Electronic Communications in Probability

Absolute continuity of complex martingales and of solutions to complex smoothing equations

Abstract

Let $X$ be a $\mathbb{C}$-valued random variable with the property that $X \ \text{ has the same law as } \ \sum _{j\ge 1} T_j X_j$ where $X_j$ are i.i.d. copies of $X$, which are independent of the (given) $\mathbb{C}$-valued random variables $(T_j)_{j\ge 1}$. We provide a simple criterion for the absolute continuity of the law of $X$ that requires, besides the known conditions for the existence of $X$, only finiteness of the first and second moment of $N$ - the number of nonzero weights $T_j$. Our criterion applies in particular to Biggins’ martingale with complex parameter.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 60, 12 pp.

Dates
Accepted: 21 July 2018
First available in Project Euclid: 12 September 2018

https://projecteuclid.org/euclid.ecp/1536718013

Digital Object Identifier
doi:10.1214/18-ECP155

Mathematical Reviews number (MathSciNet)
MR3863916

Zentralblatt MATH identifier
1401.60068

Citation

Damek, Ewa; Mentemeier, Sebastian. Absolute continuity of complex martingales and of solutions to complex smoothing equations. Electron. Commun. Probab. 23 (2018), paper no. 60, 12 pp. doi:10.1214/18-ECP155. https://projecteuclid.org/euclid.ecp/1536718013

References

• [1] John D. Biggins, Uniform convergence of martingales in the branching random walk, Ann. Probab. 20 (1992), no. 1, 137–151.
• [2] John D. Biggins and David R. Grey, Continuity of limit random variables in the branching random walk, J. Appl. Probab. 16 (1979), no. 4, 740–749.
• [3] Brigitte Chauvin, Quansheng Liu, and Nicolas Pouyanne, Limit distributions for multitype branching processes of $m$-ary search trees, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 2, 628–654.
• [4] Ron A. Doney, A limit theorem for a class of supercritical branching processes, J. Appl. Probability 9 (1972), 707–724.
• [5] Lars Hörmander, An introduction to complex analysis in several variables, revised ed., North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973, North-Holland Mathematical Library, Vol. 7.
• [6] Margarete Knape and Ralph Neininger, Pólya urns via the contraction method, Combin. Probab. Comput. 23 (2014), no. 6, 1148–1186.
• [7] Konrad Kolesko and Matthias Meiners, Convergence of complex martingales in the branching random walk: the boundary, Electron. Commun. Probab. 22 (2017), 14 pp.
• [8] Kevin Leckey, On Densities for Solutions to Stochastic Fixed Point Equations, To appear in Random Structures & Algorithms (2018+).
• [9] Quansheng Liu, Asymptotic properties and absolute continuity of laws stable by random weighted mean, Stochastic Process. Appl. 95 (2001), no. 1, 83–107.
• [10] Russell Lyons, A simple path to Biggins’ martingale convergence for branching random walk, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 217–221.
• [11] Mark M. Meerschaert and Hans-Peter Scheffler, Limit distributions for sums of independent random vectors, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 2001, Heavy tails in theory and practice.
• [12] Matthias Meiners and Sebastian Mentemeier, Solutions to complex smoothing equations, Probab. Theory Related Fields 168 (2017), no. 1–2, 199–268.
• [13] Bernt P. Stigum, A theorem on the Galton-Watson process, Ann. Math. Statist. 37 (1966), 695–698.
• [14] Kôsaku Yosida, Functional analysis, sixth ed., Grundlehren der Mathematischen Wissenschaften, vol. 123, Springer-Verlag, Berlin, 1980.