Corrected Discrete Approximations for Multiple Window Scan Statistics of One-Dimensional Poisson Processes

A stochastic model of a dynamic marker array in which markers could disappear, duplicate, and move relative to its original position is constructed to reflect on the nature of long DNA sequences. The sequence changes of deletions, duplications, and displacements follow the stochastic rules: (i) the original distribution of the marker array {…, X −2, X −1, X 0, X 1, X 2, …} is a Poisson process on the real line; (ii) each marker is replicated l times; replication or loss of marker points occur independently; (iii) each replicated point is independently and randomly displaced by an amount Y relative to its original position, with the Y displacements sampled from a continuous density g(y). Limiting distributions for the maximal and minimal statistics of the r-scan lengths (collection of distances between r + 1 successive markers) for the l-shift model are derived with the aid of the Chen-Stein method and properties of Poisson processes.

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A problem of discrete approximations to an arbitrarily varying one-dimensional seismic model

Geophysical Journal International ◽

10.1111/j.1365-246x.1984.tb01958.x ◽

1984 ◽

Vol 78 (2) ◽

pp. 431-437

Author(s):

S. H. Gray

Keyword(s):

Seismic Model ◽

Discrete Approximations ◽

One Dimensional

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Corrected discrete approximations for the conditional and unconditional distributions of the continuous scan statistic

Journal of Applied Probability ◽

10.1017/jpr.2016.101 ◽

2017 ◽

Vol 54 (1) ◽

pp. 304-319 ◽

Cited By ~ 1

Author(s):

Yi-Ching Yao ◽

Daniel Wei-Chung Miao ◽

Xenos Chang-Shuo Lin

Keyword(s):

Poisson Process ◽

Order Approximation ◽

Order Term ◽

Time Discretization ◽

Discrete Analogue ◽

Discrete Approximation ◽

Discrete Approximations ◽

Scan Statistic ◽

One Dimensional ◽

Richardson’S Extrapolation

AbstractThe (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). Using a change of measure argument, we derive the first-order term of the discrete approximation which involves some functionals of the Poisson process. Richardson's extrapolation is then applied to yield a corrected (second-order) approximation. Numerical results are presented to compare various approximations.

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Derivation of a one-dimensional von Kármán theory for viscoelastic ribbons

Nonlinear Differential Equations and Applications NoDEA ◽

10.1007/s00030-021-00745-0 ◽

2022 ◽

Vol 29 (2) ◽

Author(s):

Manuel Friedrich ◽

Lennart Machill

Keyword(s):

Metric Spaces ◽

Three Dimensional ◽

Gradient Flows ◽

Dimensional Model ◽

Discrete Approximations ◽

Three Dimensional Model ◽

One Dimensional ◽

Von Karman ◽

Von Kármán ◽

Appl Math

AbstractWe consider a two-dimensional model of viscoelastic von Kármán plates in the Kelvin’s-Voigt’s rheology derived from a three-dimensional model at a finite-strain setting in Friedrich and Kružík (Arch Ration Mech Anal 238: 489–540, 2020). As the width of the plate goes to zero, we perform a dimension-reduction from 2D to 1D and identify an effective one-dimensional model for a viscoelastic ribbon comprising stretching, bending, and twisting both in the elastic and the viscous stress. Our arguments rely on the abstract theory of gradient flows in metric spaces by Sandier and Serfaty (Commun Pure Appl Math 57:1627–1672, 2004) and complement the $$\Gamma $$ Γ -convergence analysis of elastic von Kármán ribbons in Freddi et al. (Meccanica 53:659–670, 2018). Besides convergence of the gradient flows, we also show convergence of associated time-discrete approximations, and we provide a corresponding commutativity result.

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