On the number of occurrences of all short factors in almost all words

We previously proved that almost all words of length n over a finite alphabet A with m letters contain as factors all words of length k(n) over A as n → ∞, provided lim supn→∞ k(n)/ log n < 1/ log m. In this note it is shown that if this condition holds, then the number of occurrences of any word of length k(n) as a factor into almost all words of length n is at least s(n), where limn→∞ log s(n)/ log n = 0. In particular, this number of occurrences is bounded below by C log n as n →∞, for any absolute constant C > 0.