Fixed points of the smoothing transform: two-sided solutions

@article{Alsmeyer2010FixedPO,
  title={Fixed points of the smoothing transform: two-sided solutions},
  author={Gerold Alsmeyer and Matthias Meiners},
  journal={Probability Theory and Related Fields},
  year={2010},
  volume={155},
  pages={165-199}
}
Given a sequence (C, T) = (C, T1, T2, . . .) of real-valued random variables with Tj ≥ 0 for all j ≥ 1 and almost surely finite N = sup{j ≥ 1 : Tj > 0}, the smoothing transform associated with (C, T), defined on the set $${\mathcal{P}(\mathbb R)}$$ of probability distributions on the real line, maps an element $${P \in \mathcal{P}(\mathbb R)}$$ to the law of $${C + \sum_{j \geq 1} T_j X_j}$$ , where X1, X2, . . . is a sequence of i.i.d. random variables independent of (C, T) and with… 
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