Geodesic
Generalization of Portmanteau Theorem with Respect to the Pseudoweak Convergence of Random Closed Sets
Teoriâ veroâtnostej i ee primeneniâ, Tome 54 (2009) no. 1, pp. 97-115 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The main result of this paper is a theorem of portmanteau type for pseudoweak convergent sequences of capacity functionals for a sequence of random closed sets. For that purpose the classical Lebesgue integral had been substituted with a more general one, known as general pseudo-integral, and there is introduced the pseudoweak convergence of capacity functionals. A connection between weak convergence of a sequence of probability measures induced by the sequence of random closed sets and convergence of pseudo-integral with respect to the corresponding sequence of capacity functionals is given.
Keywords: pseudo-operations, pseudo-integral, random closed set, capacity functional, pseudoweak convergence.
Mots-clés : portmanteau theorem 
@article{TVP_2009_54_1_a5,
     author = {T. Grbi\'c and E. Pap},
     title = {Generalization of {Portmanteau} {Theorem} with {Respect} to the {Pseudoweak} {Convergence} of {Random} {Closed} {Sets}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {97--115},
     year = {2009},
     volume = {54},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2009_54_1_a5/}
}
TY  - JOUR
AU  - T. Grbić
AU  - E. Pap
TI  - Generalization of Portmanteau Theorem with Respect to the Pseudoweak Convergence of Random Closed Sets
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2009
SP  - 97
EP  - 115
VL  - 54
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TVP_2009_54_1_a5/
LA  - en
ID  - TVP_2009_54_1_a5
ER  - 
%0 Journal Article
%A T. Grbić
%A E. Pap
%T Generalization of Portmanteau Theorem with Respect to the Pseudoweak Convergence of Random Closed Sets
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2009
%P 97-115
%V 54
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_2009_54_1_a5/
%G en
%F TVP_2009_54_1_a5
T. Grbić; E. Pap. Generalization of Portmanteau Theorem with Respect to the Pseudoweak Convergence of Random Closed Sets. Teoriâ veroâtnostej i ee primeneniâ, Tome 54 (2009) no. 1, pp. 97-115. http://geodesic.mathdoc.fr/item/TVP_2009_54_1_a5/

[1] Benvenuti P., Mesiar R., Vivona D., “Monotone set functions-based integrals”, Handbook of Measure Theory, v. 2, ed. E. Pap, North-Holland, Amsterdam, 2002, 1329–1379 | MR | Zbl

[2] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977, 351 pp. | MR

[3] Choquet G., “Theory of capacities”, Ann. Inst. Fourier, 5 (1953), 131–295 | MR

[4] Denneberg D., Non-Additive Measure and Integral, Kluwer, Dordrecht, 1994, 178 pp. | MR | Zbl

[5] Fuks L., Chastichno uporyadochennye algebraicheskie sistemy, Mir, M., 1965, 342 pp. | MR

[6] Grbić T., Pap E., “Pseudo-weak convergence of the random sets defined by a pseudo-integral based on non-additive measure”, International Workshop “Idempotent and Tropical Mathematics and Problems of Mathematical Physics”, v. I (Moscow, 2007), eds. G. L. Litvinov, V. P. Maslov, and S. N. Sergeev, Independent University of Moscow, Moscow, 2007, 72–77

[7] Goutsias J., Modeling random shapes: an introduction to the random closed set theory, Technical Report JHU/ECE, 1990

[8] Jupp D. L. B., Strahler A. H., Woodcock C. E., “Autocorrelation and regularization in digital images. I. Basic theory”, IEEE Trans. Geoscience and Remote Sensing, 26 (1988), 463–473 | DOI

[9] Jupp D. L. B., Strahler A. H., Woodcock C. E., “Autocorrelation and regularization in digital images. II. Simple image models”, IEEE Trans. Geoscience and Remote Sensing, 27 (1989), 247–258 | DOI | MR

[10] Kendall D. G., “Foundations of a theory of random sets”, Stochastic Geometry, eds. E. F. Harding and D. G. Kendall, Wiley, London, 1974, 322–376 | MR

[11] Klein E., Thompson A. C., Theory of Correspondences, Wiley, New York, 1984, 256 pp. | MR

[12] Klement E. P., Mesiar R., Pap E., Triangular Norms, Kluwer, Dordrecht, 2000, 385 pp. | MR | Zbl

[13] Kuich W., Salomaa A., Semirings, Automata, Languages, Springer-Verlag, Berlin, 1986, 374 pp. | MR | Zbl

[14] Litvinov G. L., “The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction”, Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, 2005, 1–17 | MR | Zbl

[15] Litvinov G. L., Maslov V. P. (eds.), Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Amer. Math. Soc., Providence, 2005 | MR

[16] Maslov V. P., Samoborskiĭ S. N. (eds.), Idempotent Analysis, Adv. in Soviet Math., 13, Amer. Math. Soc., Providence, 1992 | MR

[17] Materon Zh., Sluchainye mnozhestva i integralnaya geometriya, Mir, M., 1978, 318 pp. | MR

[18] Molchanov I. S., “Ob odnom obobschenii teoremy Shoke dlya sluchainykh mnozhestv s dannym klassom realizatsii”, Teoriya veroyatn. i matem. statist., 28 (1984), 99–106 | Zbl

[19] Molchanov I. S., “Empiricheskoe otsenivanie kvantilei raspredelenii sluchainykh zamknutykh mnozhestv”, Teoriya veroyatn. i ee primen., 35:3 (1990), 586–592 | MR

[20] Molchanov I. S., “Sostoyatelnaya otsenka parametrov bulevykh modelei sluchainykh zamknutykh mnozhestv”, Teoriya veroyatn. i ee primen., 36:3 (1991), 580–587 | MR

[21] Molchanov I. S., Limit Theorems for Unions of Random Closed Sets, Lecture Notes in Math., 1561, Springer-Verlag, Berlin, 1993, 157 pp. | MR | Zbl

[22] Molchanov I., Theory of Random Sets, Springer-Verlag, London, 2005, 488 pp. | MR

[23] Nguyen H. T., “Choquet weak convergence of capacity functionals of random sets”, Soft Methodology and Random Information Systems, Springer-Verlag, Berlin, 2004, 19–31 | MR | Zbl

[24] Nguyen H. T., Bouchon-Meunier B., “Random sets and large deviations principle as a foundation for possibility measures”, Soft Computing, 8 (2004), 61–70

[25] Pap E., Null-Additive Set Functions, Kluwer, Dordrecht, 1995, 315 pp. | MR

[26] Pap E., “Pseudo-additive measures and their applications”, Handbook of Measure Theory, v. II, ed. E. Pap, North-Holland, Amsterdam, 2002, 1403–1468 | MR | Zbl

[27] Pap E., Grbić T., Nedović Lj., Ralević N. M., “Weak convergence of random sets”, 3rd Serbian-Hungarian Joint Symposium on Intelligent Systems (Subotica, 2005), 73–80

[28] Puhalskii A., Large Deviations and Idempotent Probability, Chapman Hall/CRC, Boca Raton, 2001, 500 pp. | MR | Zbl

[29] Serra J., Image Analysis and Mathematical Morphology, Academic Press, London, 1984, 610 pp. | MR