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Tail behavior of solutions of linear recursions on trees

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  • Olvera-Cravioto, Mariana
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    Abstract

    Consider the linear nonhomogeneous fixed-point equation R=D∑i=1NCiRi+Q, where (Q,N,C1,C2,…) is a random vector with N∈{0,1,2,3,…}∪{∞},Ci≥0 for all i∈N, P(|Q|>0)>0, and {Ri}i∈N is a sequence of i.i.d. random variables independent of (Q,N,C1,C2,…) having the same distribution as R. It is known that R will have a heavy-tailed distribution under several different sets of assumptions on the vector (Q,N,C1,C2,…). This paper investigates the settings where either ZN=∑i=1NCi or Q are regularly varying with index −α<−1 and E[∑i=1NCiα]<1. This work complements previous results showing that P(R>t)∼Ht−α provided there exists a solution α>0 to the equation E[∑i=1N|Ci|α]=1, and both Q and ZN have lighter tails.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 4 ()
    Pages: 1777-1807

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1777-1807

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    Related research

    Keywords: Stochastic fixed-point equations; Weighted branching processes; Regular variation; Stochastic recursions; Large deviations; Random difference equations; Multiplicative cascades;

    References

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    1. Iksanov, Aleksander M., 2004. "Elementary fixed points of the BRW smoothing transforms with infinite number of summands," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 27-50, November.
    2. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
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