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The product-decomposability of probability measures on Abelian metrizable groups

Seria
Rozprawy Matematyczne tom/nr w serii: 351 wydano: 1996
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Warianty tytułu
Abstrakty

Introduction.............................................................5
I. Preliminaries.........................................................6
   1.1. Semigroups........................................7
   1.2. Algebraic groups..................................7
   1.3. Additive operators in Abelian groups and linear operators in linear spaces................................8
   1.4. Abelian metrizable groups........................10
   1.5. Locally compact Abelian groups...................13
   1.6. Transformation groups............................15
   1.7. Locally convex spaces............................16
   1.8. The space $ℝ^∞$..................................18
II. Basic properties of probability measures............................19
   2.1. Probability measures on metrizable spaces................................19
   2.2. Probability measures on transformation groups............................20
   2.3. Probability measures on Abelian metrizable groups........................22
   2.4. Invariant subgroups of probability measures..............................26
III. Borel decomposability semigroups of probability measures...................................29
   3.1. Additive measurable operators in Abelian metrizable groups...............29
   3.2. Borel decomposability semigroups of probability measures.................33
   3.3. Additive projections in Borel decomposability semigroups of probability measures.........................35
   3.4. Additive projections in Borel decomposability semigroups of probability measures without idempotent factors....................................41
IV. Product-decomposability of probability measures.....................46
   4.1. Basic definitions and results....................46
   4.2. Gaussian measures in the sense of Gnedenko...............................47
   4.3. Gaussian measures in the sense of Gnedenko without idempotent factors....49
   4.4. Product-atoms in Borel decomposability semigroups of probability measures without idempotent factors.....55
   4.5. Product-atomless probability measures without idempotent factors.........57
   4.6. Canonical product-decomposition of probability measures..................60
V. Product-decomposability of probability measures on locally convex metrizable spaces..........61
   5.1. Strong product-decomposability of probability measures on metrizable linear spaces.......................61
   5.2. Infinitely divisible probability measures on locally convex metrizable spaces............................65
   5.3. Gaussian measures on locally convex metrizable spaces....................67
   5.4. Product-atomless probability measures on locally convex metrizable spaces................................71
   5.5. Canonical product-decomposition and canonical strong product-decomposition of probability measures on locally convex metrizable spaces.........72
VI. Product decomposability of probability measures on LCA metrizable groups....................73
   6.1. Initial results on probability measures..........73
   6.2. Gaussian measures................................76
   6.3. Product-atomless probability measures............78
   6.4. Canonical product-decomposition of probability measures..................80
References..............................................................81
Index of symbols........................................................83
Subject index...........................................................85
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 351
Liczba stron
86
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLI
Daty
wydano
1996
otrzymano
1994-07-18
poprawiono
1994-12-22
Twórcy
  • Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, krakow@math.uni.wroc.pl
Bibliografia
  • [1] S. Banach, Théorie des opérations linéaires, Monograf. Mat. 1, Warszawa, 1932.
  • [2] J. F. Berglund, H. D. Junghenn and P. Milnes, Compact Right Topological Semigroups and Generalizations of Almost Periodicity, Lecture Notes in Math. 663, Springer, Berlin, 1984.
  • [3] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
  • [4] G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1972.
  • [5] E. Dettweiler, Grenzwertsätze für Wahrscheinlichkeitsmaße auf Badrikianschen Räumen, Z. Warsch. Verw. Gebiete 34 (1976), 285-311.
  • [6] G. M. Fel'dman, Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups, Transl. Math. Monographs 116, Amer. Math. Soc., Providence, R.I., 1993.
  • [7] L. Fuchs, Infinite Abelian Groups I, Academic Press, New York, 1970.
  • [8] B. V. Gnedenko, On a theorem of S. N. Bernshteĭn, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 97-100 (in Russian).
  • [9] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, New York, 1979.
  • [10] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1963.
  • [11] H. Heyer, Probability Measures on Locally Compact Groups, Springer, Berlin, 1977.
  • [12] B. Jessen and A. Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), 48-88.
  • [13] R. Kershner and A. Wintner, On symmetric Bernoulli convolutions, Amer. J. Math. 57 (1935), 541-548.
  • [14] A. I. Khinchin, Contributions à l'arithmétique des lois de distribution, Bull. Mosk. Univ. Sect. A 1 (1) (1937), 6-17.
  • [15] G. Köthe, Topological Linear Spaces I, Springer, Berlin, 1969.
  • [16] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967.
  • [17] W. Rudin, Independent perfect sets in groups, Michigan Math. J. 5 (1958), 159-161.
  • [18] W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962.
  • [19] H. H. Schaefer, Topological Vector Spaces, Springer, New York, 1971.
  • [20] L. Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Tata Inst. fund. Res. Stud. math. 6, Oxford Univ. Press, Bombay, 1973.
  • [21] A. Tortrat, Convolutions dénombrables équitendues dans un groupe topologique X, in: Les Probabilités sur les Structures Algébriques, Colloques Internationaux du CNRS, Clermont-Ferrand 1969, Éditions du Centre National de la Recherche Scientifique, Paris, 1970, 327-343.
  • [22] A. Tortrat, Structure des lois indéfiniment divisibles (μ ∈ I=I(X)) dans un espace vectoriel topologique (séparé) X, in: Symposium on Probability Methods in Analysis, Lecture Notes in Math. 31, Springer, 1967, 299-328.
  • [23] K. Urbanik, Lévy's probability measures on Euclidean spaces, Studia Math. 44 (1972), 119-148.
  • [24] K. Urbanik, Operator semigroups associated with probability measures, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 75-76.
  • [25] N. Vakhania, Probability Distributions in Linear Spaces, Mecniereba, Tbilisi, 1971 (in Russian).
  • [26] N. Vakhania, S. A. Chobanjan and V. I. Tarieladze, Probability Distributions on Banach Spaces, Reidel, Dordrecht, 1987.
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Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: 60B11, 60B15, 60E99.
Identyfikator YADDA
bwmeta1.element.zamlynska-d55d833c-4cfb-4b6a-b8ef-ea42c3a787d4
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ISSN
0012-3862
Kolekcja
DML-PL
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