Abstract
Given a nonincreasing null sequence \(T = (T_j)_{j \geqslant 1}\) of nonnegative random variables satisfying some classical integrability assumptions and \({\mathbb {E}}(\sum _{j}T_{j}^{\alpha })=1\) for some \(\alpha >0\), we characterize the solutions of the well-known functional equation
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 \({\mathbb {E}}\big (\sum _{j\geqslant 1}T_{j}^{\alpha }\log T_{j}\big )=0\);
-
(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.
Similar content being viewed by others
Data Availability
This manuscript was produced with no additional data.
References
Aïdékon, E.: Convergence in law of the minimum of a branching random walk. Ann. Probab. 41(3A), 1362–1426 (2013)
Aldous, D.J., Bandyopadhyay, A.: A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15(2), 1047–1110 (2005)
Alsmeyer, G., Biggins, J.D., Meiners, M.: The functional equation of the smoothing transform. Ann. Probab. 40(5), 2069–2105 (2012)
Alsmeyer, G., Iksanov, A.: A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks. Electron. J. Probab. 14(10), 289–312 (2009)
Biggins, J.D.: Martingale convergence in the branching random walk. J. Appl. Probab. 14(1), 25–37 (1977)
Biggins, J.D., Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36(2), 544–581 (2004)
Biggins, J.D., Kyprianou, A.E.: Fixed points of the smoothing transform: the boundary case. Electron. J. Probab. 10, 609–631 (2005). (electronic)
Chen, X.: A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk. Adv. Appl. Probab. 47(3), 741–760 (2015)
Choquet, G., Deny, J.: Sur l’équation de convolution \(\mu =\mu \ast \). C. R. Acad. Sci. Paris 250, 799–801 (1960)
Doney, R.A.: The Martin boundary and ratio limit theorems for killed random walks. J. Lond. Math. Soc. (2) 58(3), 761–768 (1998)
Durrett, R., Liggett, T.M.: Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64(3), 275–301 (1983)
Gut, A.: Stopped Random Walks. Limit Theorems and Applications. Springer Series in Operations Research and Financial Engineering, 2nd edn., p. 2009. Springer, New York (2009)
Kahane, J.-P., Peyrière, J.: Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22(2), 131–145 (1976)
Kyprianou, A.E.: Martingale convergence and the stopped branching random walk. Probab. Theory Related Fields 116(3), 405–419 (2000)
Liu, Q.: Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Probab. 30(1), 85–112 (1998)
Lyons, R.: A simple path to Biggins’ martingale convergence for branching random walk. In: Classical and Modern Branching Processes. (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, pp. 217–221. Springer, New York (1997)
Peyrière, J.: Turbulence et dimension de Hausdorff. C. R. Acad. Sci. Paris Sér. A 278, 567–569 (1974)
Spitzer, F.: Principles of Random Walks. Graduate Texts in Mathematics, vol. 34, 2nd edn. Springer, New York (1976)
Tanaka, H.: Time reversal of random walks in one-dimension. Tokyo J. Math. 12(1), 159–174 (1989)
Acknowledgements
Most of this work was done during a visit of the second author in March 2019 at the University of Münster. Financial support and kind hospitality are gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Gerold Alsmeyer: Funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure. Bastien Mallein: Partially funded by the ANR project 16-CE93-0003 (ANR MALIN)
Rights and permissions
About this article
Cite this article
Alsmeyer, G., Mallein, B. A Simple Method to Find All Solutions to the Functional Equation of the Smoothing Transform. J Theor Probab 35, 2569–2599 (2022). https://doi.org/10.1007/s10959-021-01151-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-021-01151-z
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