18,484 research outputs found

    On the Diffusion Property of Iterated Functions

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    For vectorial Boolean functions, the behavior of iteration has consequence in the diffusion property of the system. We present a study on the diffusion property of iterated vectorial Boolean functions. The measure that will be of main interest here is the notion of the degree of completeness, which has been suggested by the NESSIE project. We provide the first (to the best of our knowledge) two constructions of (n,n)(n,n)-functions having perfect diffusion property and optimal algebraic degree. We also obtain the complete enumeration results for the constructed functions

    Maximum-likelihood estimation for diffusion processes via closed-form density expansions

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    This paper proposes a widely applicable method of approximate maximum-likelihood estimation for multivariate diffusion process from discretely sampled data. A closed-form asymptotic expansion for transition density is proposed and accompanied by an algorithm containing only basic and explicit calculations for delivering any arbitrary order of the expansion. The likelihood function is thus approximated explicitly and employed in statistical estimation. The performance of our method is demonstrated by Monte Carlo simulations from implementing several examples, which represent a wide range of commonly used diffusion models. The convergence related to the expansion and the estimation method are theoretically justified using the theory of Watanabe [Ann. Probab. 15 (1987) 1-39] and Yoshida [J. Japan Statist. Soc. 22 (1992) 139-159] on analysis of the generalized random variables under some standard sufficient conditions.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1118 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Coupling iterated Kolmogorov diffusions

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    The Kolmogorov (1934) diffusion is the two-dimensional diffusion generated by real Brownian motion B and its time integral integral B d t. In this paper we construct successful co-adapted couplings for iterated Kolmogorov diffusions defined by adding iterated time integrals integral integral B d s d t,... as further components to the original Kolmogorov diffusion. A Laplace-transform argument shows it is not possible successfully to couple all iterated time integrals at once; however we give an explicit construction of a successful co-adapted coupling method for (B, integral B d t, integral integral B d s d t); and a more implicit construction of a successful co-adapted coupling method which works for finite sets of iterated time integrals

    Fractional diffusion equations and processes with randomly varying time

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    In this paper the solutions uν=uν(x,t)u_{\nu}=u_{\nu}(x,t) to fractional diffusion equations of order 0<ν≤20<\nu \leq 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order ν=12n\nu =\frac{1}{2^n}, n≥1,n\geq 1, we show that the solutions u1/2nu_{{1/2^n}} correspond to the distribution of the nn-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order ν=23n\nu =\frac{2}{3^n}, n≥1,n\geq 1, is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that uνu_{\nu} coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions uνu_{\nu} and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.Comment: Published in at http://dx.doi.org/10.1214/08-AOP401 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Brownian subordinators and fractional Cauchy problems

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    A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involve subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes, and consequently, an equivalence between these two families of partial differential equations.Comment: 18 pages, minor spelling correction

    Patterns in Sinai's walk

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    Sinai's random walk in random environment shows interesting patterns on the exponential time scale. We characterize the patterns that appear on infinitely many time scales after appropriate rescaling (a functional law of iterated logarithm). The curious rate function captures the difference between one-sided and two-sided behavior.Comment: Published in at http://dx.doi.org/10.1214/11-AOP724 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org
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