scholarly journals The difference between the product and the convolution product of distribution functions in Rn

2011 ◽  
Vol 89 (103) ◽  
pp. 19-36 ◽  
Author(s):  
E. Omey ◽  
R. Vesilo
Keyword(s):  
Distribution Function ◽  
Distribution Functions ◽  
Convolution Product ◽  
Random Vectors ◽  
The Difference

Assume that X? and Y? are independent, nonnegative d-dimensional random vectors with distribution function (d.f.) F(x?) and G(x?), respectively. We are interested in estimates for the difference between the product and the convolution product of F and G, i.e., D(x?) = F(x?)G(x?) ? F ? G(x?). Related to D(x?) is the difference R(x?) between the tail of the convolution and the sum of the tails: R(x?) = (1 ? F ? G(x?))?(1 ? F(x?) + 1 ? G(x?)). We obtain asymptotic inequalities and asymptotic equalities for D(x?) and R(x?). The results are multivariate analogues of univariate results obtained by several authors before.

A Simplified Lattice Boltzmann Method without Evolution of Distribution Function

2016 ◽  
Vol 9 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Z. Chen ◽  
C. Shu ◽  
Y. Wang ◽  
L. M. Yang ◽  
D. Tan
Keyword(s):  
Distribution Function ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Equilibrium Distribution ◽  
Numerical Test ◽  
Distribution Functions ◽  
Navier Stokes ◽  
Equilibrium Distribution Function ◽  
The Difference ◽  
Boltzmann Method

AbstractIn this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

scholarly journals Non-uniform Estimates in the Approximation by the Irwin Law

2016 ◽  
Vol 55 (1) ◽  
pp. 112-118
Author(s):  
Kazimieras Padvelskis ◽  
Ruslan Prigodin
Keyword(s):  
Distribution Function ◽  
Remainder Term ◽  
Cumulative Distribution Function ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Type Condition ◽  
Uniform Estimates ◽  
Cumulative Distribution Functions ◽  
Cumulant Method ◽  
The Difference

We consider an approximation of a cumulative distribution function F(x) by the cumulative distributionfunction G(x) of the Irwin law. In this case, a function F(x) can be cumulative distribution functions of sums (products) ofindependent (dependent) random variables. Remainder term of the approximation is estimated by the cumulant method.The cumulant method is used by introducing special cumulants, satisfying the V. Statulevičius type condition. The mainresult is a nonuniform bound for the difference |F(x)-G(x)| in terms of special cumulants of the symmetric cumulativedistribution function F(x).

scholarly journals THE STUDY OF THE DEPENDENCE OF SIZES OF MEASURED MICROSCOPIC OBJECTS AT THE POSITION OF THE SLICE PLANE

2019 ◽  
Vol 27 (4) ◽  
pp. 55-60
Author(s):  
A. E. Lun'kov ◽  
U. A. Gladilin ◽  
K. E. Ibragimova
Keyword(s):  
Distribution Function ◽  
Standard Deviation ◽  
Cross Sections ◽  
Mean Value ◽  
Distribution Functions ◽  
Distribution Law ◽  
The Mean ◽  
The Cross ◽  
The Difference ◽  
The Given

For microscopic objects in the form of spheres of different radii have been calculated the functions of distribution of the cross sections radii, taking into account the dependence on the position of the plane of the slice. Taking into account this dependence, the distribution functions of the cross sections radii of the spheres whose radii were given by the normal distribution law with the variation of its parameters were calculated. It is found that the difference between the given distribution function of the radii of spheres and the distribution function of their sections in the plane of the slice depends on the ratio of the standard deviation to the mean value of the radii. Depending on this relation, two simple algorithms are proposed to reconstruct the distribution function of the radii of objects by the distribution function of the radii of their sections. It is shown that these algorithms can be used to correct the experimental curve of the size distribution of micro-objects in the form of ellipsoid.

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

Extended two-dimensional belief function based on divergence measurement

2020 ◽  
pp. 1-8
Author(s):  
Jianping Fan ◽  
Jing Wang ◽  
Meiqin Wu
Keyword(s):  
Belief Function ◽  
Distribution Functions ◽  
Divergence Measure ◽  
Uncertain Information ◽  
Belief Functions ◽  
Two Dimensional ◽  
Probability Distribution Functions ◽  
Basic Probability ◽  
The Difference ◽  
Supporting Degree

The two-dimensional belief function (TDBF = (mA, mB)) uses a pair of ordered basic probability distribution functions to describe and process uncertain information. Among them, mB includes support degree, non-support degree and reliability unmeasured degree of mA. So it is more abundant and reasonable than the traditional discount coefficient and expresses the evaluation value of experts. However, only considering that the expert’s assessment is single and one-sided, we also need to consider the influence between the belief function itself. The difference in belief function can measure the difference between two belief functions, based on which the supporting degree, non-supporting degree and unmeasured degree of reliability of the evidence are calculated. Based on the divergence measure of belief function, this paper proposes an extended two-dimensional belief function, which can solve some evidence conflict problems and is more objective and better solve a class of problems that TDBF cannot handle. Finally, numerical examples illustrate its effectiveness and rationality.

scholarly journals Some Remarks on a Recent Paper by Borch

1961 ◽  
Vol 1 (5) ◽  
pp. 265-272 ◽  
Author(s):  
Paul Markham Kahn
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Fixed Number ◽  
Point Of View ◽  
Optimum Amount ◽  
Measurable Transformation ◽  
The Difference ◽  
Image Position ◽  
Stop Loss ◽  
Condition B

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such thatwhere c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

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