Pattern correlation matrices and their properties (Q5935365)

In analyzing sequences of symbols such as text, the pattern correlation polynomials of \textit{L. J. Guibas} and \textit{A. M. Odlyzko} [J. Comb. Theory, Ser. A 30, 183-208 (1981; Zbl 0454.68109)] are used to analyze the probabilities of counts of occurrences of strings given probabilities of individual symbols. These involve counts of overlaps between the given set of strings. \textit{M. Régnier} and \textit{W. Szpankowski} [Algorithmica 22, No. 4, 631-649 (1998; Zbl 0918.68108)] formed a pattern correlation matrix from these. This paper gives a Jordan block representation, a structure for the row and column spaces, and derives results on the covariance matrix of the joint distribution of frequencies of these strings, and derives a particular statistic with a chi-square distribution.