Pattern statistics and Vandermonde matrices

In this paper, we determine some limit distributions of pattern statistics in rational stochastic models. We present a general approach to analyze these statistics in rational models having an arbitrary number of strongly connected components. We explicitly establish the limit distributions in most significant cases; they are characterized by a family of unimodal density functions defined by means of confluent Vandermonde matrices.

[1]  Philippe Flajolet,et al.  Motif Statistics , 1999, ESA.

[2]  Alberto Bertoni,et al.  On the number of occurrences of a symbol in words of regular languages , 2003, Theor. Comput. Sci..

[3]  F. Csaki,et al.  Some notes on the inversion of confluent Vandermonde matrices , 1975 .

[4]  Diego de Falco,et al.  Frequency of symbol occurrences in bicomponent stochastic models , 2004, Theor. Comput. Sci..

[5]  Wojciech Szpankowski,et al.  On The Collapse of the q-Gram Filtration , 1998 .

[6]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[7]  Jérémie Bourdon,et al.  Generalized Pattern Matching Statistics , 2002 .

[8]  Jean Berstel,et al.  Rational series and their languages , 1988, EATCS monographs on theoretical computer science.

[9]  Karla Rost,et al.  Matrix exponentials and inversion of confluent Vandermonde matrices. , 2004 .

[10]  Egon Börger,et al.  Trends in theoretical computer science , 1988 .

[11]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[12]  Mireille Régnier,et al.  On Pattern Frequency Occurrences in a Markovian Sequence , 1998, Algorithmica.

[13]  Philippe Flajolet,et al.  An introduction to the analysis of algorithms , 1995 .

[14]  Esko Ukkonen,et al.  Two Algorithms for Approximate String Matching in Static Texts , 1991, MFCS.

[15]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[16]  E. Seneta.  Non-negative Matrices and Markov Chains , 2008 .

[17]  Leo J. Guibas,et al.  Maximal Prefix-Synchronized Codes , 1978 .

[18]  Arto Salomaa,et al.  Automata-Theoretic Aspects of Formal Power Series , 1978, Texts and Monographs in Computer Science.

[19]  Boris Gnedenko,et al.  Theory of Probability , 1963 .

[20]  Leonidas J. Guibas,et al.  String Overlaps, Pattern Matching, and Nontransitive Games , 1981, J. Comb. Theory A.

[21]  Bernard Prum,et al.  Finding words with unexpected frequencies in deoxyribonucleic acid sequences , 1995 .

[22]  Michel Rigo,et al.  Additive Functions with Respect to Numeration Systems on Regular Languages , 2003 .

[23]  Massimiliano Goldwurm.  Probabilistic Estimation of the Number of Prefixes of a Trace , 1992, Theor. Comput. Sci..

[24]  Valerie Isham,et al.  Non‐Negative Matrices and Markov Chains , 1983 .