Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts
U
= (N(S,w
1), …, N(S,w
d
)), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of
U
by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of
U
is nonsingular, the error bounds decay at rate
O
((log n) / √n) in the i.i.d. case and
O
((log n)3 / √n) in the Markov case. In order for
U
to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.
From error bounds to the complexity of first-order descent methods for convex functions
