Maximums on trees

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:125

Article

Jelenkovic, Predrag R.¹  Olvera-Cravioto, Mariana² 

[1] Columbia Univ, Dept Elect Engn, New York, NY 10027 USA

[2] Columbia Univ, Dept Ind Engn & Operat Res, New York, NY 10027 USA

关键词: High-order Lindley equation;    Stochastic fixed-point equations;    Weighted branching processes;    Branching random walk;    Power law distributions;    Large deviations;    Cramer-Lundberg approximation;    Random difference equations;    Maximum recursion;   

DOI  :  10.1016/j.spa.2014.09.004

来源: Elsevier

【 摘 要 】

We study the minimal/endogenous solution R to the maximum recursion on weighted branching trees given by R (D) double under bar (V-i=1(N) CiRi) (sic) Q, where (Q, N, C-1, C-2,...) is a random vector with N is an element of N boolean OR {infinity}, P(vertical bar Q vertical bar > 0) > 0 and nonnegative weights {C-i}, and {R-i}(i is an element of N) is a sequence of i.i.d. copies of R independent of (Q, N, C-1, C-2,..); (D) double under bar denotes equality in distribution. Furthermore, when Q > 0 this recursion can be transformed into its additive equivalent, which corresponds to the maximum of a branching random walk and is also known as a highorder Lindley equation. We show that, under natural conditions, the asymptotic behavior of R is power-law, i.e., P (vertical bar R vertical bar > x) similar to Hx(-alpha), for some alpha > 0 and H > 0. This has direct implications for the tail behavior of other well known branching recursions. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

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