Citations: The distribution of subword counts is usually normal

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E. A. Bender and F. Kochman. The Distribution of Subwords Counts is Usually Normal. European Journal of Combinatorics, 14:265--275, 1993.

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Rare Events in Random Strings - Régnier, Denise (Correct)

....in random texts. So far, the rst moments , e.g. the mean and the variance, have been extensively studied by various authors under di erent probability models and di erent counting schemes [Wat95, R eg00, Szp01] Moreover, it is well known that this number converges, in law, to the normal law [BK93, PRdT95, RS97a, NSF99, FGSV01] when the size n of the text grows to in nity. Nevertheless, very few results are known out of the convergence domain, also called the central domain. We study here the tail distribution. First, we consider a single given word, H 1 . In [RS97a, NSF99] a large ....

....is valid for di erent counting models, we restrict here to the most commonly used, e.g. the overlapping model [Wat95] When several words are searched simultaneously, one needs some additional conditions on this set of words, H. It is generally assumed that the set H is reduced. De nition 2. 6 [BK93] A set of words is reduced if no word in this set is a proper factor of another word. The two words H 1 and H 2 do not play the same role in the conditional counting problem. One can partially relax the reduction condition: De nition 2.7 A couple of words (H 1 ; H 2 ) is reduced i the set fH ....

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Edward A. Bender and Fred Kochman. The Distribution of Subwords Counts is Usually Normal. European Journal of Combinatorics, 14:265-275, 1993.

Hidden Word Statistics - Flajolet, Szpankowski, Vallée (Correct)

....t n (of length n) The algorithms by Knuth Morris Pratt and Boyer Moore [11] provide ecient ways of nding such occurrences. Accordingly, the number of string occurrences in a random text has been intensively studied over the last two decades, with signi cant progress in this area being reported [5, 20, 21, 32, 33, 34, 41]. For instance Guibas and Odlyzko [20, 21] have revealed the fundamental r ole played by autocorrelation vectors and their associated polynomials. R egnier and Szpankowski [33, 34] established that the number of occurrences of a string is asymptotically normal under a diversity of models that ....

....the mean and the variance become of linear growth. To visualize the dependency of (W) on W , we observe that, when all the d j equal 1, the problem further reduces to traditional string matching , which was extensively studied in the past as witnessed by the (incomplete) list of references: [5, 20, 21, 32, 33, 34, 41]. It is well known that for string matching the variance coecient is a function of the so called autocorrelation of the string. In the general case of hidden pattern matching, the autocorrelation must be replaced by a more complex quantity that depends on the way pairs of constrained ....

[Article contains additional citation context not shown here]

E. A. Bender and F. Kochman, The distribution of subword counts is usually normal. European Journal of Combinatorics 14 (1993), 265-275.

Hidden Pattern Statistics - Flajolet, Guivarc'h, Szpankowski.. (2001) (6 citations) (Correct)

....of length n. The algorithms by Knuth Morris Pratt and Boyer Moore [7] provide efficient ways of finding such occurrences. Accordingly, the number of string occurrences in a random text has been intensively studied over the last two decades, with significant progress in this area being reported [3, 9, 10, 15 17, 24]. For instance Guibas and Odlyzko [9, 10] have revealed the fundamental role played by autocorrelation vectors and their associated polynomials. Regnier and Szpankowski [16, 17] established that the number of occurrences of a string is asymptotically normal under a diversity of models that include ....

....1, while the mean and the variance become of linear growth. To visualize the dependency of 2 (W) of W , we observe that, when all the d j equal 1, the problem reduces to traditional string matching that was extensively studied in the past as witnessed by the (incomplete) list of references: [3, 9, 10, 15 17, 24]. It is well known that for string matching the variance coefficient is a function of the so called autocorrelation of the string. In the 4 general case of hidden pattern matching, the autocorrelation must be replaced by a more complex quantity that depends on the way pairs of constrained ....

[Article contains additional citation context not shown here]

E. Bender and F. Kochman, The Distribution of Subword Counts is Usually Normal, European Journal of Combinatorics, 14, 265-275, 1993.

Hidden Pattern Statistics - Flajolet, Guivarc'h, al. (2001) (6 citations) (Correct)

....length n. The algorithms by Knuth Morris Pratt and Boyer Moore [3, 9] provide efficient ways of finding such occurrences. Accordingly, the number of string occurrences in a random text has been intensively studied over the last two decades, with significant progress in this area being reported [4, 14, 15, 24, 26, 27, 32]. For instance Guibas and Odlyzko [14, 15] have revealed the fundamental role played by autocorrelation vectors and their associated polynomials. Regnier and Szpankowski [26, 27] established that the number of occurrences of a string is asymptotically normal under a diversity of models that ....

....1, while the mean and the variance become of linear growth. To visualize the dependency of 2 (w) of w, we observe that, when all the d j equal 1, the problem reduces to the traditional string matching that was extensively studied in the past as witnessed by the (incomplete) list of references: [4, 14, 15, 24, 26, 27, 32]. It is well known that for string matching the variance coefficient is a function of the so called autocorrelation of the string. In the general case of hidden pattern matching, the autocorrelation must be replaced by a more complex quantity that depends on the way pairs of constrained ....

E. Bender and F. Kochman, The Distribution of Subword Counts is Usually Normal, European Journal of Combinatorics, 14, 265-275, 1993.

Motif Statistics - Nicodème, Salvy, Flajolet (1999) (4 citations) (Correct)

....statistics is vast. It originates largely with the introduction of correlation polynomials by #Guibas Odlyzko 1981# in the case of patterns de#ned by one word. The case of several words was studied by many authors, including #Guibas Odlyzko 1981#, #Flajolet, Kirschenhofer Tichy 1988#, and #Bender Kochman 1993#. Finite sets of words in Bernoulli or Markov texts are further considered by#R#egnier 1998; R#egnier Szpankowski 1998#. As a result of these works, the number of occurrences of any #nite set of patterns in a random Bernoulli or Markov text is known to be asymptotically normal; see also the ....

Bender, E. A., and Kochman, F. 1993. The distribution of subword counts is usually normal. European Journal of Combinatorics 14:265#275.

A Unified Approach to Word Statistics - Regnier (1998) (2 citations) (Correct)

....construction of the confidence interval for pattern occurrences. The problem of pattern occurrences in a random string is a classical one, Fel68, GO81, Li80, BWZ85, GM89, BK77, GGS 95, PRdT95, Sch95, Wat95] In this paper, frequency of pattern occurrences is fully characterized. It is known [BK93] that the limiting distribution is usually normal . Results below allow an easy computation of all moments, using for instance a symbolic computation system. The computation of the probability of occurrences in the finite range also follows. Moreover, most parameters of interest (average number ....

....7; 9; 12. All these occurrences are valid, and (NH 1 ; NH 2 ) 3; 4) This is the general scheme in the search of words that occur with unexpectedly high or low frequencies. A possible application is notably the search of tandem repeats [BT93, Ben98] The problem has been extensively studied in [BK93, PBM91, RS97a, RS97b, Lun90, Wat95] In the renewal model, studied in [BWZ85, TA97] two overlapping occurrences cannot be valid simultaneously. In a chain of overlapping occurrences, the first occurrence is always valid. An other occurrence is valid iff it does not overlap on the left with a ....

[Article contains additional citation context not shown here]

Edward A. Bender and Fred Kochman. The Distribution of Subwords Counts is Usually Normal. European Journal of Combinatorics, 14:265--275, 1993.

Motif Statistics - Nicodème, Salvy, Flajolet (1999) (4 citations) (Correct)

....statistics is vast. It originates largely with the introduction of correlation polynomials by (Guibas Odlyzko 1981) in the case of patterns defined by one word. The case of several words was studied by many authors, including (Guibas Odlyzko 1981) Flajolet, Kirschenhofer Tichy 1988) and (Bender Kochman 1993). Finite sets of words in Bernoulli or Markov texts are further considered by (R egnier 1998; R egnier Szpankowski 1998) As a result of these works, the number of occurrences of any finite set of patterns in a random Bernoulli or Markov text is known to be asymptotically normal; see also the ....

Bender, E. A., and Kochman, F. 1993. The distribution of subword counts is usually normal. European Journal of Combinatorics 14:265--275.

Assessing the significance of Sets of Words - Valentina Boeva Julien (2005) (Correct)

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E. A. Bender and F. Kochman. The Distribution of Subwords Counts is Usually Normal. European Journal of Combinatorics, 14:265--275, 1993.

Journal of Bioinformatics and Computational Biology - Imperial College Press (Correct)