Intermediate Regular and Π Variation

Abstract

A generalization of regular variation is discussed which is intermediate to extended regular variation and O-regular variation. Analogous to this intermediate regular variation is intermediate n-variation, a generalization of Pi-variation. Paralleling the theories of regular variation and Fl-variation, we demonstrate uniform convergence and representation theorems. We also prove a Karamata theorem and a Tauberian theorem for intermediate regular variation and in so doing we include an interesting extension to the corresponding results for O-regular variation.Contained in our proofs is the resolution of a measurability problem extant in other discussions of generalized regular variation.for all A e [l, A].(1.3)Evidently, these classes are nested: ERczIR <=OR (see Corollary 1.2). The inclusions are proper, as we will show by example and characterization. A thorough treatment of the classes ER and OR, and of related classes, is given by Bingham, Goldie, and Teugels in [8, with additional references. Details not included there are provided by Bingham and Goldie in [6,7]. The class IR fills a gap between ER and OR by allowing the power function bounds, X d and A c in (1.1), to be replaced with others which are also continuous at 1.

Precise large deviations for sums of random variables with consistently varying tails

“…The regularity property in (2.1) of the tail probability F was first introduced and named 'intermediate regular varying property' by Cline (1994). Some closely related discussions of the class C can be found in Cline and Samorodnitsky (1994).…”

Asymptotics for the Finite Time Ruin Probability in the Renewal Model with Consistent Variation

“…The regularity property in (2.1) was first introduced and called ''intermediate regularly varying'' property by Cline [9] . This class has been used in different studies of applied probability such as queueing system and ruin theory; see, for example, Schlegel [27] , Sec.…”

The impact of the service discipline on delay asymptotics

“…We make two assumptions: one regarding the distribution of the causal variable and one regarding S(τ). This assumption can be relaxed to distributions of intermediate regular variation, a class introduced by Cline [29], without invalidating Theorem 4.1, see [47]. …”

“…The coefficients of this polynomial can be obtained recursively by substitution into (29); in particular, p k−1 (0) = E{B k }. Taking κ = m we may use (24) and (30) With the FBPS discipline, the customers which so far have received the least amount of service share equally in the total capacity.…”