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Overshoots and undershoots of Lèvy processes
No full textWe obtain a new fluctuation identity for a general Lévy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Klüppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766–1801] concerning asymptotic overshoot distribution of a particular class of Lévy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying Lévy process is spectrally one side
A GSVD formulation of a domain decomposition method for planar eigenvalue problems
No full textIn this article we present a modification of the domain
decomposition method of Descloux and Tolley for planar eigenvalue problems. Instead of formulating a generalized eigenvalue problem our method is based on the generalized singular value decomposition. This approach is robust and at the same time highly accurate. Furthermore, we give an improved convergence analysis based on results from complex approximation theory. Several examples show the effectiveness of our method
Texture Classification and Verification Using Bispectral Estimates at All the Frequencies on a Lattice
No full textDigitized texture images can often be considered as realisations of stationary
random fields and their estimated normalised bispectra at diagonal frequencies
have been used for classification. In this paper, we devise a classifier and a
hypothesis test using bispectral estimates at all the discrete frequencies
on a lattice. For flexibility textures in each class are allowed to have a certain
amount of variation in theoretical normalised bispectra from that of a single
training sample for the class, and we also consider the case where the texture
to be classified does not belong to any of the classes. The novelty of this
paper is in a) the flexible formulation of the problem,
b) the known asymptotic distribution of the classifier,
c) the inclusion of a verification stage and
d) the complete coverage of frequencies on a lattice.
The methodology is extendable to 4th and higher order spectra and is applicable
to random field data other than texture images
Delay differential equations driven by Lévy processes: Stationarity and Feller properties
Get PDFWe consider a stochastic delay differential equation driven by a general Lévy process. Both the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. We show that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space. The existence of an invariant measure is shown by proving tightness of the segments using semimartingale characteristics and the Krylov–Bogoliubov method. A counterexample shows that the stationary solution in completely general situations may not be unique, but in more specific cases uniqueness is established
Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations
Get PDFMany applications give rise to nonlinear eigenvalue problems
with an underlying structured matrix polynomial.
In this paper several useful classes of structured polynomial
(e.g., palindromic, even, odd) are identified
and the relationships between them explored.
A special class of linearizations
that reflect the structure of these polynomials,
and therefore preserve symmetries in their spectra,
is introduced and investigated.
We analyze the existence and uniqueness of such linearizations,
and show how they may be systematically constructed
Modelling of Covariance Structures in Generalised Estimating Equations for Longitudinal Data
No full textWhen used for modelling longitudinal data generalised estimating equations specify a
working structure for the within-subject covariance matrices, aiming to produce efficient
parameter estimators. However, misspecification of the working covariance structure may
lead to a large loss of efficiency of the estimators of the mean parameters. In this paper
we propose an approach for joint modelling of the mean and covariance structures of
longitudinal data within the framework of generalised estimating equations. The resulting
estimators for the mean and covariance parameters are shown to be consistent and
asymptotically Normally distributed. Real data analysis and simulation studies show that
the proposed approach yields efficient estimators for both the mean and covariance
parameters
Multivariate Non-Linear Regression with Applications: A Frequency Domain Approach
Get PDFIn this paper we consider estimating the parameters of a multivariate multiple nonlinear regression
model with correlated errors, through the use of Finite Fourier Transforms. Consistency and asymp-
totic normality of the weighted least squares estimates are established under various conditions on
the regressor variables. These conditions involve different types of scalings, and such scaling factors
are obtained explicitly for various nonlinear regression models including an interesting model which
requires estimating the frequencies. This is a very classical problem in signal processing and is also
of great interest in many other areas. We illustrate our techniques on the time-series data of polar
motion (which is now widely known as "Chandlers Wobble") where one has to estimate the drift parameters, the offset parameters and the two periodicities associated with elliptical motion. The data
was first analyzed by Arato, Kolmogorov and Sinai who treat it as bivariate time series data satisfying
a finite order time series model. They estimate the periodicities using the coefficients of the models.
Our analysis shows that the two dominant frequencies are 12 hours and 410 days and that the errors
exhibit some long-range dependence
A three-phase mixture theory for particle size segregation in shallow granular free-surface flows
Get PDFParticle-size segregation within granular materials is of great technological significance yet it is still very poorly understood. There are several causes of segregation, but this paper focuses on kinetic sieving which is the dominant mechanism in dense gravity-driven shallow free-surface flows, or, granular avalanches. The segregation model is derived from a three-phase mixture theory composed of large particles, small particles and a passive interstitial fluid. Steady-state solutions are constructed for a normally graded inflow in a steady uniform flow field. This problem is of fundamental interest, because it shows how an unstably stratified layer readjusts into a stable configuration. Expansion fans and concentration shocks are generated and sufficiently far downstream inversely graded segregated layers form, with the larger particles overlying the finer ones. This a good approximation for segregation in flows with weak diffusive remixing. The distance for complete segregation to occur is shown to increase with rising fluid density and tends to infinity as its density approaches that of the grains. If the particles are buoyant then the initial configuration is stable. An exact time-dependent two-dimensional solution is constructed for plug flow, which exploits the uncoupling of material columns of grains in the absence of shear. This yields insight into the nature of more complex numerical solutions for strong shear, which are computed with a high-resolution shock-capturing numerical scheme