Error bounds on multivariate Normal approximations for word count statistics
Haiyan Huang · Advances in Applied Probability · 2002
Given a sequenceSand a collection Ω ofdwords, it is of interest in many applications to characterize the multivariate distribution of the vector of countsU= (N(S,w1), …,N(S,wd)), whereN(S,w) is the number of times a wordw∈ Ω appears in the sequenceS. We obtain an explicit bound on the error made when approximating the multivariate distribution ofUby 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 ofUis nonsingular, the error bounds decay at rateO((logn) / √n) in the i.i.d. case andO((logn)3/ √n) in the Markov case. In order forUto have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is notfull, that is, that Ω is not the collection of all possible words of some lengthkover the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.