期刊论文

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

中文

中国科学.数学

1674-7216

2019

49

11

1687-1706

10.1360/N012019-00045

国家自然科学基金(批准号:11731012和11571052) 湖南省自然科学基金(批准号:2017JJ2271) 中南财经政法大学中央高校基本科研业务费专项资金(批准号:2722019JCG067) Centre Henri Lebesgue,France(批准号:ANR-11-LABX-0020-01)资助项目。

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假设(a k, b k )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)~ ∞ π_(k – 1) b_k的渐近性质,其中 π_0 = 1, π_k = ∏_(i=1)~k a_i. 当该级数几乎必然收敛时,它是由随机线性递归方程X_n = a_n X_)n – 1) + b_n满足初始条件X_0 = x ∈ R所定义的随机序列(X_n )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) =(a_1, b_1 )与X相互独立. 本文给出使加权矩E( | X |~α ℓ( | 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 = ∑_(i=1)~N A_i Z_i解的加权矩存在准则,其中(N, ...

Let (a_k, b_k ) be a sequence of independent and identically distributed R~2 -valued random variables. We consider asymptotic properties of the random series X = ∑_(k=1)~∞ π_(k – 1) b_k, where π_0 = 1, π_k = ∏_(i=1)~k a_i. When the series converges almost surely, it is the limit in law of the sequence (X_n ) defined by the recursive stochastic linear equations X_n = a_n X_(n – 1) + b_n, starting with X_0 = x ∈ R, and is the unique solution of the distributional equation X d = aX + b (equality in law), where (a, b) = (a_1, b_1 ) is inde...