By Ph Barbe, W. P. Mccormick
The authors determine a few asymptotic expansions for limitless weighted convolution of distributions having on a regular basis various tails. purposes to linear time sequence types, tail index estimation, compound sums, queueing concept, branching methods, infinitely divisible distributions and implicit brief renewal equations are given.A noteworthy function of the technique taken during this paper is that during the advent of gadgets, which the authors name the Laplace characters, a hyperlink is demonstrated among tail quarter expansions and algebra. through advantage of this illustration process, a unified approach to identify expansions throughout various difficulties is gifted and, additionally, the strategy will be simply programmed in order that a working laptop or computer algebra package deal makes implementation of the strategy not just possible yet easy.
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Additional resources for Asymptotic Expansions for Infinite Weighted Convolutions of Heavy Tail Distributions and Applications (Memoirs of the American Mathematical Society)
Let p = (p0 , . . , pm ) be the coeﬃcients of the expansion of F in the asymptotic scale ei , 0 i m. Then, the coeﬃcients of FC in this asymptotic scale are given by q = LFC ,m L−1 Mc F,m Mc p . c∈C So, it suﬃces to prove that we can ﬁnd F such that q = e0 . Note that there is no loss of generality in assuming all the ﬁrst m moments of F ﬁxed, because ﬁxing these moments does not put any restriction on the vector p. Hence the ﬁrst m moments of FC are ﬁxed. In other words both the Laplace characters of F and FC can be taken ﬁxed.
It is therefore not clear that second-order regular variation provides the right framework for studying second-order expansions of FC . In particular, for some exceptional sequences of constants and some distributions, higher-order regular variation will be needed to obtain the exact second order. We also would like to point out that smooth variation of ﬁnite order is far easier to check than second-order regular variation, and that it holds for most — if not all — heavy tail distributions used in practical applications.
Take p such that the exponent of t is less than −m but such that EN m+α+p+ is ﬁnite. 1 hold. To conclude, note that for i positive and less than N , the equalities Mwi F = F and FW \wi = F (N −1) hold. As in Barbe and McCormick (2004), using the Laplace transforms of X1 and N allows one to derive a rather neat expression. Indeed, setting ΛX (t) = Ee−tX and ΛN (t) = Ee−tN , 52 4. 1) in Barbe and McCormick (2004) yields EN LF (N −1) ,m =− 0 j m 1 dj ΛN − log ΛX (u) j! duj ΛX (u) Dj . 1) u=0 Then, the technique explained in Barbe and McCormick (2004) allows for eﬃcient computation using computer algebra packages.