Up-smooth Samples of Geometric Variables

Abstract

We study samples $\mathrm{\Gamma }=({\mathrm{\Gamma }}_1,\dots ,{\mathrm{\Gamma }}_n)$ of length $n$ where the letters ${\mathrm{\Gamma }}_i$ are independently generated according to the geometric distribution $\mathbb{P}({\mathrm{\Gamma }}_j=i)=p{q}^{i-1}$, for $1\le j\le n$, with $p+q=1$ and $0<p<1$. An $up-smoothsample$ $\mathrm{\Gamma }$ is a sample such that ${\mathrm{\Gamma }}_{i+1}-{\mathrm{\Gamma }}_i\le 1$. We find generating functions for the probability that a sample of $n$ geometric variables is up-smooth, with or without a specified first letter. We also extend the up-smooth results to words over an alphabet of $k$ letters and to compositions of integers. In addition, we study smooth samples $T$ of geometric random variables, where the condition now is $|{\mathrm{\Gamma }}_{i+1}-{\mathrm{\Gamma }}_i|\le 1$.