We consider words or strings of characters $a_1a_2a_3 \ldots a_n$ of length $n$, where the letters $a_i \in \mathbb{Z}$ are independently generated with a geometric probability $\mathbb{P} \{ X=k \} = pq^{k-1}$ where $p+q=1$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more if $a_{i+1} \geq a_i+d$. We determine the mean, variance and limiting distribution of the number of ascents of size $d$ or more in a random geometrically distributed word.