Counterbalancing steps at random in a random walk
Counterbalancing steps at random in a random walk
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随机游走中随机平衡步骤
DOI：
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发表时间：
2020
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通讯作者：
J. Bertoin
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作者：
J. Bertoin
具有平衡步骤的随机游走是一个部分和 $check S(n)=check X_1+ cdots + check X_n$ 的过程，其步骤 $check X_n$ 递归地给出如下。对于每个 $ngeq 2$，具有固定概率 $p$，$check X_n$ 是来自某个固定法则 $mu$ 的新独立样本，并且具有互补概率 $1-p$，$check X_n= -check X_{v(n)}$ 抵消了先前的步骤，$v(n)$ 是从 ${1, ldots, n-1}$ 中均匀随机选择的。我们根据 $p$ 和 $mu$ 的前两个矩确定 $check S(n)$ 的渐近行为。由于 H.A.，我们的方法依赖于与强化算法的耦合。 Simon，以及随机递归树和欧拉数的属性，这可能是独立的兴趣。该方法可以适应步长分布$mu$属于稳定律吸引域的情况。
A random walk with counterbalanced steps is a process of partial sums $check S(n)=check X_1+ cdots + check X_n$ whose steps $check X_n$ are given recursively as follows. For each $ngeq 2$, with a fixed probability $p$, $check X_n$ is a new independent sample from some fixed law $mu$, and with complementary probability $1-p$, $check X_n= -check X_{v(n)}$ counterbalances a previous step, with $v(n)$ a uniform random pick from ${1, ldots, n-1}$. We determine the asymptotic behavior of $check S(n)$ in terms of $p$ and the first two moments of $mu$. Our approach relies on a coupling with a reinforcement algorithm due to H.A. Simon, and on properties of random recursive trees and Eulerian numbers, which may be of independent interest. The method can be adapted to the situation where the step distribution $mu$ belongs to the domain of attraction of a stable law.