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  1. arXiv:1704.04406 (Published 2017-04-14)

    Two-time correlation and occupation time for the Brownian bridge and tied-down renewal processes

    Claude Godrèche
    Comments: 17 pages, 5 figures

    Tied-down renewal processes are generalisations of the Brownian bridge, where an event (or a zero crossing) occurs both at the origin of time and at the final observation time $t$. We give an analytical derivation of the two-time correlation function for such processes in the Laplace space of all temporal variables. This yields the exact asymptotic expression of the correlation in the Porod regime of short separations between the two times. We also investigate other quantities, such as the backward and forward recurrence times, as well as the occupation time of the process, which are determined exactly. Physical implications of these results for the Poland Scheraga and related models are given. These results also give exact answers to questions posed in the past in the context of stochastically evolving surfaces.

  2. arXiv:math/0701058 (Published 2007-01-02, updated 2012-08-14)

    Dependence on the Dimension for Complexity of Approximation of Random Fields

    N. Serdyukova
    Comments: 18 pages. The published in Theory Probab. Appl. (2010) extended English translation of the original paper "Zavisimost slozhnosti approximacii sluchajnyh polej ot rasmernosti", submitted on 15.01.2007 and published in Theor. Veroyatnost. i Primenen. 54:2, 256-270
    Journal: Theory Probab. Appl. (2010) 54:2, 272-284
    Categories: math.PR
    Subjects: 41A25, 41A63, 60G60

    We consider an \eps-approximation by n-term partial sums of the Karhunen-Lo\`eve expansion to d-parametric random fields of tensor product-type in the average case setting. We investigate the behavior, as d tends to infinity, of the information complexity n(\eps,d) of approximation with error not exceeding a given level \eps. It was recently shown by M.A. Lifshits and E.V. Tulyakova that for this problem one observes the curse of dimensionality (intractability) phenomenon. The aim of this paper is to give the exact asymptotic expression for the information complexity n(\eps,d).