arXiv 2004.089662020-04math.PR

树上极大值的重要抽样

Importance sampling for maxima on trees

Bojan Basrak, Michael Conroy, Mariana Olvera-Cravioto, Zbigniew Palmowski

摘要：

我们考虑分布不动点方程：$$R \stackrel{\mathcal{D}}{=} Q \vee \left( \bigvee_{i=1}^N C_i R_i \right),$$其中$\{R_i\}$是i.i.d.~ $R$的拷贝，独立于载体$(Q, N, \{C_i\})$，其中$N \in \mathbb{N}$、$Q, \{C_i\} \geq 0$和$P(Q > 0) > 0$。通过设置$W = \log R$、$X_i = \log C_i$、$Y = \log Q$，它等价于高阶Lindley方程$$W \stackrel{\mathcal{D}}{=} \max\left\{ Y, \, \max_{1 \leq i \leq N} (X_i + W_i) \right\}.$$。已知在Kesten假设下，$$P(W > t) \sim H e^{-αt}, \qquad t \to \infty,$$，其中$α>0$求解Cramér-Lundberg方程$E \left[ \sum_{j=1}^N C_i ^α\right] = E\left[ \sum_{i=1}^N e^{αX_i} \right] = 1$。本文的主要目标是提供一个明确的表示$P(W > t)$，它可以直接连接到底层的加权分支过程$W$的构造，并可用于构建无偏和强有效的估计所有$t$。此外，我们展示了如何使用Alsmeyer的马尔可夫更新定理，这种新的表示可以直接分析，产生一个替代的表示常数$H$。我们提供数值例子说明使用这种新算法。

英文摘要：

We consider the distributional fixed-point equation: $$R \stackrel{\mathcal{D}}{=} Q \vee \left( \bigvee_{i=1}^N C_i R_i \right),$$ where the $\{R_i\}$ are i.i.d.~copies of $R$, independent of the vector $(Q, N, \{C_i\})$, where $N \in \mathbb{N}$, $Q, \{C_i\} \geq 0$ and $P(Q > 0) > 0$. By setting $W = \log R$, $X_i = \log C_i$, $Y = \log Q$ it is equivalent to the high-order Lindley equation $$W \stackrel{\mathcal{D}}{=} \max\left\{ Y, \, \max_{1 \leq i \leq N} (X_i + W_i) \right\}.$$ It is known that under Kesten assumptions, $$P(W > t) \sim H e^{-αt}, \qquad t \to \infty,$$ where $α>0$ solves the Cramér-Lundberg equation $E \left[ \sum_{j=1}^N C_i ^α\right] = E\left[ \sum_{i=1}^N e^{αX_i} \right] = 1$. The main goal of this paper is to provide an explicit representation for $P(W > t)$, which can be directly connected to the underlying weighted branching process where $W$ is constructed and that can be used to construct unbiased and strongly efficient estimators for all $t$. Furthermore, we show how this new representation can be directly analyzed using Alsmeyer's Markov renewal theorem, yielding an alternative representation for the constant $H$. We provide numerical examples illustrating the use of this new algorithm.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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