随机线性自返分布方程解的加权矩及应用 献给余家荣教授100华诞

假设(ak,bk)为一列独立同分布的取值于R^2的随机变量.考虑随机级数X=∑∞k=1^πk-1^bk的渐近性质,其中π0=1,πk=∏^ki=1^ai.当该级数几乎必然收敛时,它是由随机线性递归方程Xn=anXn-1+bn满足初始条件X0=x∈R所定义的随机序列(Xn)的极限分布,且是随机线性自返分布方程Xd=aX+b(分布相等)的唯一解,其中(a,b)=(a1,b1)与X相互独立.本文给出使加权矩E(|X|αl(|X|)存在的准则,其中α> 0,l是一个无穷远处的缓变函数.作为该结论的一个应用,本文得到光滑变换不动点方程Z=∑^Ni=1AiZi解的加权矩存在准则,其中(N,A1,A2,…)是一列随机变量,N∈N∪{∞},Ai∈R+,(Zi)是一列独立并与Z同分布的随机变量,且与(N,A1,A2,…)独立.本文也给出该准则在一般分枝过程和分枝随机游动中的应用,并证明任意一个具有有限均值的光滑变换的不动点可以从具有相同均值的初始分布出发由光滑变换迭代的极限得到.

Weighted moments of solutions of the stochastic linear recursive distributional equation and applications

Let(ak,bk) be a sequence of independent and identically distributed R2-valued random variables.We consider asymptotic properties of the random series X=∑∞k=1^πk-1^bk,where π0=1,πk=∏^ki=1^ai.When the series converges almost surely,it is the limit in law of the sequence(Xn) defined by the recursive stochastic linear equations Xn=anXn-1+bn,starting with X0=x∈R,and is the unique solution of the distributional equation Z=∑^Ni=1AiZi■(equality in law),where(a,b)=(a1,b1) is independent of X.We give a criterion for the existence of weighted moments of the form E(|X|αl(|X|),where α> 0,l is a function slowly varying at ∞.As an application,we obtain a criterion for the existence of weighted moments of solutions of the fixed point equation Xd=aX+b of the smoothing transform,where(N,A1,A2,...) is a given sequence of random variables with N ∈ N∪ {∞} and Ai ∈ R+,Zi are random variables independent of each other and independent of(N,A1,A2,...),each of which has the same distribution as Z which is unknown.Applications are also given to limit variables of general branching processes and branching random walks.We also give a proof of the fact that each fixed point with mean 1 of the smoothing transform can be obtained as the limit of iterations of the smoothing transform starting with an initial probability distribution of the same mean.