Summary
Between the operations which produce partial maxima and partial sums of a sequenceY 1,Y 2, ..., lies the inductive operation:X n =X n-1∨(αX n-1+Y n ),n≧1, for 0<α<1. If theY n are independent random variables with common distributionF, we show that the limiting behavior of normed sequences formed from {X n ,n≧1}, is, for 0<α<1, parallel to the extreme value case α=0. ForF∈D(Φγ) we give a full proof of the convergence, whereas forF∈D(Ψγ)∪D(Λ), we only succeeded in proving tightness of the involved sequence. The processX n is interesting for some applied probability models.
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Greenwood, P.E., Hooghiemstra, G. On the domain of attraction of an operator between supremum and sum. Probab. Th. Rel. Fields 89, 201–210 (1991). https://doi.org/10.1007/BF01366906
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DOI: https://doi.org/10.1007/BF01366906