Ascents of Size Less Than $d$ in Samples of Geometric Random Variables: Mean, Variance, and Distribution

Abstract

We consider words ${\pi }_1{\pi }_2{\pi }_3\dots {\pi }_n$ of length $n$, where ${\pi }_i\in \mathbb{N}$ are independently generated with a geometric probability

$$P(\pi =k)=p(q{)}^{k-1}\text{where p + q = 1}.$$

Let $d$ be a fixed non-negative integer. We say that we have an ascent of size $d$ or more, an ascent of size less than $d$, a level, and a descent if ${\pi }_{i+1}\ge {\pi }_i+d$, ${\pi }_{i+1}{\pi }_{i+1}$, respectively.We determine the mean and variance of the number of ascents of size less than $d$ in a random geometrically distributed word. We also show that the distribution is Gaussian as $n$ tends to infinity.