Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Regularly varying sequences

Authors: J. Galambos and E. Seneta
Journal: Proc. Amer. Math. Soc. 41 (1973), 110-116
MSC: Primary 26A12
MathSciNet review: 0323963
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A simple necessary and sufficient condition is developed for a sequence $ \{ \theta (n)\} ,n = 0,1,2, \cdots $, of positive terms, to satisfy $ \theta (n) = R(n),n \geqq 0$, where $ R( \cdot )$ is a regularly varying function on $ [0,\infty )$. The condition (2.1), below, leads to a Karamata-type exponential representation for $ \theta (n)$. Various associated difficulties are also discussed. (The results are of relevance in connection with limit theorems in various branches of probability theory.)

References [Enhancements On Off] (What's this?)

  • [J] Aczél, Vorlesungen über Funktionalgleichungen und ihre Anwendungen, Lehrbücker und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, Band 25, Birkhäuser Verlag, Basel, 1961; English transl., Math. in Sci. and Engineering, vol. 19, Academic Press, New York, 1966. MR 23 #A1959; 34 #8020. MR 0124647 (23:A1959)
  • [R] Bojanić and E. Seneta, Slowly varying functions and asymptotic relations, J. Math. Anal. Appl. 34 (1971), 302-315. MR 43 #438. MR 0274676 (43:438)
  • [N] G. de Bruijne, Pairs of slowly oscillating functions occurring in asymptotic problems concerning the Laplace transform, Nieuw. Arch. Wisk. (3) 7 (1959), 20-26. MR 21 #5847. MR 0107120 (21:5847)
  • [L] de Haan, On regular variation and its application to the weak convergence of sample extremes, Mathematical Centre Tracts, 32, Mathematisch Centrum, Amsterdam, 1970. MR 44 #3370. MR 0286156 (44:3370)
  • [I] A. Ibragimov and Ju. V. Linnik, Independent and stationarily connected variables, Nauka, Moscow, 1965; English transl., Noordhoff, Groningen, 1971. MR 34 #2049. MR 0202176 (34:2049)
  • [J] Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38-53.
  • 1. -, Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France 61 (1933), 55-62. MR 1504998
  • [J] P. Kubilius, Probabilistic methods in the theory of numbers, Gos. Izdat. Polit. Naučn. Lit. Litovsk. SSR, Vilna, 1962; English transl., Transl. Math. Monographs, vol. 11, Amer. Math. Soc., Providence, R.I., 1964. MR 26 #3691 ; 28 #3956. MR 0131415 (24:A1266)
  • [R] Schmidt, Über divergente Folgen und lineare Mittelbildungen, Math. Z. 22 (1925), 89-152. MR 1544717
  • [I] Schur, Zur theorie der Césaroschen und Holderschen Mittelwerte, Math. Z. 31 (1930), 391-407. MR 1545122

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A12

Retrieve articles in all journals with MSC: 26A12

Additional Information

Keywords: Regularly varying functions, regularly varying sequences, imbedding sequences in functions, exponential representations of sequences and functions
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society