A Simple Method to Find All Solutions to the Functional Equation of the Smoothing Transform

Gerold Alsmeyer () and Bastien Mallein ()
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Gerold Alsmeyer: University of Münster
Bastien Mallein: LAGA, UMR 7539 Université Paris 13 - Sorbonne Paris Cité Université Paris 8

Journal of Theoretical Probability, 2022, vol. 35, issue 4, 2569-2599

Abstract: Abstract Given a nonincreasing null sequence $$T = (T_j)_{j \geqslant 1}$$ T = ( T j ) j ⩾ 1 of nonnegative random variables satisfying some classical integrability assumptions and $${\mathbb {E}}(\sum _{j}T_{j}^{\alpha })=1$$ E ( ∑ j T j α ) = 1 for some $$\alpha >0$$ α > 0 , we characterize the solutions of the well-known functional equation $$\begin{aligned} f(t)\,=\,\textstyle {\mathbb {E}}\left( \prod _{j\geqslant 1}f(tT_{j})\right) ,\quad t\geqslant 0, \end{aligned}$$ f ( t ) = E ∏ j ⩾ 1 f ( t T j ) , t ⩾ 0 , related to the so-called smoothing transform and its min-type variant. In order to do so within the class of nonnegative and nonincreasing functions, we provide a new three-step method whose merits are that (i) it simplifies earlier approaches in some relevant aspects; (ii) it works under weaker, close to optimal conditions in the so-called boundary case when $${\mathbb {E}}\big (\sum _{j\geqslant 1}T_{j}^{\alpha }\log T_{j}\big )=0$$ E ( ∑ j ⩾ 1 T j α log T j ) = 0 ; (iii) it can be expected to work as well in more general setups like random environment. At the end of this article, we also give a one-to-one correspondence between those solutions that are Laplace transforms and thus correspond to the fixed points of the smoothing transform and certain fractal random measures. The latter are defined on the boundary of a weighted tree related to an associated branching random walk.

Keywords: Stochastic fixed-point equation; Distributional fixed point; Smoothing transform; Branching random walk; Multiplicative martingales; Choquet–Deny-type lemma; Fractal random measure; Disintegration; Many-to-one lemma; Primary 39B22; Secondary 60E05; 60J85; 60G42 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-021-01151-z

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