The fragment assembly problem is that of reconstructing a DNA sequence from a collection of randomly sampled fragments. Traditionally the objective of this problem has been to produce the shortest string that contains all the fragments as substrings, but in the case of repetitive target sequences this objective produces answers that are overcompressed. In this paper, the problem is reformulated as one of finding a maximum-likelihood reconstruction with respect to the 2-sided Kolmogorov-Smirnov statistic, and it is argued that this is a better formulation of the problem. Next the fragment assembly problem is recast in graph-theoretic terms as one of finding a non-cyclic subgraph with certain properties and the objectives of being shortest or maximally-likely are also recast in this framework. Finally, a series of graph reduction transformations are given that dramatically reduce the size of the graph to be explored in practical instances of the problem. This reduction is ...

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56> kR n k p = p kff n k (p=2) converges almost surely to a finite positive explicit constant. We also extend our result to a more general Bahadur--Kiefer process. Keywords. Empirical process, quantile process, Bahadur--Kiefer representation, L p - modulus of continuity for Brownian motion, Brownian bridge, Kiefer process. 1991 Mathematics Subject Classification. Primary 60F25; Secondary 62G30. Research supported by an NSERC Canada Grant at Carleton University, Ottawa, and by a Paul Erdos Visiting Professorship of the Paul Erdos Summer Research Center of Mathematics, Budapest. Research supported by a Fellowship of the Paul Erdos Summer Research Center of Mathematics, Budapest. -- 1 -- 1. Introduction Let fU i g<F48.69

Several classical results on boundary-crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain “clumps ” around high-level crossings. We also discuss how these results are related to the Poisson clumping heuristic and tube formulas of Gaussian random fields, and describe their applications to laws of the iterated logarithm in the form of the Kolmogorov– Erdős–Feller integral tests.

The asymptotic power of the one-sided and two-sided Kolmogorov tests of goodness of fit of a hypothesis distribution H(x) against sequences of alternatives Gn(x) for 'which s ~ JnIH(x)-Gn(x)I tends to a limit is investigated by application of Doobls "heuristic procedure"; bounds on the power are found, and some numerical examples provided.