On the distribution of the number of missing words in random texts

Determining the distribution of the number of empty urns after a number of balls have been thrown randomly into the urns is a classical and well understood problem. We study a generalization: Given a finite alphabet of size sigma and a word length q, what is the distribution of the number X of words (of length q) that do not occur in a random text of length n + q - 1 over the given alphabet? For q = 1, X is the number Y of empty urns with sigma urns and n balls. For q greater than or equal to 2, X is related to the number Y of empty urns with sigma(q) urns and n balls, but the law of X is more complicated because successive words in the text overlap. We show that, perhaps surprisingly, the laws of X and Y are not as different as one might expect, but some problems remain currently open.