## On the characterization of certain point processes.(English)Zbl 0645.60057

This paper consists of two parts. First, a characterization is obtained for a class of infinitely divisible point processes on $${\mathbb{R}}\times {\mathbb{R}}_+'=(-\infty$$, $$\infty)\times (0,\infty)$$. Second, the result is applied to identify the weak limit of the point process $$N_ n$$ with points (j/n, $$u_ n^{-1}(\xi_ j))$$, $$j=0$$, $$\pm 1$$, $$\pm 2$$,..., where $$\{\xi_ j\}$$ is a stationary sequence satisfying a certain mixed condition $$\Delta$$, and $$\{u_ n\}$$ is a sequence of nonincreasing functions on (0,$$\infty)$$ such that $\lim_{n\to \infty}P\{\max_{1\leq j\leq n}\xi_ j\leq u_ n(\tau)\}=e^{- \tau},\quad \tau >0.$ This application extends a result of T. Mori [Yokohama Math. J. 25, 155-168 (1977; Zbl 0374.60010)], which assumes that $$\{\xi_ j\}$$ is $$\alpha$$-mixing, and that the distribution of $$\max_{1\leq j\leq n}\xi_ j$$ can be linearly normalized to converge to a maximum stable distribution.

### MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F05 Central limit and other weak theorems 60G10 Stationary stochastic processes

Zbl 0374.60010
Full Text:

### References:

 [1] Adler, R.J., Weak convergence results for extremal processes generated by dependent random variables, Ann. probab., 6, 660-667, (1978) · Zbl 0377.60027 [2] Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201 [3] Davis, R.; Resnick, S., Limit theory for moving averages of random variables with regularly varying tail probabilities, Ann. probab., 13, 179-195, (1986) · Zbl 0562.60026 [4] Dwass, M., Extremal processes, Ann. math. statist., 35, 1718-1725, (1964) · Zbl 0171.38801 [5] Hsing, T., Point processes associated with extreme value theory, () [6] Hsing, T.; Hüsler, J.; Leadbetter, M.R., On the exceedance point process for a stationary sequence, () · Zbl 0619.60054 [7] Hsing, T., On the extreme order statistics for a stationary sequence, () · Zbl 0654.60021 [8] Kallenberg, O., Random measures, (1983), Akademie-Verlag Berlin, Academic Press, London, New York · Zbl 0288.60053 [9] Lamperti, J., On extreme order statistics, Ann. math. statist., 35, 1726-1737, (1964) · Zbl 0132.39502 [10] Leadbetter, M.R., On extreme value in stationary sequences, Z. wahrsch. verw. geb., 28, 289-303, (1974) · Zbl 0265.60019 [11] Leadbetter, M.R., Weak convergence of high level exceedances by a stationary sequence, Z. wahrsch. verw. geb., 34, 11-15, (1976) · Zbl 0339.60028 [12] Leadbetter, M.R.; Lindgren, G.; Rootzén, H., Extremes and related properties of random sequencees and processes, () · Zbl 0518.60021 [13] Matthes, K.; Kerstan, J.; Mecke, J., Infinitely divisible point processes, (1978), Wiley New York · Zbl 0383.60001 [14] Mori, T., Limit distributions of two-dimensional point processes generated by strong mixing sequences, Yokohama math. J., 25, 155-168, (1977) · Zbl 0374.60010 [15] O’Brien, G.L., Extreme values for stationary and Markov sequences, Ann. probab., 15, 281-289, (1987) · Zbl 0619.60025 [16] Parthasarathy, K.R., Probability measures on metric spaces, (1967), Academic Press New York · Zbl 0153.19101 [17] Pickands, J., The two dimensional Poisson process and external processes, J. appl. prob., 8, 745-756, (1971) · Zbl 0242.62024 [18] Resnick, S.I., Weak convergence to extreme processes, Ann. probab., 3, 951-960, (1975) · Zbl 0322.60024 [19] Rootzén, H., Extremes of moving averages of stable processes, Ann. probab., 6, 847, (1978) · Zbl 0394.60025 [20] Rootzén, H., Extreme value theory for moving average processes, Ann. probab., 14, 612-652, (1986) · Zbl 0604.60019 [21] Wiessman, I., Multivariate extremal processes generated by independent non-identically distributed variables, J. appl. prob., 12, 477-487, (1975)
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