Representation of the density functions of a multidimensional strictly stable distributions by series of generalized functions

Density Functions ◽

Lizorkin Space ◽

Starting Point ◽

Abel Summation

В статье рассматриваются многомерные строго устойчивые распределения. Как известно, функции плотности этих законов не представляются в явном виде за исключением известных законов Гаусса и Коши. Отправным пунктом для исследований являются характеристические функции. Имеется несколько различных форм их представления. В статье выбирается форма, предложенная в [1]. Применение обратного преобразования Фурье совместно с суммированием интегралов по Абелю позволило получить разложения функций плотности многомерных устойчивых распределений (см.[1], [12]). Основным результатом статьи являются представления этих функций с помощью рядов обобщенных функций над пространством Лизоркина. Они позволяют определить порядок убывания главного члена разложения на бесконечности для любого радиального направления. Кроме того, выведенные формулы дают возможность увидеть структуру формирования слагаемых в разложениях. В следствии приводятся примеры для различных случаев носителей спектральной меры многомерных устойчивых законов. The article discusses multidimensional strictly stable distributions. As is known, the density functions of these laws are not represented in closed form, with the exception of the well-known laws of Gauss and Cauchy. Characteristic functions are the starting point for research. There are several different forms of their presentation. The article chooses the form proposed in [1]. The application of the inverse Fourier transform together with the Abel summation of the integrals made it possible to obtain expansions of the density functions of multidimensional stable distributions (see [1], [12]). The main result of the article is the representation of these functions using series of generalized functions over the Lizorkin space. They make it possible to determine the order of decay of the principal term of the expansion at infinity for any radial direction. In addition, the derived formulas make it possible to see the structure of the formation of terms in expansions. In the corollary, examples are given for various cases of the support of the spectral measure of multidimensional stable laws.

The hierarchy of multi-point probability density functions and characteristic functions in compressible turbulence

Physics of Fluids ◽

10.1063/5.0010584 ◽

2020 ◽

Vol 32 (6) ◽

pp. 066102

Author(s):

Divya Sri Praturi ◽

Dominik Plümacher ◽

Martin Oberlack

Keyword(s):

Probability Density ◽

Compressible Turbulence ◽

Density Functions

NEARLY EXACT OPTION PRICE SIMULATION USING CHARACTERISTIC FUNCTIONS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024912500471 ◽

2012 ◽

Vol 15 (07) ◽

pp. 1250047 ◽

Cited By ~ 9

Author(s):

CAROLE BERNARD ◽

ZHENYU CUI ◽

DON MCLEISH

Keyword(s):

Characteristic Function ◽

Probability Density ◽

Option Price ◽

Heston Model ◽

Density Functions ◽

New Approach ◽

Black Scholes ◽

Unbiased Estimates

This paper presents a new approach to perform a nearly unbiased simulation using inversion of the characteristic function. As an application we are able to give unbiased estimates of the price of forward starting options in the Heston model and of continuously monitored Parisian options in the Black-Scholes framework. This method of simulation can be applied to problems for which the characteristic functions are easily evaluated but the corresponding probability density functions are complicated.

On Probability Density Functions Which are Their Own Characteristic Functions

Theory of Probability and Its Applications ◽

10.1137/1140065 ◽

1996 ◽

Vol 40 (3) ◽

pp. 577-581 ◽

Cited By ~ 1

Author(s):

K. Schladitz ◽

H. J. Engelbert

Keyword(s):

Probability Density ◽

Density Functions

Generalized Probability Functions

Advances in Mathematical Physics ◽

10.1155/2009/206176 ◽

2009 ◽

Vol 2009 ◽

pp. 1-13 ◽

Cited By ~ 5

Author(s):

Alexandre Souto Martinez ◽

Rodrigo Silva González ◽

César Augusto Sangaletti Terçariol

Keyword(s):

Exponential Function ◽

Error Function ◽

Generalized Functions ◽

Density Functions ◽

Logarithmic Function ◽

Rank Distribution ◽

Exponential Functions ◽

Generalized Exponential Function ◽

Generalized Probability

From the integration of nonsymmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. Motivated by the mathematical curiosity, we show that these generalized functions are suitable to generalize some probability density functions (pdfs). A very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one- and two-tail stretched exponential functions. We obtain, as particular cases, the generalized error function, the Zipf-Mandelbrot pdf, the generalized Gaussian and Laplace pdf. Their cumulative functions and moments were also obtained analytically.

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

The characteristic functions of functions

10.1098/rspa.1968.0192 ◽

1968 ◽

Vol 307 (1490) ◽

pp. 317-334 ◽

Cited By ~ 4

Keyword(s):

Multivariate Analysis ◽

Characteristic Function ◽

Gaussian Model ◽

Random Variable ◽

Stable Distributions ◽

Real Characteristic ◽

Vector Random Variable

The characteristic functions of various functions of a real or vector random variable are expressed in terms of the characteristic function of that variable. In the examples there is special emphasis on the stable distributions that have real characteristic functions. Some of the results suggest the practicability of generalizing traditional multivariate analysis beyond the multi-Gaussian model.

An Objective Prior from a Scoring Rule

Entropy ◽

10.3390/e23070833 ◽

2021 ◽

Vol 23 (7) ◽

pp. 833

Author(s):

Stephen G. Walker ◽

Cristiano Villa

Keyword(s):

Prior Distribution ◽

Score Function ◽

Information Criterion ◽

Scoring Rules ◽

Bregman Divergence ◽

Density Functions ◽

Objective Prior ◽

Starting Point ◽

Information Divergence ◽

Do So

In this paper, we introduce a novel objective prior distribution levering on the connections between information, divergence and scoring rules. In particular, we do so from the starting point of convex functions representing information in density functions. This provides a natural route to proper local scoring rules using Bregman divergence. Specifically, we determine the prior which solves setting the score function to be a constant. Although in itself this provides motivation for an objective prior, the prior also minimizes a corresponding information criterion.

Some Exact Distribution Results for the Partially Restricted Reduced form Estimator

Econometric Theory ◽

10.1017/s0266466600008276 ◽

1994 ◽

Vol 10 (1) ◽

pp. 140-171 ◽

Cited By ~ 1

Author(s):

Terrence W. Kinal ◽

John L. Knight

Keyword(s):

Fractional Calculus ◽

Linear Combination ◽

Exact Distribution ◽

Reduced Form ◽

Density Functions ◽

Finite Sample ◽

Invariant Polynomials ◽

Variable Model ◽

Finite Sample Properties

This paper considers some finite sample properties of the partially restricted reduced form estimators in a general (n + 1) endogenous variable model. In particular, the characteristic functions, density functions, and moments are examined for both the vector of estimators and a linear combination. The approach utilizes both invariant polynomials of matrix argument (see Chikuse and Davis [4] and Davis [6]) and fractional calculus techniques (see Phillips [20,25,26]).