We consider words or strings of characters a1a2a3⋯an of length n, where the letters ai ∈ℤ are independently generated with a geometric probability ℙ{X=k}=pqk-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 ai+1 ≥ai+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.