论文标题
随机步行中随机平衡步骤
Counterbalancing steps at random in a random walk
论文作者
论文摘要
带有平衡步骤的随机步行是部分总和的过程$ \ check s（n）= \检查x_1 + \ cdots + \ check x_n $，其步骤$ \ check x_n $被递归如下。对于每个$ n \ geq 2 $，带有固定概率$ p $，$ \ check x_n $是来自某些固定定律$ $ $的新独立示例，并且具有互补的概率$ 1-p $，$ \ check x_n = - \ check x_n = x_ \ check x_ {v（n）} $以上步骤，与$ v（n）$ a y rystry pick $ n cyt-ust-l \ c \ c \ c \ c \ c。我们确定$ p $的$ \ check s（n）$的渐近行为和$μ$的前两个时刻。我们的方法依赖于H.A.引起的增强算法的耦合。西蒙（Simon），以及随机递归树和欧拉（Eulerian）数字的性质，这可能具有独立感兴趣。该方法可以适应台阶分布$μ$属于稳定法律的领域的情况。
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 $n\geq 2$, with a fixed probability $p$, $\check X_n$ is a new independent sample from some fixed law $μ$, 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 $μ$. 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 $μ$ belongs to the domain of attraction of a stable law.
