zbMATH — the first resource for mathematics

Random capacities and their distributions. (English) Zbl 0581.60042
We formalize the notion of an increasing and outer continuous random process, indexed by a class of compact sets, that map the empty set on zero. Existence and convergence theorems for distributions of such processes are proved. These results generalize or are similar to those known in the special cases of random measures, random (closed) sets and random (upper) semicontinuous functions. For the latter processes infinite divisibility under the maximum is introduced and characterized.
Our result generalizes known characterizations of infinite divisibility for random sets and max-infinite divisibility for random vectors. Also discussed is the convergence in distribution of the row-wise maxima of a null-array of random semicontinuous functions.

60G99 Stochastic processes
60E07 Infinitely divisible distributions; stable distributions
60F05 Central limit and other weak theorems
60G57 Random measures
Full Text: DOI
[1] Balkema, A.A., Resnick, S.I.: Max-infinite divisibility. J. Appl. Probab. 14, 309-319 (1977) · Zbl 0366.60025 · doi:10.2307/3213001
[2] Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968 · Zbl 0172.21201
[3] Billingsley, P.: Probability and Measure. New York: Wiley 1979 · Zbl 0411.60001
[4] Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131-295 (1953) · Zbl 0064.35101
[5] Choquet, G.: Lectures on analysis, vol. I, New York: W. A. Benjamin 1969 · Zbl 0181.39602
[6] Dolecki, S., Salinetti, G., Wets, R.J.-B.: Convergence of functions; equi-semicontinuity. Trans. Am. Math. Soc. 276, 409-429 (1983) · Zbl 0504.49006 · doi:10.1090/S0002-9947-1983-0684518-7
[7] Fell, J.M.G.: A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Proc. Am. Math. Soc. 13, 472-476 (1962) · Zbl 0106.15801 · doi:10.1090/S0002-9939-1962-0139135-6
[8] Gerritse, G.: Supremum self-decomposable random vectors. Dep. Math., Catholic University, Nijmegen. Report 8341 (1983) (Published in Probab. Th. Rel. Fields 72, 17-33 (1986)) · Zbl 0601.60017
[9] Graf, S.: A Radon-Nikodym theorem for capacities. J. Reine Angew. Math. 320, 192-214 (1980) · Zbl 0434.31004 · doi:10.1515/crll.1980.320.192
[10] Kallenberg, O.: Random measures. Berlin: Academie-Verlag, and London: Academic Press 1983 · Zbl 0544.60053
[11] Kisy?ski, J.: On the generation of tigh measures. Studia Math. 30, 141-151 (1968)
[12] Matheron, G.: Random sets and integral geometry. New York: Wiley 1975 · Zbl 0321.60009
[13] Norberg, T.: Convergence and existence of random set distributions. Ann. Probab. 12, 726-732 (1984) · Zbl 0545.60021 · doi:10.1214/aop/1176993223
[14] Salinetti, G., Wets, R.J.-B.: On the convergence in distribution of measurable multifunctions, normal integrands, stochastic processes and stochastic infima. International Institute for Applied Systems Analysis. Report CP-82-87 (1982). To appear in Math. Oper. Res.
[15] Verwaat, W.: The Structure of limit theorems in probability. Lect. Notes 1981
[16] Verwaat, W.: Random upper semicontinuous functions and extremal processes. University of Nijmegen 1982
[17] Weissman, I.: Extremal processes generated by independent nonidentically distributed random variables. Ann. Probab. 3, 172-177 (1975) · Zbl 0303.60031 · doi:10.1214/aop/1176996459
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.