995). A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables. Ann. Prob. 23, 938–947.CrossRefGoogle Scholar
Kratz, M. (2014). Normex, a new method for evaluating the distribution of aggregated heavy tailed risks. Extremes 17, 661–691.CrossRefGoogle Scholar
Kratz, M. and Prokopenko, E. (2023). Multi-normex distributions for the sum of random vectors. Rates of convergence. Extremes 26, 509–544.CrossRefGoogle Scholar
Krizmanić, D. (2020). On joint weak convergence of partial sum and maxima processes. Stochastics 92, 876–899.CrossRefGoogle Scholar
Leonenko, N., Macci, C. and Pacchiarotti, B. (2021). Large deviations for a class of tempered subordinators and their inverse processes. Proc. R. Soc. Edinburgh Sect. A 151, 2030–2050.CrossRefGoogle Scholar
Lynch, J. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Prob. 15, 610–627.CrossRefGoogle Scholar
Müller, U. K. (2019). Refining the central limit theorem approximation via extreme value theory. Statist. Prob. Lett. 155, 108564.CrossRefGoogle Scholar
Puhalskii, A. (1991). On functional principle of large deviations. In New Trends in Probability and Statistics, Vol. 1, eds V. Sazonov and T. Shervashidze, VSP, Moklas, pp. 198–218.Google Scholar
Qeadan, F., Kozubowski, T. J. and Panorska, A. K. (2012). The joint distribution of the sum and the maximum of IID exponential random variables. Commun. Statist. Theory Meth. 41, 544–569.CrossRefGoogle Scholar