Unbounded Random Variables Research Articles

Correction to: On the equivalence of conglomerability and disintegrability for unbounded random variables

Correction to: On the equivalence of conglomerability and disintegrability for unbounded random variables

Disintegration property of coherent upper conditional previsions with respect to Hausdorff outer measures for unbounded random variables

In a metric space , coherent upper conditional previsions are defined by Hausdorff outer measures on the linear space of all absolutely Choquet integrable random variables. They are proven to satisfy the disintegration property on every non-null partition if Ω is a set with positive and finite Hausdorff outer measure in its Hausdorff dimension. Coherent upper conditional probabilities are obtained as restrictions to the indicators function and it is proven they can be extended as finitely additive probabilities in the sense of Dubins. Examples are given in the discrete metric space and in the Euclidean metric space.

Weak approximation of transformed stochastic gradient MCMC

Stochastic gradient Langevin dynamics (SGLD) is a computationally efficient sampler for Bayesian posterior inference given a large scale dataset and a complex model. Although SGLD is designed for unbounded random variables, practical models often incorporate variables within a bounded domain, such as non-negative or a finite interval. The use of variable transformation is a typical way to handle such a bounded variable. This paper reveals that several mapping approaches commonly used in the literature produce erroneous samples from theoretical and empirical perspectives. We show that the change of random variable in discretization using an invertible Lipschitz mapping function overcomes the pitfall as well as attains the weak convergence, while the other methods are numerically unstable or cannot be justified theoretically. Experiments demonstrate its efficacy for widely-used models with bounded latent variables, including Bayesian non-negative matrix factorization and binary neural networks.

Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay

In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented.

Berry–Esseen bounds in the inhomogeneous Curie–Weiss model with external field

We study the inhomogeneous Curie–Weiss model with external field, where the inhomogeneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their weights. In this model, the sum of the spins obeys a central limit theorem outside the critical line. We derive a Berry–Esseen rate of convergence for this limit theorem using Stein’s method for exchangeable pairs. For this, we, amongst others, need to generalize this method to a multidimensional setting with unbounded random variables.

Solving random mean square fractional linear differential equations by generalized power series: Analysis and computing

This paper deals with solving the general random (Caputo) fractional linear differential equation under general assumptions on random input data (initial condition, forcing term and diffusion coefficient). Our contribution extends, in two directions, the results presented in a recent contribution by the authors. In that paper, a mean square random generalized power series solution has been constructed in the case that the fractional order, say α , of the Caputo derivative lies on the interval ] 0 , 1 ] and assuming that the diffusion coefficient belongs to a class, ℭ , of random variables that contains all bounded random variables. However, significant families of unbounded random variables, such as Gaussian and Exponential, for example, do not fall into class ℭ . Now, in this contribution we first enlarge the class of random variables to which the diffusion coefficient belongs and we prove that the constructed random generalized power series solution is mean square convergent too. We show that any bounded random variable and important unbounded random variables, including Gaussian and Exponential ones, are allowed to play the role of the diffusion coefficient as well. Secondly, we construct a mean square random generalized power series solution in the case that α parameter lies on the larger interval ] 0 , 2 ] . As a consequence, the results established in our previous contribution are fairly generalized. It is particularly enlightening, the numerical study of the convergence of the approximations to the mean and the standard deviation of the solution stochastic process in terms of α parameter and on the type of the probability distribution chosen for the diffusion coefficient.

Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series

The aim of this paper is to solve a class of non-autonomous linear fractional differential equations with random inputs. A mean square convergent series solution is constructed in the case that the fractional order α of that Caputo derivative lies in ] 0 , 1 ] using a random Fröbenius approach. The analysis is conducted by using the so-called mean square random calculus. The mean square convergence of the series solution is established assuming mild conditions on random inputs (diffusion coefficient and initial condition). We show that these conditions are satisfied for a variety of unbounded random variables. In addition, explicit expressions to approximate the mean, the variance and the covariance functions of the random series solution are given. Two full illustrative examples are shown.

Error Analysis of ERM Algorithm with Unbounded and Non-Identical Sampling

A standard assumption in the literature of learning theory is the samples which are drawn independently from an identical distribution with a uniform bounded output. This excludes the common case with Gaussian distribution. In this paper we extend these assumptions to a general case. To be precise, samples are drawn from a sequence of unbounded and non-identical probability distributions. By drift error analysis and Bennett inequality for the unbounded random variables, we derive a satisfactory learning rate for the ERM algorithm.

On the equivalence of conglomerability and disintegrability for unbounded random variables

We extend a result of Dubins (Ann Probab 3:89-99, 1975) from bounded to unbounded random variables. Dubins showed that a finitely additive expectation over the collection of bounded random variables can be written as an integral of conditional expectations (disintegrability) if and only if the marginal expectation is always within the smallest closed interval containing the conditional expectations (conglomerability). We give a sufficient condition to extend this result to collections ofrandomvariablesthathavefiniteexpectedvalueandwhoseconditionalexpectations are finite and have finite expected value.

On the distribution of sums of random variables with copula-induced dependence

We investigate distributional properties of the sum of d possibly unbounded random variables. The joint distribution of the random vector is formulated by means of an absolutely continuous copula, allowing for a variety of different dependence structures between the summands. The obtained expression for the distribution of the sum features a separation property into marginal and dependence structure contributions typical for copula approaches. Along the same lines we obtain the formulation of a conditional expectation closely related to the expected shortfall common in actuarial and financial literature. We further exploit the separation to introduce new numerical algorithms to compute the distribution and quantile function, as well as this conditional expectation. A comparison with the most common competitors shows that the discussed Path Integration algorithm is the most suitable method for computing these quantities. In our example, we apply the theory to compute Value-at-Risk forecasts for a trivariate portfolio of index returns.

Robust mechanism synthesis with truncated dimension variables and interval clearance variables

The dimension variables of a mechanism vary randomly within their tolerance ranges. Likewise, the positions of the pins in joints also vary randomly, and the joint distribution of the position coordinates of a pin is usually unknown. In traditional robust mechanism synthesis, however, dimension variables are usually considered as unbounded random variables, and the joint distributions of the clearance variables are commonly assumed to be known. This work intends to make robust mechanism synthesis more practical by treating dimension variables as truncated random variables and clearance variables as interval variables. A new robust methodology is then proposed to accommodate truncated dimension variables and interval clearance variables. A quality loss function is created with the mean and standard deviation of the maximal motion error with respect to interval clearance variables. Then minimizing the expected quality loss function leads to reductions in both nominal motion error and the variation of the motion error. This ensures that the mechanism is robust against uncertainties in both dimension variables and clearance variables. The robust synthesis methodology is applied to a four-bar function generating mechanism.

Characterization of left-monotone risk aversion in the RDEU model

We extend the characterization of the left-monotone risk aversion developed by Ryan (2006) to the case of unbounded random variables. The notion of weak convergence is insufficient for such an extension. It requires the solution of a host of delicate convergence problems. To this end, some further intrinsic properties of the location independent risk order are investigated. The characterization of the right-monotone risk aversion for unbounded random variables is also mentioned. Moreover, we remove the gap in the proof of the main result in Ryan (2006).

On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Unbounded Random Variables. II

Let {Y n , n ≥ 1} be a sequence of i.i.d. random variables and let l and L denote the essential infimum of Y 1 and the essential supremum of Y 1, respectively. The set ℂ of almost sure limit points of where b > 1 is investigated. The new findings are for the case where Y 1 is unbounded and are as follows: (i) If ℂ ∩ ℝ ≠ ∅, then ℂ = [l, L]; (ii) If l ∈ ℝ, then either ℂ = {∞} or ℂ = [l, ∞]; (iii) If l = − ∞ and L = ∞, then either ℂ = {∞}, ℂ = {− ∞}, ℂ = [− ∞, ∞], or ℂ = {− ∞, ∞}. Illustrative examples are referenced or provided showing that each of the various alternatives can hold. The current work is a continuation of the investigations of Li et al. [3, 4] wherein the set ℂ is identified, respectively, for bounded Y 1 as the spectrum of the distribution function of and for unbounded Y 1 with 𝔼(log (max {|Y 1|, e})) < ∞ as [l, L].

On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Unbounded Random Variables

For a sequence of i.i.d. unbounded random variables {Y n , n ≥ 1} and a constant b > 1, it is shown for that if 𝔼(log (max {|Y 1|, e})) < ∞, then and for almost every ω ∈ Ω, where l and L are the essential infimum of Y 1 and the essential supremum of Y 1, respectively. For the case where 𝔼(log (max {|Y 1|, e})) = ∞, examples are given wherein the limit point set ℂ is identified and it is not necessarily the interval [l, L]. The current work is a follow-up to the investigation of Li, Qi, and Rosalsky (Stochastic Analysis and Applications, 2008, 28:86–102) identifying the limit point set of W n when Y 1 is bounded; the results for unbounded Y 1 are structurally different from those for bounded Y 1 and are thus not merely simple extensions of the bounded case.

Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers

We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain the rate of convergence n−1/2βnlog 3/2n in which βn can be particularly taken as (log log n)1/σ with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n−1/2(log log n)1/2log 2n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi:10.1155/2008/385362 (2008) for the case mentioned above, and derive the convergence rate n−1/2βnlog 1/2n for the above βn under the given covariance function, which improves the relevant one n−1/2(log log n)1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results, which extend and improve some known results.

On convex risk measures on L p -spaces

Much of the recent literature on risk measures is concerned with essentially bounded risks in L∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on Lp spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on Lp we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for Lp risks.

Addendum to “An extension of an inequality of Hoeffding to unbounded random variables”: The non-i.i.d. case

Let Sn = X1 + ⋯ + Xn be a sum of independent random variables such that 0 ⩽ Xk ⩽ 1 for all k. Write {ie237-01} and q = 1 − p. Let 0 < t < q. In our recent paper [3], we extended the inequality of Hoeffding ([6], Theorem 1) {fx237-01} to the case where Xk are unbounded positive random variables. It was assumed that the means {ie237-02} of individual summands are known. In this addendum, we prove that the inequality still holds if only an upper bound for the mean {ie237-03} is known and that the i.i.d. case where {ie237-04} dominates the general non-i.i.d. case. Furthermore, we provide upper bounds expressed in terms of certain compound Poisson distributions. Such bounds can be more convenient in applications. Our inequalities reduce to the related Hoeffding inequalities if 0 ⩽ Xk ⩽ 1. Our conditions are Xk ⩾ 0 and {ie237-05}. In particular, Xk can have fat tails. We provide as well improvements comparable with the inequalities in Bentkus [2]. The independence of Xk can be replaced by super-martingale type assumptions. Our methods can be extended to prove counterparts of other inequalities in Hoeffding [6] and Bentkus

Block-diagonal preconditioning for spectral stochastic finite-element systems

Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate partial differential equations (PDEs) whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random fields with prescribed statistics. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions are averaged to obtain statistical properties of the solution variables. Alternatively, so-called stochastic finite-element methods (SFEMs) discretize the probabilistic dimension of the PDE directly leading to a single structured linear system. The latter approach is becoming extremely popular but its computational cost is still perceived to be problematic as this system is orders of magnitude larger than for the corresponding deterministic problem. A simple block-diagonal preconditioning strategy incorporating only the mean component of the random field coefficient and based on incomplete factorizations has been employed in the literature and observed to be robust, for problems of moderate variance, but without theoretical analysis. We solve the stochastic Darcy flow problem in primal formulation via the spectral SFEM and focus on its efficient iterative solution. To achieve optimal computational complexity, we base our block-diagonal preconditioner on algebraic multigrid. In addition, we provide new theoretical eigenvalue bounds for the preconditioned system matrix. By highlighting the dependence of these bounds on all the SFEM parameters, we illustrate, in particular, why enriching the stochastic approximation space leads to indefinite system matrices when unbounded random variables are employed.

An extension of the Hoeffding inequality to unbounded random variables

Let S = X1 + ⋯ + Xn be a sum of independent random variables such that 0 ⩽ Xk ⩽ 1 for all k. Write p = E S/n and q = 1 − p. Let 0 < t < q. In this paper, we extend the Hoeffding inequality [16, Theorem 1] $$ \mathbb{P}\left\{ {S \geqslant nt + np} \right\} \leqslant H^n \left( {t,p} \right), {\rm H}\left( {t,p} \right) = \left( {\frac{p} {{p + t}}} \right)^{p + t} \left( {\frac{q} {{q - t}}} \right)^{q - t} , $$ , to the case where Xk are unbounded positive random variables. Our inequalities reduce to the Hoeffding inequality if 0 ⩽ Xk ⩽ 1. Our conditions are Xk ⩾ 0 and E S < ∞. We also provide improvements comparable with the inequalities of Bentkus [5]. The independence of Xk can be replaced by supermartingale-type assumptions. Our methods can be extended to prove counterparts of other inequalities of Hoeffding [16] and Bentkus [5].

Cokriging particle size fractions of the soil

SummaryIt is often necessary to predict the distribution of mineral particles in soil between size fractions, given observations at sample sites. Because the contents in each fraction necessarily sum to 100%, these values constitute a composition, which we may assume is drawn from a random compositional variate. Elements of a D‐component composition are subject to non‐stochastic constraints; they are constrained to lie on a D– 1 dimensional simplex. This means we cannot treat them as realizations of unbounded random variables such as the multivariate Gaussian. For this reason, there are theoretical reasons not to use ordinary cokriging (or ordinary kriging) to map particle size distributions. Despite this, the compositional constraints on data on particle size fractions are not always accounted for by soil scientists.The additive log‐ratio (alr) transform can be used to transform data from a compositional variate into a form that can be treated as a realization of an unbounded random variable. Until now, while soil scientists have made use of the alr transform for the spatial prediction of particle size, there has been concern that the simple back‐transform of the optimal estimate of the alr‐transformed variables does not yield the optimal estimate of the composition. A numerical approximation to the conditional expectation of the composition has been proposed, but we are not aware of examples of its application and it has not been used in soil science.In this paper, we report two case studies in which we predicted clay, silt and sand contents of the soil at test sites by ordinary cokriging of the alr‐transformed data followed by both the direct (biased) back‐transform of the estimates and the unbiased back‐transform. We also computed estimates by ordinary cokriging of the untransformed data (which ignores the compositional constraints on the variables) for comparison.In one of our case studies, the benefit of using the alr transform was apparent, although there was no consistent advantage in using the unbiased back‐transform. In the other case study, there was no consistent advantage in using the alr transform, although the bias of the simple back‐transform was apparent. The differences between these case studies could be explained with respect to the distribution on the simplex of the particle size fractions at the two sites.