Let {Yn}n∈Z+ be a sequence of random variables in Rd and let A ⊂ Rd. Then P{Yn ∈ A for some n} is the hitting probability of the set A by the sequence {Yn}. We consider the asymptotic behavior, as m → ∞, of P{Yn ∈ mA, some n} = P{hitting mA} whenever (1) the probability law of Yn/n satisfies the large deviation principle and (2) the central tendency of Yn/n is directed away from the given set A. For a particular function Ĩ,weshowP{Yn∈mA, some n} ≈e−mĨ(A). 1

We consider local alignments without gaps of two independent Markov chains from a finite alphabet, and we derive sufficient conditions for the number of essentially different local alignments with a score exceeding a high threshold to be asymptotically Poisson distributed. From the Poisson approximation a Gumbel approximation of the maximal local alignment score is obtained. The results extend those obtained by Dembo, Karlin and Zeitouni [Ann. Probab. 22 (1994) 2022–2039] for independent sequences of i.i.d. variables. 1. Introduction. Local

We develop techniques to estimate the statistical significance of gap-free alignments between two genomic DNA sequences, using human–mouse alignments as an example. The sequences are assumed to be sufficiently similar that some but not all of the neutrally evolving regions (i.e., those under no evolutionary constraint) can be reliably aligned. Our goal is to model the situation in which the neutral rate of evolution, and hence the extent of the aligning intervals, varies across the genome. In some cases, this permits the weaker of two matches to be judged as less likely to have arisen by chance, provided it lies in a genomic interval with a high level of background divergence. We employ a hidden Markov model to capture variations in divergence rates and assign probability values to gap-free alignments using techniques of Dembo and Karlin, which are related to those used for the same purpose by BLAST. Our methods are illustrated in detail using a 1.49 Mb genomic region. Results obtained from the analysis of human chromosome 22 using these techniques are also provided.

Three results on hitting a rare set by the increments of an IR d valued random process with stationary independent increments are presented: the first time that it occurs, the duration of such a segment and the typical trajectory during the segment. 1 Introduction Large exceedances in Markov processes are of theoretical and applied relevance, especially in the context of biomolecular (DNA and protein) data, for assessing statistical significance of a sequence segment composition [KA90, KDK90]. In the context of sequential decision procedures, the false alarm rate in detection of change points by the commonly used CUSUM method corresponds to the location of the first segment with cumulative log-likelihood score exceeding the decision threshold, cf. [Sie85]. Another example pertains to one--server light traffic queues where the event Partially supported by grants NIH 8R01HG00335--04, NSF DMS86--06244, NSF DMS92-09712, and by a US-- ISRAEL BSF grant y Partially supported by grants...

Abstract: We investigate the maximal non-critical cluster in a big box in various percolation-type models. We investigate its typical size, and the fluctuations around this typical size. The limit law of these fluctuations are related to maxima of independent random variable with law described by a single cluster.

A "semi-probabilistic" alignment algorithm which combines ideas from Smith-Waterman and probabilistic alignment is proposed and studied in detail. It is predicted that the score statistics of this "hybrid" algorithm is of the universal Gumbel form, with the key Gumbel parameter taking on a fixed asymptotic value for a wide variety of scoring parameters. We have also characterized the "extremal ensemble", i.e., the collection of sequence pairs exhibiting similarities that a given scoring system is most sensitive to. Based on this extremal ensemble, a simple recipe for the computation of the "relative entropy", and from it the correction to due to -finite sequence length is also given. This allows us to assign p-values to the alignment results for arbitrary scoring parameters and gap costs. The predictions compare well with direct numerical simulations for a broad range of sequence lengths with various choices of the substitution scores and affine gap parameters.

We define the reflection of a random walk at a general barrier and derive, in case the increments are light tailed and have negative mean, a necessary and sufficient criterion for the global maximum of the reflected process to be finite a.s. If it is finite a.s., we show that the tail of the distribution of the global maximum decays exponentially fast and derive the precise rate of decay. Finally, we discuss an example from structural biology that motivated the interest in the reflection at a general barrier. 1. Introduction. The

Abstract. Current numerical methods for assessing the statistical significance of local alignments with gaps are time consuming. Analytical solutions thus far have been limited to specific cases. Here, we present a new line of attack to the problem of statistical significance assessment. We combine this new approach with known properties of the dynamics of the global alignment algorithm and high performance numerical techniques and present a novel method for assessing significance of gaps within practical time scales. The results and performance of these new methods test very well against tried methods with drastically less effort.

. We describe a new method of processing similarity queries of a proteic multiple alignment with a set (database) of protein sequences, or similarity queries of a protein sequence with a set of protein alignments. We use a representation of multiple alignments as alignment-graphs. Comparisons with different classical methods is made. This new method allows the detection of subtle similarities which are not found by the other methods. It has direct applications for similarities querying with the database of protein domains ProDom. BlastMultAl: une extension de Blast pour la recherche de similarit'es au moyen de graphes d'alignement. R'esum'e. Nous d'ecrivons une nouvelle m'ethode de recherche de similarit'es d'un alignement multiple de prot'eines avec un ensemble (une base de donn'ees) de s'equences prot'eiques, ou de recherche de similarit'es d'une s'equence prot'eique avec un ensemble d'alignements multiples de prot'eines. Nous comparons cette approche avec des approches classiques...

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