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

Quansheng Liu

Stochastic Processes and their Applications, 2001, vol. 95, issue 1, 83-107

Abstract: We study properties of stable-like laws, which are solutions of the distributional equation where (N,A1,A2,...) is a given random variable with values in {0,1,...}x[0,[infinity])x[0,[infinity])x..., 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[less-than-or-equals, slant]x) and those of the characteristic function EeitZ; we prove that the distribution of Z is absolutely continuous on (0,[infinity]), and that its support is the whole half-line [0,[infinity]). Solutions which are not necessarily positive are also considered.

Keywords: Multiplicative; cascades; Branching; processes; Crump-Mode-Jagers; Branching; random; walks; Smoothing; processes; Martingales; Functional; equations; Moments; of; negative; orders; Left; tails; Decay; rate; of; characteristic; function; Absolute; continuity; Support (search for similar items in EconPapers)
Date: 2001
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Citations: View citations in EconPapers (11)

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