分支线性递归的有效模拟
Efficient Simulation for Branching Linear Recursions
摘要：
本文考虑$$R^{(k+1)}\stackrel{\mathcal D}{=}\sum_{i=1}^{N}C_iR^{(k)}_i+Q,$$形式的线性递归，其中$(Q,N,C_1,C_2,\dots)$是具有$N\in\mathbb{N}=\{0, 1, 2, \dots\}$的实值随机向量，$\{R^{(k)}_i\}_{i\in\mathbb{N}}$是i.i.d. $R^{(k)}$的拷贝数，独立于$(Q,N,C_1,C_2,\dots)$，而$\stackrel{\mathcal{D}}{=}$表示分布相等。对于合适的向量$(Q,N,C_1,C_2,\dots)$并且假设$R^{(0)}$的初始分布是良好的，已知过程$R^{(k)}$收敛到对应的随机不动点方程的内生解，其出现在信息排序算法的分析中，例如，PageRank，以及分而治之算法的复杂性分析，例如快速排序。基于分支递归的朴素蒙特卡罗模拟$R^{(k)}$在$k$中具有指数级的复杂性，因此需要高效的方法。本文提出了一种迭代自举算法，具有线性复杂度，可用于近似采样$R^{(k)}$。我们证明了我们所提出的算法的基础上估计的一致性。
英文摘要：
We consider a linear recursion of the form $$R^{(k+1)}\stackrel{\mathcal D}{=}\sum_{i=1}^{N}C_iR^{(k)}_i+Q,$$ where $(Q,N,C_1,C_2,\dots)$ is a real-valued random vector with $N\in\mathbb{N}=\{0, 1, 2, \dots\}$, $\{R^{(k)}_i\}_{i\in\mathbb{N}}$ is a sequence of i.i.d. copies of $R^{(k)}$, independent of $(Q,N,C_1,C_2,\dots)$, and $\stackrel{\mathcal{D}}{=}$ denotes equality in distribution. For suitable vectors $(Q,N,C_1,C_2,\dots)$ and provided the initial distribution of $R^{(0)}$ is well-behaved, the process $R^{(k)}$ is known to converge to the endogenous solution of the corresponding stochastic fixed-point equation, which appears in the analysis of information ranking algorithms, e.g., PageRank, and in the complexity analysis of divide and conquer algorithms, e.g. Quicksort. Naive Monte Carlo simulation of $R^{(k)}$ based on the branching recursion has exponential complexity in $k$, and therefore the need for efficient methods. We propose in this paper an iterative bootstrap algorithm that has linear complexity and can be used to approximately sample $R^{(k)}$. We show the consistency of estimators based on our proposed algorithm.
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