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.
Mathematics Subject Classification (2010): Primary 60F10; Secondary 26A12
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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.
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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
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DOI: https://doi.org/10.1007/978-3-642-33549-5_7
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