This article considers the random walk over R^p, with p ≥ 2, where the directions taken by the individual steps follow either the isotropic or the vonMises–Fisher distributions. Saddlepoint approximations to the density and to upper tail probabilities of the total distance covered by the random walk, i.e., of the length of the resultant, are derived. The saddlepoint approximations are onedimensional and simple to compute, even though the initial problem is p-dimensional. Numerical illustrations of the high accuracy of the proposed approximations are provided.

中文翻译：

---

多元各向同性和von Mises–Fisher随机游动的总距离分布的鞍点近似

本文讨论在R中的随机游动^p，与p ≥2，其中，由各个步骤所采取的方向跟随或者各向同性或vonMises -费希尔分布。得出了随机游走所覆盖的总距离的密度和上尾部概率的鞍点近似值，即结果长度的近似值。鞍点近似为一维且易于计算，即使最初的问题是p维。提供了所提出的近似值的高精度的数值图示。