Asymptotic properties and absolute continuity of laws stable by random weighted mean

By Quansheng Liu. 2001

112 citations recorded by the source corpus.

Abstract We study properties of stable-like laws, which are solutions of the distributional equation Z = d ∑ i=1 N A i Z i , where (N,A1,A2,…) is a given random variable with values in {0,1,…}×[0,∞)×[0,∞)×…, and Z,Z1,Z2,… are identically distributed positive random variables, independent of each other and independent of (N,A1,A2,…). Examples of such laws contain the laws of the well-known limit random variables in: (a) the Galton–Watson process or general branching processes, (b) branching random walks, (c) multiplicative processes, and (d) smoothing processes. For any solution Z (with finite or infinite mean), we find asymptotic properties of the distribution function P(Z⩽x) and those of the characteristic function EeitZ; we prove that the distribution of Z is absolutely continuous on (0,∞), and that its support is the whole half-line [0,∞). Solutions which are not necessarily positive are also considered.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a845fe29123edaf79dfb002edf47e40b3ebb38ee

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Citations in the corpus

The singularity spectrum of Lévy processes in multifractal time (2007, background): “The singularity spectrum of Lévy processes in multifractal time”
Random walk on sparse random digraphs (2015, background): “A more general version of (7) has been studied by R¨osler [38], Liu [30, 29, 31, 32] and Barral [4, 5].”
Branching random walks (2015, methodology): “For the existence4 of such a function Φ under our assumption (ψ(0) > 0, ψ(1) = 0 and ψ′(1) ≤ 0), see Liu [167].”
A survey of max-type recursive distributional equations (2004, background): “later. Where the state space is R+, the key ideas are from [28] which assumed N is nonrandom; the extensions to random N (which is a frequent setting for our max-type examples) have been developed in [49, 50]. Here is a typical result (Corollaries 1.5 and 1.6 of [49]; the case of nonrandom N is in [”
Moments, moderate and large deviations for a branching process in a random environment (2010, background): “Lemma (Liu, 2001) Let A be a positive random variable such that for some 0 < p < 1, t0 ≥ 0 and all t > t0, φ(t) ≤ pEφ(At).”
Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees (2007, background): “Under the additional assumption that {u, |u|= 1} contains at least two elements almost surely, it is possible (Liu [28]) to improve (3.”