The Distribution of Subword Counts is Usually Normal

Make the set of all n-long words from a finite alphabet into a probability space with a Bernoulli distribution. The joint probability distribution for `independent' counts of subwords from a finite set usually satisfies a central limit theorem, with means and covariances growing asymptotically with n. This usually remains true even when we condition on the values of other word counts, including the possibility of excluding certain words entirely. A local limit theorem also often holds. Practical formulas are given for computing the parameters when there is no conditioning. Impractical formulas are given for the general case. We correct errata in Mood's covariance matrices for runs count statistics.