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Tail behavior of solutions of linear recursions on trees

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  • Olvera-Cravioto, Mariana

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.

Suggested Citation

  • Olvera-Cravioto, Mariana, 2012. "Tail behavior of solutions of linear recursions on trees," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1777-1807.
  • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1777-1807
    DOI: 10.1016/j.spa.2012.01.003
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    References listed on IDEAS

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    1. Liu, Quansheng, 2000. "On generalized multiplicative cascades," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 263-286, April.
    2. 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.
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