Fixed Points of the Smoothing Transform: the Boundary Case

By J. Biggins, Andreas E. Kyprianou. 2005

119 citations recorded by the source corpus.

Let $A=(A_1,A_2,A_3,\ldots)$ be a random sequence of non-negative numbers that are ultimately zero with $E[\sum A_i]=1$ and $E \left[\sum A_{i} \log A_i \right] \leq 0$. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation $\Phi(\psi)= E \left[ \prod_{i} \Phi(\psi A_i) \right], $ where $\Phi$ is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when $E\left[\sum A_{i} \log A_i \right]<0$. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where $E\left[\sum A_{i} \log A_i \right]=0$, are obtained.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fb643f126161c3280e8e7f2f4d11af0d253bbb72

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Citations in the corpus

Fixed points of the smoothing transform: two-sided solutions (2010, background): “3), there is a substantial literature, [2,6,7, 9,10,14,19,22], and relatively complete results.”
Elementary fixed points of the BRW smoothing transforms with infinite number of summands (2003, methodology): “Fixed points of the BRW smoothing transforms with almost surely finite number of summands were studied (without using the name) by Biggins (1977), Biggins and Kyprianou (1996, 1997, 2001, 2003), Kyprianou (1998), Liu (2000).”
Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees (2007, background): “The following question is raised in Biggins and Kyprianou [9]: are there deterministic normalizers (an) such that Wn an converges? We aim at answering this question.”
The functional equation of the smoothing transform (2009, background): “This basic idea goes back at least to Neveu [36] and is used more recently in [14, 16] and [8].”
The Seneta–Heyde scaling for the branching random walk (2011, background): “The branching random walk is then said to be in the boundary case (Biggins and Kyprianou [13]).”
Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012, background): “In the context of branching random walks, the derivative martingale was introduced in the study of the fixed points of the smoothing transform at criticality (the smoothing transform is a generalization of Mandelbrot’s ⋆-equation for discrete multiplicative cascades; see also [16]).”
Gaussian multiplicative chaos and applications: A review (2013, background): “6) has an exact equivalent in the context of branching random walks [60, 29, 108], the solutions of which are called ”the fixed points of the smoothing transform”.”
Convergence in law of the minimum of a branching random walk (2013, unknown): “[7] Biggins, J.”
Branching random walks (2015, methodology): “Most of the material in this chapter can be found in Biggins and Kyprianou [58].”
On the invariant measure of the random difference equation $X_n=A_n X_{n-1}+ B_n$ in the critical case (2008, unknown): “n as Z. There exists an extensive literature, where the problems of existence, uniqueness and asymptotic behavior of Z were studied, see e.g. Durrett and Liggett [14], Liu [25], Biggins and Kyprianou [4]. The case of solutions having ﬁnite mean has been completely described, however in the boundary ”
Convergence in law of the minimum of a branching random walk (2011, background): “We assume that we are in the boundary case (in the sense of [8])”