Matrix variate generalizations of Pareto distributions are proposed. Several properties of these distributions including cumulative distribution functions, characteristic functions and relationship ...to matrix variate beta type I and matrix variate type II distributions are studied.
Let ˆ μ be the Fourier transform of a Borel probability measure μ on Rd . We look at the closed abelian subgroup Γ ( μ ) of Rd , which consists of the periods of the function ˆ μ . We prove the ...following dichotomy: i) The support of μ is non-degenerate if and only if Γ ( μ ) is a lattice. ii) The support of μ is degenerate if and only if Γ ( μ ) contains a linear subspace ≠ { 0 } of Rd . A similar dichotomy is also discussed for the period group of the function | ˆ μ | .
We introduce quantum tomography on locally compact Abelian groups G. A linear map from the set of quantum states on the C⁎-algebra A(G) generated by the projective unitary representation of G to the ...space of characteristic functions is constructed. The dual map determining symbols of quantum observables from A(G) is derived. Given a characteristic function of a state the quantum tomogram consisting a set of probability distributions is introduced. We provide three examples in which G=R (the optical tomography), G=Zn (corresponding to measurements in mutually unbiased bases) and G=T (the tomography of the phase). As an application we have calculated the quantum tomogram for the output states of quantum Weyl channels.
•Quantum tomography on locally compact groups G is introduced.•Three examples are given in which G=R such that the corresponding tomogram is optical, G=Zn, where the tomogram is reduced to measurements in mutually unbiased bases, and G=T for the phase measurement.•As an application the tomogram of the output state for quantum Weyl channels is calculated.
In this paper, we are concerned with a doubly nonlinear anisotropic parabolic equation, in which the diffusion coefficient and the variable exponent depend on the time variable t. Under certain ...conditions, the existence of weak solution is proved by applying the parabolically regularized method. Based on a partial boundary value condition, the stability of weak solution is also investigated.
Numerous facets of scientific research implicitly or explicitly call for the estimation of probability densities. Histograms and kernel density estimates (KDEs) are two commonly used techniques for ...estimating such information, with the KDE generally providing a higher fidelity representation of the probability density function (PDF). Both methods require specification of either a bin width or a kernel bandwidth. While techniques exist for choosing the kernel bandwidth optimally and objectively, they are computationally intensive, since they require repeated calculation of the KDE. A solution for objectively and optimally choosing both the kernel shape and width has recently been developed by Bernacchia and Pigolotti (2011). While this solution theoretically applies to multidimensional KDEs, it has not been clear how to practically do so.
A method for practically extending the Bernacchia–Pigolotti KDE to multidimensions is introduced. This multidimensional extension is combined with a recently-developed computational improvement to their method that makes it computationally efficient: a 2D KDE on 105 samples only takes 1 s on a modern workstation. This fast and objective KDE method, called the fastKDE method, retains the excellent statistical convergence properties that have been demonstrated for univariate samples. The fastKDE method exhibits statistical accuracy that is comparable to state-of-the-science KDE methods publicly available in R, and it produces kernel density estimates several orders of magnitude faster. The fastKDE method does an excellent job of encoding covariance information for bivariate samples. This property allows for direct calculation of conditional PDFs with fastKDE. It is demonstrated how this capability might be leveraged for detecting non-trivial relationships between quantities in physical systems, such as transitional behavior.
•A multidimensional, fast, and robust kernel density estimation is proposed: fastKDE.•fastKDE has statistical performance comparable to state-of-the-science kernel density estimate packages in R.•fastKDE is demonstrably orders of magnitude faster than comparable, state-of-the-science density estimate packages in R.•A Python-based implementation of fastKDE is available at https://bitbucket.org/lbl-cascade/fastkde.
We study the analytical properties of the Laplace transform of the lognormal distribution. Two integral expressions for the analytic continuation of the Laplace transform of the lognormal ...distribution are provided, one of which takes the form of a Mellin–Barnes integral. As a corollary, we obtain an integral expression for the characteristic function. We present two approximations for the Laplace transform of the lognormal distribution, both valid in ℂ∖(−∞,0. In the last section, we discuss how one may use our results to compute the density of a sum of independent lognormal random variables.
This paper aims to provide an effective method for pricing forward starting options under the double fractional stochastic volatilities mixed-exponential jump-diffusion model. The value of a forward ...starting option is expressed in terms of the expectation of the forward characteristic function of log return. To obtain the forward characteristic function, we approximate the pricing model with a semimartingale by introducing two small perturbed parameters. Then, we rewrite the forward characteristic function as a conditional expectation of the proportion characteristic function which is expressed in terms of the solution to a classic PDE. With the affine structure of the approximate model, we obtain the solution to the PDE. Based on the derived forward characteristic function and the Fourier transform technique, we develop a pricing algorithm for forward starting options. For comparison, we also develop a simulation scheme for evaluating forward starting options. The numerical results demonstrate that the proposed pricing algorithm is effective. Exhaustive comparative experiments on eight models show that the effects of fractional Brownian motion, mixed-exponential jump, and the second volatility component on forward starting option prices are significant, and especially, the second fractional volatility is necessary to price accurately forward starting options under the framework of fractional Brownian motion.
This paper models the messages embedded by spatial least significant bit (LSB) matching as independent noises to the cover image, and reveals that the histogram of the differences between pixel gray ...values is smoothed by the stego bits despite a large distance between the pixels. Using the characteristic function of difference histogram (
DHCF
), we prove that the center of mass of
DHCF
(
DHCF COM
) decreases after messages are embedded. Accordingly, the
DHCF COMs
are calculated as distinguishing features from the pixel pairs with different distances. The features are calibrated with an image generated by average operation, and then used to train a support vector machine (SVM) classifier. The experimental results prove that the features extracted from the differences between nonadjacent pixels can help to tackle LSB matching as well.
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