Efficient Simulation for Branching Linear Recursions

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.Comment: submitted to WSC 201

Information Ranking and Power Laws on Trees

We study the situations when the solution to a weighted stochastic recursion has a power law tail. To this end, we develop two complementary approaches, the first one extends Goldie's (1991) implicit renewal theorem to cover recursions on trees; and the second one is based on a direct sample path large deviations analysis of weighted recursive random sums. We believe that these methods may be of independent interest in the analysis of more general weighted branching processes as well as in the analysis of algorithms