Random variables, monotone relations, and convex analysis

In the present paper, monotone relations and maximal monotone relations from an Hadamard space to its linear dual space are investigated. Fitzpatrick transform of monotone relations in Hadamard spaces is introduced. It is shown that Fitzpatrick transform of a special class of monotone relations is proper, convex and lower semi-continuous. Finally, a representation result for monotone relations is given.

Let X 1,…, X n be i.i.d. random variables from an unknown cumulative distribution function F defined on the real line ℝ. No assumptions are made on the unknown F. The problem is to predict the empirical distribution function of a future sample Y = (Y 1,…,Y m ) from the distribution function F on the basis of the observations X 1,…, X n . To solve the problem, a decision theoretic approach is used. In certain situations, when for example a weighted quadratic loss function is used in the prediction problem, the problem of finding the minimax predictor of the empirical distribution function can be reduced to the problem of minimax estimation of the cumulative distribution function F under a composite weighted quadratic loss function. In such a case, a general method of finding minimax predictors of is presented. The method is also useful when no equalizer rule is available. The problem of prediction considered refers also to prediction of the empirical lifetime distribution function and may have some practical significance. For example, the results enable to predict, under the minimax criterion, the number of devices (from among all the m ones) which will fail up to any fixed time t 0.

<p>Risk measures have been studied for several decades in the financial literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some space X to the (extended) real line. It is difficult to give a reliable assessment of financial risk without a suitable model space X where the financial position lies there. </p> <p>In this work, we assume the domain <em>X</em> for the risk measure <em>ρ : X</em> → (−∈,∈] is a subset of <em>L</em><sup>1</sup> that contains the space of simple random variables L and has the Fatou property. We focus on dilatation monotonicity as desirable preferences of the decision-maker over risks. We discuss continuity properties of dilatation monotone risk measures on <em>X</em>. We prove that on a non-atomic probability space, every dilatation monotone convex risk measure on <em>X</em> can extend uniquely to a σ(<em>L</em><sup>1</sup>,L)-lower semicontinuous dilatation monotone convex risk measure. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. </p> <p>In the second part of this thesis, we overview various properties of the Haezendonck–Goovaerts risk measure as a dilatation monotone risk measure. In the special case <em>Φ(x) = x</em><sup><em>2</em></sup>, we develop an explicit expression for the Haezendonck–Goovaerts risk measure when the risk variable <em>X</em> follows some specific distributions and propose an empirical algorithm to estimate this risk measure. Moreover, we investigate the performance of some members of this class of risk measures on real data as a possible application of the Haezendonck-Goovaerts risk measure. </p>

We study the class of random entire functions given by power series, in which the coefficients are formed as the product of an arbitrary sequence of complex numbers and two sequences of random variables. One of them is the Rademacher sequence, and the other is an arbitrary complex-valued sequence from the class of sequences of random variables, determined by a certain restriction on the growth of absolute moments of a fixed degree from the maximum of the module of each finite subset of random variables. In the paper we prove sharp Wiman–Valiron’s type inequality for such random entire functions, which for given p∈(0;1) holds with a probability p outside some set of finite logarithmic measure. We also considered another class of random entire functions given by power series with coefficients, which, as above, are pairwise products of the elements of an arbitrary sequence of complex numbers and a sequence of complex-valued random variables described above. In this case, similar new statements about not improvable inequalities are also obtained.

<p><span>Measurement uncertainty analysis based on combining the state-of-knowledge distributions requires evaluation of the probability density function (PDF), the cumulative distribution function (CDF), and/or the quantile function (QF) of a random variable reasonably associated with the measurand. This can be derived from the characteristic function (CF), which is defined as a Fourier transform of its probability distribution function. Working with CFs provides an alternative and frequently much simpler route than working directly with PDFs and/or CDFs. In particular, derivation of the CF of a weighted sum of independent random variables is a simple and trivial task. However, the analytical derivation of the PDF and/or CDF by using the inverse Fourier transform is available only in special cases. Thus, in most practical situations, a numerical derivation of the PDF/CDF from the CF is an indispensable tool. In metrological applications, such approach can be used to form the probability distribution for the output quantity of a measurement model of additive, linear or generalized linear form. In this paper we propose new original algorithmic implementations of methods for numerical inversion of the characteristic function which are especially suitable for typical metrological applications. The suggested numerical approaches are based on the Gil-Pelaez inverse formulae and on using the approximation by discrete Fourier transform and the fast Fourier transform (FFT) algorithm for computing PDF/CDF of the univariate continuous random variables. As illustrated here, for typical metrological applications based on linear measurement models the suggested methods are an efficient alternative to the standard Monte Carlo methods.</span></p>

The method of reliability estimation with hybrid uncertainties including random and fuzzy variables is proposed. By means of random set theory, random variables and fuzzy variables are transformed into random set, and then a unified model of reliability is proposed in the unified framework of random set theory. The limit-state function of a structure with hybrid uncertainties is mapped into a random set by using the extension principle of function based on random set, and then the limit cumulative distribution functions (CDFs) of reliability are determined. According to the upper and the lower CDF, the interval estimation of reliability can be obtained. The traditional reliability theory based on probability can be viewed as one of the sub-cases of the method proposed. The proposed method is demonstrated with an example.

The reliability-based design optimization (RBDO) using performance measure approach for problems with correlated input variables requires a transformation from the correlated input random variables into independent standard normal variables. For the transformation with correlated input variables, the two most representative transformations, the Rosenblatt and Nataf transformations, are investigated. The Rosenblatt transformation requires a joint cumulative distribution function (CDF). Thus, the Rosenblatt transformation can be used only if the joint CDF is given or input variables are independent. In the Nataf transformation, the joint CDF is approximated using the Gaussian copula, marginal CDFs, and covariance of the input correlated variables. Using the generated CDF, the correlated input variables are transformed into correlated normal variables and then the correlated normal variables are transformed into independent standard normal variables through a linear transformation. Thus, the Nataf transformation can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications. This paper develops a PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula. Several numerical examples show that the correlated random input variables significantly affect RBDO results.

This article compares the distributions of integer-valued random variables and Poisson random variables. It considers the total variation and the Wasserstein distance and provides, in particular, explicit bounds on the pointwise difference between the cumulative distribution functions. Special attention is dedicated to estimating the difference when the cumulative distribution functions are evaluated at 0. This permits to approximate the minimum (or maximum) of a collection of random variables by a suitable random variable in the Kolmogorov distance. The main theoretical results are obtained by combining the Chen-Stein method with size-bias coupling and a generalization of size-bias coupling for integer-valued random variables developed herein. A wide variety of applications are then discussed with a focus on stochastic geometry. In particular, transforms of the minimal circumscribed radius and the maximal inradius of Poisson-Voronoi tessellations as well as the minimal inter-point distance of the points of a Poisson process are considered and bounds for their Kolmogorov distances to extreme value distributions are derived.

In this paper, the k-μ-g random variable is considered. The k-μ-g random process arises from the κ-μ random process when the power of the k-μ random variable follows Gamma distribution. The closed form expressions for probability density function (PDF) and cumulative distribution function (CDF) of k-μ-g random variable are determined. Then, PDFs of the moment of n-th order of k-μ-g random variable, the ratio and product of two k-μ-g random variables are calculated in the closed form. Random variables are of crucial significance, not only for the statistical but also for the deterministic modeling of wireless mobile radio channels. The main contribution of using derived expressions is possibility to make easier performance analysis of wireless communication systems in the presence of k-μ-g short term fading, Gamma shadowing and k-μ-g cochannel interference.

Rate outage probability is an important performance metric to measure the level of quality of service (QoS) in the 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> Generation (4G) broadband access networks. Thus, in this paper, we calculate a closed form expression of the rate outage probability for a given user in a down-link multi-hop OFDMA-based system encountered as a result of links' channel variations. The channel random behavior on different subcarriers allocated to a given user are assumed to follow independent non-identical Nakagami-m distributions. Besides the rate outage probability formulas, we also derive a novel closed form formulas for the moment generating function, probability distribution function (pdf), and the cumulative distribution function (cdf) of a product of independent non-identical Gamma distributed random variables (RVs). These RVs are functions of the attainable signal-to-noise power ratio (SNR) on the allocated group of subcarriers.

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.

The authors investigate four approaches that can be used to compute the distribution of a sum of independent lognormal random variables (RVs). Simulation results for the complementary cumulative distribution function (CDF) of sums of independent lognormal RVs are used for verification and comparison. The aim is to determine which method is best for computing the complementary cumulative distribution function. The problem is formally stated and some observations regarding different approaches to the problem are given. Three related methods based on lognormal approximation of the sum CDF are described and assessed. An assessment of a different approach. Farley's method, that has been reported in the literature is described. An intuitive basis for the method is also given. A comparative examination of the different methods is presented. This leads to an explanation of the differences of the approximations. >

Reliability is one of the main indicators of structural elements mechanical safety. The choice of stochastic models is an important task in reliability analysis for describing the variability of random variables with aleatory and epistemic uncertainty. The article proposes a method for the reliability analysis of RHS (rectangular hollow sections) steel truss joints based on p-boxes approach. The p-boxes consist of two boundary distribution functions that create an area of possible distribution functions of a random variable. The using of p-boxes make possible to model random variables without making unreasonable assumptions about the exact cumulative distribution functions (CDF) or the exact values of the CDF parameters. The developed approach allows to give an interval estimate of the non-failure probability of the truss joints, which is necessary for a comprehensive (system) reliability analysis of the entire truss.

In this paper, ration of Weibull random variable and α - μ random variable is considered. This ratio can be used for evaluation performances wireless communication system operating in Weibull multipath fading channel and the α - μ channel interference is presented. In this paper, probability density function of ratio Weibull random variable and α - μ random variable, cumulative distribution function of ratio Weibull random variable and α - μ random variable and moments of ratio Weibull random variable and α - μ random variable are calculated.

In this work, a physics-based first-order reliability method (FORM) algorithm is proposed for the flutter reliability analysis of an aircraft wing in the frequency domain. The limit state function, which is an implicit function of random variables, is defined in terms of the damping ratio of the aeroelastic system in a conditional sense on flow velocity. Two aeroelastic cases, namely, an airfoil section model and a cantilever wing model, are considered for carrying out the studies. These aeroelastic models have well separated mean bending and mean torsional modal frequencies. The geometric, structural, and aerodynamic parameters of airfoil and wing systems are modeled as independent Gaussian random variables. The effects of these on the statistics of frequency and damping ratio, and the cumulative distribution functions (CDFs) of flutter velocity are studied. In the case of the wing, the effects of modeling stiffness parameters as Gaussian random fields on the CDFs of flutter velocity are also studied. Here, spectral stochastic finite element method (SFEM) based on Karhunen–Loeve (K–L) expansion is used to discretize the random fields into random variables. From the study of an airfoil system, it is observed that parameters like torsional stiffness, elastic axis location, free stream density, and mass moment of inertia are more sensitive as compared with other parameters. However, in the case of the wing parameters such as torsional stiffness, free stream density, mass moment of inertia, and mass are observed to be more sensitive. The CDFs of flutter velocity obtained using the proposed algorithm are compared with Monte Carlo simulations (MCS) and found to be accurate. A comparative study of aeroelastic reliability for the wing is also carried out by treating stiffness parameters as random variables and random fields. It is observed that the CDFs of flutter velocity in the tail region are conservative when stiffness parameters are treated as random variables.