Generic subadditive functions

Bingham, N.; Ostaszewski, A.

doi:10.1090/S0002-9939-08-09504-X

Generic subadditive functions
HTML articles powered by AMS MathViewer

by N. H. Bingham and A. J. Ostaszewski PDF

Proc. Amer. Math. Soc. 136 (2008), 4257-4266 Request permission

Abstract:

We prove a generalization of the ‘Subadditive Limit Theorem’ and of the corresponding Berz Theorem in a class of functions that includes both the measurable functions and the ‘Baire functions’. The generic subadditive functions are defined by a combinatorial property previously introduced by the authors for the study of the foundations of regular variation. By specialization we provide the previously unknown Baire variants of the fundamental theorems of subadditive functions, answering an old question posed by Bingham, Goldie, and Teugels in 1987.

References

F. Bernstein and G. Doetsch, Zur Theorie der konvexen Funktionen, Math. Ann. 76 (1915), no. 4, 514–526 (German). MR 1511840, DOI 10.1007/BF01458222
Edgar Berz, Sublinear functions on $R$, Aequationes Math. 12 (1975), no. 2-3, 200–206. MR 387862, DOI 10.1007/BF01836548
N. H. Bingham and Charles M. Goldie, Extensions of regular variation. I. Uniformity and quantifiers, Proc. London Math. Soc. (3) 44 (1982), no. 3, 473–496. MR 656246, DOI 10.1112/plms/s3-44.3.473
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871, DOI 10.1017/CBO9780511721434
N. H. Bingham and A. J. Ostaszewski, Foundations of Regular Variation, CDAM Research Report Series, LSE-CDAM-2006-22.
N. H. Bingham and A. J. Ostaszewski, Beyond the Theorems of Steinhaus and Ostrowski: Combinatorial Versions, LSE-CDAM Report, LSE-CDAM-2007-15.
N. H. Bingham and A. J. Ostaszewski, Beyond Lebesgue and Baire: Generic Regular Variation, LSE-CDAM Report, LSE-CDAM-2007-16.
N. H. Bingham and A. J. Ostaszewski, Duality and the Kestelman-Borwein-Ditor Theorem, LSE-CDAM Report, LSE-CDAM-2007-29.
I. Csiszár and P. Erdös, On the function $g(t)=\lim \sup _{x\rightarrow \infty }(f(x+t)-f(x))$, Magyar Tud. Akad. Kut. Int. Közl. A 9 (1964), 603-606.
D. J. Daley, Upper bounds for the renewal function via Fourier methods, Ann. Probab. 6 (1978), no. 5, 876–884. MR 0494547
William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. MR 944909
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
J. F. C. Kingman, Subadditive processes, École d’Été de Probabilités de Saint-Flour, V–1975, Lecture Notes in Math., Vol. 539, Springer, Berlin, 1976, pp. 167–223. MR 0438477
Marek Kuczma, An introduction to the theory of functional equations and inequalities, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 489, Uniwersytet Śląski, Katowice; Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985. Cauchy’s equation and Jensen’s inequality; With a Polish summary. MR 788497
Thomas M. Liggett, Interacting particle systems, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276, Springer-Verlag, New York, 1985. MR 776231, DOI 10.1007/978-1-4613-8542-4
Janusz Matkowski and Tadeusz Świątkowski, On subadditive functions, Proc. Amer. Math. Soc. 119 (1993), no. 1, 187–197. MR 1176072, DOI 10.1090/S0002-9939-1993-1176072-2
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
R. A. Rosenbaum, Sub-additive functions, Duke Math. J. 17 (1950), 227–247. MR 36796