18,484 research outputs found
On the Diffusion Property of Iterated Functions
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 -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
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
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
In this paper the solutions to fractional diffusion
equations of order are analyzed and interpreted as densities of
the composition of various types of stochastic processes. For the fractional
equations of order , we show that the solutions
correspond to the distribution of the -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 , 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 coincides with the distribution of Brownian
motion with random time or of different processes with a Brownian time. The
interplay between the solutions 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
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
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
- âŚ