论文标题
几何分布的单词中不同相邻对的数量：概率和组合分析
The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis
论文作者
论文摘要
$ n $随机变量具有几何分布的弦的分析最近引起了新的兴趣：Archibald等。考虑几何分布的单词中不同相邻对的数量。在不同且相同的对的情况下，他们获得了此数字的渐近（$ n \ rightarrow \ infty $）的平均值。在本文中，在相同情况下，在不同情况下的渐近方差和在两种情况下的渐近分布中，我们对所有渐近矩都感兴趣。我们使用两种方法：第一种方法，即概率方法，在同一情况下的所有时刻以及两种情况下的分布中都导致差异和某些猜想。第二种方法是组合方法，依赖于多元模式匹配技术，为第一和第二矩提供了精确的公式。我们使用诸如Mellin Transforms，Analytic Comminatorics，Markov链等工具。
The analysis of strings of $n$ random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic ($n\rightarrow\infty$) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains.
