$\Pi$-regular variation
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- by J. L. Geluk PDF
- Proc. Amer. Math. Soc. 82 (1981), 565-570 Request permission
Abstract:
A function $U:{R^ + } \to {R^ + }$ is said to be $\Pi$-regularly varying with exponent $\alpha$ if $U(x){x^{ - \alpha }}$ is nondecreasing and there exists a positive function $L$ such that \[ \frac {{U(\lambda x)/{\lambda ^\alpha } - U(x)}} {{{x^\alpha }L(x)}} \to \log \lambda \quad (x \to \infty ){\text {for}}\lambda {\text { > 0}}{\text {.}}\] Suppose \[ \hat {U}(t)=\int _0^\infty {e^{ - tx}}dU(x){\text {exists for }}t > 0.\] We prove that $U$ is $\Pi$-regularly varying iff $U$ is $\Pi$-regularly varying.References
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Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 565-570
- MSC: Primary 40E05; Secondary 26A12
- DOI: https://doi.org/10.1090/S0002-9939-1981-0614879-X
- MathSciNet review: 614879