A stochastic fixed point equation for weighted minima and maxima

Alsmeyer, Gerold, and Rösler, Uwe. "A stochastic fixed point equation for weighted minima and maxima." Annales de l'I.H.P. Probabilités et statistiques 44.1 (2008): 89-103. <http://eudml.org/doc/77965>.

@article{Alsmeyer2008,
abstract = {Given any finite or countable collection of real numbers Tj, j∈J, we find all solutions Fto the stochastic fixed point equation \[W\stackrel\{\mathrm \{d\}\}\{=\}\inf \_\{j\in J\}T\_\{j\}W\_\{j\},\] whereW and the Wj, j∈J, are independent real-valued random variables with distribution Fand $\stackrel\{\mathrm \{d\}\}\{=\}$ means equality in distribution. The bulk of the necessary analysis is spent on the case when |J|≥2 and all Tj are (strictly) positive. Nontrivial solutions are then concentrated on either the positive or negative half line. In the most interesting (and difficult) situation T has a characteristic exponent α given by ∑j∈JTjα=1 and the set of solutions depends on the closed multiplicative subgroup of ℝ&gt;=(0, ∞) generated by the Tj which is either \{1\}, ℝ&gt; itself or rℤ=\{rn:n∈ℤ\} for some r&gt;1. The first case being trivial, the nontrivial fixed points in the second case are either Weibull distributions or their reciprocal reflections to the negative half line (when represented by random variables), while in the third case further periodic solutions arise. Our analysis builds on the observation that the logarithmic survival function of any fixed point is harmonic with respect to Λ=∑j≥1δTj, i.e. Γ=Γ⋆Λ, where ⋆ means multiplicative convolution. This will enable us to apply the powerful Choquet–Deny theorem.},
author = {Alsmeyer, Gerold, Rösler, Uwe},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic fixed point equation; weighted minima and maxima; weighted branching process; harmonic analysis on trees; Choquet–Deny theorem; Weibull distributions; Choquet-Deny theorem},
language = {eng},
number = {1},
pages = {89-103},
publisher = {Gauthier-Villars},
title = {A stochastic fixed point equation for weighted minima and maxima},
url = {http://eudml.org/doc/77965},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Alsmeyer, Gerold
AU - Rösler, Uwe
TI - A stochastic fixed point equation for weighted minima and maxima
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 1
SP - 89
EP - 103
AB - Given any finite or countable collection of real numbers Tj, j∈J, we find all solutions Fto the stochastic fixed point equation \[W\stackrel{\mathrm {d}}{=}\inf _{j\in J}T_{j}W_{j},\] whereW and the Wj, j∈J, are independent real-valued random variables with distribution Fand $\stackrel{\mathrm {d}}{=}$ means equality in distribution. The bulk of the necessary analysis is spent on the case when |J|≥2 and all Tj are (strictly) positive. Nontrivial solutions are then concentrated on either the positive or negative half line. In the most interesting (and difficult) situation T has a characteristic exponent α given by ∑j∈JTjα=1 and the set of solutions depends on the closed multiplicative subgroup of ℝ&gt;=(0, ∞) generated by the Tj which is either {1}, ℝ&gt; itself or rℤ={rn:n∈ℤ} for some r&gt;1. The first case being trivial, the nontrivial fixed points in the second case are either Weibull distributions or their reciprocal reflections to the negative half line (when represented by random variables), while in the third case further periodic solutions arise. Our analysis builds on the observation that the logarithmic survival function of any fixed point is harmonic with respect to Λ=∑j≥1δTj, i.e. Γ=Γ⋆Λ, where ⋆ means multiplicative convolution. This will enable us to apply the powerful Choquet–Deny theorem.
LA - eng
KW - stochastic fixed point equation; weighted minima and maxima; weighted branching process; harmonic analysis on trees; Choquet–Deny theorem; Weibull distributions; Choquet-Deny theorem
UR - http://eudml.org/doc/77965
ER -