Abstract
Karamata’s integral representation for slowly varying functions is extended to a broader class of the so-called ψ-locally constant functions, i.e. functions f(x) > 0 having the property that, for a given non-decreasing function ψ(x) and any fixed v, f(x + vψ(x)) ∕ f(x) → 1 as x → ∞. We consider applications of such functions to extending known theorems on large deviations of sums of random variables with regularly varying distribution tails.
Keywords
Mathematics Subject Classification (2010): Primary 60F10; Secondary 26A12
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bingham, N.H., Goldie, C.M., Teugels, I.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Borovkov, A.A., Borovkov, K.A.: Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions. Cambridge University Press, Cambridge (2008)
Cline, D.B.H., Hsing, T.: Large Deviations Probabilities for Sums and Maxima of random variables with Heavy or Subexponential Tails. Texas A &M University, College Station (1991), Preprint
Cline, D.B.H.: Intermediate regular and Π variation. Proc. Lond. Math. Soc. 68, 594–616 (1994)
Foss, S., Korshunov, D., Zachary, S.: An Introduction to Heavy-tailed and Subexponential Distributions. Springer, New York (2011)
Ng, K.W., Tang, Q., Yan, J.-A., Yang H.: Precise large deviation for sums of random variables with consistently varying tails. J. Appl. Prob. 41, 93–107 (1994)
Acknowledgements
Research supported by the Russian Foundation for Basic Research Grant 08–01–00962, Russian Federation President Grant NSh-3695.2008.1, and the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Borovkov, A.A., Borovkov, K.A. (2013). An Extension of the Concept of Slowly Varying Function with Applications to Large Deviation Limit Theorems. In: Shiryaev, A., Varadhan, S., Presman, E. (eds) Prokhorov and Contemporary Probability Theory. Springer Proceedings in Mathematics & Statistics, vol 33. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33549-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-33549-5_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33548-8
Online ISBN: 978-3-642-33549-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)