Stochastic comparisons of random minima and maxima

Let X1, X2,… be a sequence of independent random variables and let N be a positive integer-valued random variable which is independent of the Xi. In this paper we obtain some stochastic comparison results involving min {X1, X2,…, XN) and max{X1, X2,…, XN}.

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Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽

10.3390/math9090981 ◽

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.

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Convergence rates in the law of large numbers. II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100043103 ◽

1968 ◽

Vol 64 (2) ◽

pp. 485-488 ◽

Cited By ~ 4

Author(s):

V. K. Rohatgi

Keyword(s):

Convergence Rates ◽

Random Variables ◽

Almost Sure Convergence ◽

Random Variable ◽

Rates Of Convergence ◽

Independent Random Variables ◽

Large Numbers ◽

Present Context ◽

Image Position ◽

Uniformly Bounded

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense thatSet qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence ofto zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

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ASYMPTOTIC CRAMÉR TYPE DECOMPOSITION FOR WIENER AND WIGNER INTEGRALS

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025713500057 ◽

2013 ◽

Vol 16 (01) ◽

pp. 1350005

Author(s):

SOLESNE BOURGUIN ◽

JEAN-CHRISTOPHE BRETON

Keyword(s):

Random Variables ◽

Random Variable ◽

Gaussian Random Variable ◽

Independent Random Variables ◽

Asymptotic Decomposition ◽

Type Decomposition

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

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On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

Journal of Applied Probability ◽

10.1239/jap/1253279848 ◽

2009 ◽

Vol 46 (3) ◽

pp. 721-731 ◽

Cited By ~ 10

Author(s):

Shibin Zhang ◽

Xinsheng Zhang

Keyword(s):

Stable Distribution ◽

Stochastic Integral ◽

Random Variables ◽

Transition Density ◽

Random Variable ◽

Independent Random Variables ◽

Poisson Random Variable ◽

Compound Poisson ◽

Tempered Stable Distribution ◽

Transition Distribution

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is a C∞-function.

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Asymptotic tail behaviour of phase-type scale mixture distributions

Annals of Actuarial Science ◽

10.1017/s1748499517000136 ◽

2017 ◽

Vol 12 (2) ◽

pp. 412-432 ◽

Cited By ~ 1

Author(s):

Leonardo Rojas-Nandayapa ◽

Wangyue Xie

Keyword(s):

Random Variables ◽

Random Variable ◽

Independent Random Variables ◽

Mixture Distributions ◽

Scale Mixture ◽

Domains Of Attraction ◽

Heavy Tailed Distributions ◽

Tail Behaviour ◽

Phase Type ◽

Heavy Tailed

AbstractWe consider phase-type scale mixture distributions which correspond to distributions of a product of two independent random variables: a phase-type random variable Y and a non-negative but otherwise arbitrary random variable S called the scaling random variable. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction. Particular focus is given to phase-type scale mixture distributions where the scaling random variable S has discrete support – such a class of distributions has been recently used in risk applications to approximate heavy-tailed distributions. Our results are complemented with several examples.

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Maximal branching processes and ‘long-range percolation’

Journal of Applied Probability ◽

10.1017/s0021900200026966 ◽

1970 ◽

Vol 7 (01) ◽

pp. 89-98

Author(s):

John Lamperti

Keyword(s):

Probability Distribution Function ◽

Branching Process ◽

Branching Processes ◽

Random Variables ◽

Random Variable ◽

Independent Random Variables ◽

Transition Matrices ◽

A Chain ◽

Image Position ◽

The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

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Probabalistic modelling of the results of economic activity as a function of random values

Herald of Ternopil National Economic University ◽

10.35774/visnyk2020.01.102 ◽

2020 ◽

pp. 102-112

Author(s):

Olesya Martyniuk ◽

Stepan Popina ◽

Serhii Martyniuk

Keyword(s):

Mathematical Modeling ◽

Distribution Function ◽

Probability Distribution ◽

Economic Activity ◽

Probability Distribution Function ◽

Random Variables ◽

Economic Structure ◽

Random Variable ◽

Independent Random Variables ◽

Research Results

Introduction. Mathematical modeling of economic processes is necessary for the unambiguous formulation and solution of the problem. In the economic sphere this is the most important aspect of the activity of any enterprise, for which economic-mathematical modeling is the tool that allows to make adequate decisions. However, economic indicators that are factors of a model are usually random variables. An economic-mathematical model is proposed for calculating the probability distribution function of the result of economic activity on the basis of the known dependence of this result on factors influencing it and density of probability distribution of these factors. Methods. The formula was used to calculate the random variable probability distribution function, which is a function of other independent random variables. The method of estimation of basic numerical characteristics of the investigated functions of random variables is proposed: mathematical expectation that in the probabilistic sense is the average value of the result of functioning of the economic structure, as well as its variance. The upper bound of the variation of the effective feature is indicated. Results. The cases of linear and power functions of two independent variables are investigated. Different cases of two-dimensional domain of possible values of indicators, which are continuous random variables, are considered. The application of research results to production functions is considered. Examples of estimating the probability distribution function of a random variable are offered. Conclusions. The research results allow in the probabilistic sense to estimate the result of the economic structure activity on the basis of the probabilistic distributions of the values of the dependent variables. The prospect of further research is to apply indirect control over economic performance based on economic and mathematical modeling.

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Some exact distributions of a last one-sided exit time in the simple random walk

Journal of Applied Probability ◽

10.1017/s0021900200029648 ◽

1986 ◽

Vol 23 (02) ◽

pp. 332-340

Author(s):

Chern-Ching Chao ◽

John Slivka

Keyword(s):

Random Walk ◽

Positive Integer ◽

Exit Time ◽

Random Variables ◽

Random Variable ◽

Simple Random Walk ◽

Exact Distribution ◽

Partial Sum ◽

Exact Distributions ◽

Special Case

For each positive integer n, let Sn be the nth partial sum of a sequence of i.i.d. random variables which assume the values +1 and −1 with respective probabilities p and 1 – p, having mean μ= 2p − 1. The exact distribution of the random variable , where sup Ø= 0, is given for the case that λ > 0 and μ+ λ= k/(k + 2) for any non-negative integer k. Tables to the 99.99 percentile of some of these distributions, as well as a limiting distribution, are given for the special case of a symmetric simple random walk (p = 1/2).

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Estimation of a nonlinear functional from the probability density when optimizing nonparametric decision functions

Izmeritel`naya Tekhnika ◽

10.32446/0368-1025it.2021-1-14-20 ◽

2021 ◽

pp. 14-20

Author(s):

Aleksandr V. Lapko ◽

Vasiliy A. Lapko

Keyword(s):

Probability Density ◽

Decision Rules ◽

Random Variables ◽

Random Variable ◽

Kernel Functions ◽

Independent Random Variables ◽

Two Dimensional ◽

Probability Density Estimation ◽

Nonlinear Functional ◽

Approximation Errors

A method for estimating the nonlinear functional of the probability density of a two-dimensional random variable is proposed. It is relevant when implementing procedures for fast bandwidths selection in the problem of optimization of kernel probability density estimates. The solution of this problem allows to significantly improve the computational efficiency of nonparametric decision rules. The basis of the proposed approach is the analysis of the formula for the optimal bandwidth of the kernel probability density estimation. In this case, the bandwidth of kernel functions is represented as the product of an indeterminate parameter and the average square deviations of random variables. The main component of an undefined parameter is a nonlinear functional of the probability density. The considered functional is determined by the type of probability density and does not depend on the density parameters. For a family of two-dimensional lognormal laws of distribution of independent random variables, the approximation errors of the considered nonlinear functional from the probability density are determined. The possibility of applying the proposed methodology when evaluating nonlinear functionals of probability densities that differ from the lognormal distribution laws is investigated. An analysis is made of the effect of the resulting approximation errors on the root-mean-square criteria for restoring a non-parametric estimate of the probability density of a two-dimensional random variable.

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