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We propose an extension of the Cox-Ross-Rubinstein (CRR) model based on q-binomial (or Kemp) random walks, with application to default with logistic failure rates. This model allows us to consider time-dependent switching probabilities varying according to a trend parameter on a non-self-similar binomial tree. In particular, it includes tilt and stretch parameters that control increment sizes. Option pricing formulas are written using q-binomial coefficients, and we study the convergence of this model to a Black-Scholes type formula in continuous time. A convergence rate of order O(N-1/2) is obtained. [ABSTRACT FROM AUTHOR]

Hadamard spaces have traditionally played important roles in geometry and geometric group theory. More recently, they have additionally turned out to be a suitable framework for convex analysis, optimization and non-linear probability theory. The attractiveness of these emerging subject fields stems, inter alia, from the fact that some of the new results have already found their applications both in mathematics and outside. Most remarkably, a gradient flow theorem in Hadamard spaces was used to attack a conjecture of Donaldson in Kähler geometry. Other areas of applications include metric geometry and minimization of submodular functions on modular lattices. There have been also applications into computational phylogenetics and image processing. We survey recent developments in Hadamard space analysis and optimization with the intention to advertise various open problems in the area. We also point out several fallacies in the existing proofs. [ABSTRACT FROM AUTHOR]

The life distribution of a device subject to shocks governed by a homogeneous Poisson process is shown to have a bathtub failure rate average (BFRA) when the probabilities $\bar{P}_k$ of surviving k shocks possess the corresponding discrete property. We prove closure under the formation of weak limits for BFRA distributions and explore related moment convergence issues within the BFRA family. Similar results for increasing and decreasing failure rate average distributions are obtained either independently or as consequences of our results. We also establish some results outlining the positions of various non-monotonic ageing classes such as bathtub failure rate, increasing initially then decreasing mean residual life, new worse then better than used in expectation, and increasing initially then decreasing mean time to failure in the hierarchy. Finally, an open problem is posed and a partial solution provided. [ABSTRACT FROM AUTHOR]

A class of Cannings models is studied, with population size N having a mixed multinomial offspring distribution with random success probabilities W1, . . ., WN induced by independent and identically distributed positive random variables X1,X2, . . . via Wi :=Xi/SN, i ∈ {1, . . ., N}, where SN := X1 + ·· · + XN. The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into N subintervals of lengths W1, . . ., WN. Convergence results for the genealogy of these Cannings models are provided under assumptions that the tail distribution of X1 is regularly varying. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet [J. Math. Biol. 68 (2014), 727-761] for the case when X1 is Pareto distributed and complement those obtained by Schweinsberg [Stoch. Process. Appl. 106 (2003), 107-139] for models where sampling is performed without replacement from a supercritical branching process. [ABSTRACT FROM AUTHOR]

We study Markov chains on ℤm, m ≥ 2, that behave like a standard symmetric random walk outside of the hyperplane (membrane) H = {0} × ℤm−1. The exit probabilities from the membrane (penetration probabilities) H are periodic and also depend on the incoming direction to H, what makes the membrane H two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a m-dimensional diffusion whose first coordinate is a skew Brownian motion and the other m − 1 coordinates is a Brownian motion with a singular drift controlled by the local time of the first coordinate at 0. In the proof we utilize a martingale characterization of the Walsh Brownian motion and determine the effective permeability and slide direction. Eventually, a similar convergence theorem is established for the one-sided membrane without slides and random iid penetration probabilities. [ABSTRACT FROM AUTHOR]

For lower-semicontinuous and convex stochastic processes Zn and nonnegative random variables ∈n we investigate the pertaining random sets A(Zn, ∈n) of all ∈n-approximating minimizers of Zn. It is shown that, if the finite dimensional distributions of the Zn converge to some Z and if the ∈n converge in probability to some constant c, then the A(Zn, ∈n) converge in distribution to A(Z, c) in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular, in contrast to these argmin-theorems we do not require that the limit process has a unique minimizing point. In the non-unique case the limit-distribution is replaced by a Choquet-capacity. [ABSTRACT FROM AUTHOR]

The main contribution of this article is the verification of weak convergence of a general non-Markov (NM) state transition probability estimator by Titman, which has not yet been done for any other general NM estimator. A similar theorem is shown for the bootstrap, yielding resampling-based inference methods for statistical functionals. Formulas of the involved covariance functions are presented in detail. Particular applications include the conditional expected length of stay in a specific state, given occupation of another state in the past, and the construction of time-simultaneous confidence bands for the transition probabilities. The expected lengths of stay in a two-sample liver cirrhosis dataset are compared and confidence intervals for their difference are constructed. With borderline significance and in comparison to the placebo group, the treatment group has an elevated expected length of stay in the healthy state given an earlier disease state occupation. In contrast, the Aalen-Johansen (AJ) estimator-based confidence interval, which relies on a Markov assumption, leads to a drastically different conclusion. Also, graphical illustrations of confidence bands for the transition probabilities demonstrate the biasedness of the AJ estimator in this data example. The reliability of these results is assessed in a simulation study. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the exponential-type inequalities for maximal partial sums of negatively superadditive dependent (NSD) random variables, which extends the corresponding results for independent and negatively associated (NA) random variables. Using these inequalities, we further investigate the weak convergence of the M-estimators in the generalized linear model with NSD errors, which generalize and improve the corresponding results of the independent random errors to that of NSD random errors. [ABSTRACT FROM AUTHOR]

Nourdin and Peccati (2009a) established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we provide an optimal rate of convergence in the d2-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in Nourdin and Peccati (2015). In order to achieve our goal, we introduce a novel operator theory approach to Stein's method. The recent development in Stein's method for the Gamma distribution of Döbler and Peccati (2018) plays a pivotal role in our analysis. Several examples in the context of quadratic forms are considered to illustrate our optimal bound. [ABSTRACT FROM AUTHOR]

It is shown that in general the popular coefficient alpha estimator for reliability of multi-component measuring instruments converges almost surely to a quantity that is not equal to the population reliability coefficient. This convergence with probability 1 is a stronger statement than convergence in probability (consistency) and convergence in distribution for the alpha estimator, which have been studied in the past. In the special case of congeneric measures with uncorrelated errors and equal loadings on the common true score, the alpha estimator converges almost surely to the population reliability coefficient that equals population alpha, which implies also its consistency as a reliability estimator. When the loadings are unequal but sufficiently high and similar, the alpha estimator converges almost surely to population alpha that is essentially indistinguishable from the population reliability coefficient, which implies alpha's approximate consistency then. For the general case, the results entail that the alpha estimator is not a consistent estimator of reliability. The findings add to the critical literature on coefficient alpha in the general case, as well as to the justification of its use as a dependable measuring instrument reliability estimator in special cases and settings resulting under appropriate restrictive conditions, and are illustrated using a numerical example. [ABSTRACT FROM AUTHOR]

We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Γ-operators are used. Our results extend previous findings by Azmoodeh et al. (2015) and are applied to central and non-central convergence situations. Our methods are applied to investigate stable convergence and we exclude certain classes of random variables as target variables for sequences living in a fixed Wiener chaos. [ABSTRACT FROM AUTHOR]

The Lerch zeta-function L(λ, α, s) with a transcendental parameter α, or with rational parameters α and λ is universal, i.e., a wide class of analytic functions is approximated by shifts L(λ, α, s + iτ), τ ∈ R. The case of an algebraic irrational α is an open problem. In the paper, it is proved that for all parameters α, 0 < α < 1, and λ, 0 < λ 6 1, including an algebraic irrational α, there exists a closed non-empty set of analytic functions Fα,λ such that every function f ∈ Fα,λ can be approximated by shifts L(λ, α, s + iτ). [ABSTRACT FROM AUTHOR]

Summary: The pseudomarginal algorithm is a Metropolis–Hastings‐type scheme which samples asymptotically from a target probability density when we can only estimate unbiasedly an unnormalized version of it. In a Bayesian context, it is a state of the art posterior simulation technique when the likelihood function is intractable but can be estimated unbiasedly by using Monte Carlo samples. However, for the performance of this scheme not to degrade as the number T of data points increases, it is typically necessary for the number N of Monte Carlo samples to be proportional to T to control the relative variance of the likelihood ratio estimator appearing in the acceptance probability of this algorithm. The correlated pseudomarginal method is a modification of the pseudomarginal method using a likelihood ratio estimator computed by using two correlated likelihood estimators. For random‐effects models, we show under regularity conditions that the parameters of this scheme can be selected such that the relative variance of this likelihood ratio estimator is controlled when N increases sublinearly with T and we provide guidelines on how to optimize the algorithm on the basis of a non‐standard weak convergence analysis. The efficiency of computations for Bayesian inference relative to the pseudomarginal method empirically increases with T and exceeds two orders of magnitude in some examples. [ABSTRACT FROM AUTHOR]

We study the convergence of iterates of quadratic stochastic operators that are mean monotonic. They are defined on the convex set of probability measures concentrated on a weakly compact order interval S=[0,f] of a fixed Banach lattice F. We study their regularity and identify the limits of trajectories either as the “infimum” or “supremum” of the support of initial distributions. [ABSTRACT FROM AUTHOR]

We consider the subject of approximating tail probabilities in the general compound renewal process framework, where severity data are assumed to follow a heavy-tailed law (in that only the first moment is assumed to exist). By using the weak convergence of compound renewal processes to a-stable Lévy motion, we derive such weak approximations. Their applicability is then highlighted in the context of an existing, classical, index-linked catastrophe bond pricing model, and in doing so, we specialize these approximations to the case of a compound time-inhomogeneous Poisson process. We emphasize a unique feature of our approximation, in that it only demands finiteness of the first moment of the aggregate loss processes. Finally, a numerical illustration is presented. The behavior of our approximations is compared to both Monte Carlo simulations and first-order single risk loss process approximations and compares favorably. [ABSTRACT FROM AUTHOR]

Stochastic ambiguity provides a rich class of uncertainty models that includes those in stochastic, robust, risk-based, and semi-infinite optimization and that accounts for uncertainty about parameter values as well as incompleteness of the description of uncertainty. We provide a novel, unifying perspective on optimization under stochastic ambiguity that rests on two pillars. First, ambiguity is formulated in terms of the (cumulative) probability distribution associated with the random elements; more specifically, ambiguity is expressed by letting this distribution belong to a subfamily of distributions that might, or might not, depend on the decision variable. We derive a series of estimates by introducing a metric for the space of distribution functions based on the hypo-distance between upper semicontinuous functions. In the process, we show that this metric is consistent with convergence in distribution (= weak* convergence) of the associated probability measures. Second, we rely on the theory of lopsided convergence to establish existence, convergence, and approximation of solutions of optimization problems with stochastic ambiguity. For the first time, we estimate a distance between bifunctions and show that this leads to bounds on the solution quality for problems with stochastic ambiguity. Among other consequences, these results facilitate the study of the "price of robustness" and related quantities. [ABSTRACT FROM AUTHOR]

This paper develops a theory of randomization tests under an approximate symmetry assumption. Randomization tests provide a general means of constructing tests that control size in finite samples whenever the distribution of the observed data exhibits symmetry under the null hypothesis. Here, by exhibits symmetry we mean that the distribution remains invariant under a group of transformations. In this paper, we provide conditions under which the same construction can be used to construct tests that asymptotically control the probability of a false rejection whenever the distribution of the observed data exhibits approximate symmetry in the sense that the limiting distribution of a function of the data exhibits symmetry under the null hypothesis. An important application of this idea is in settings where the data may be grouped into a fixed number of 'clusters' with a large number of observations within each cluster. In such settings, we show that the distribution of the observed data satisfies our approximate symmetry requirement under weak assumptions. In particular, our results allow for the clusters to be heterogeneous and also have dependence not only within each cluster, but also across clusters. This approach enjoys several advantages over other approaches in these settings. [ABSTRACT FROM AUTHOR]

Two-sided infinite systems of Brownian particles with rank-dependent dynamics, indexed by all integers, exhibit different properties from their one-sided infinite counterparts, indexed by positive integers, and from finite systems. Consider the gap process, which is formed by spacings between adjacent particles. In stark contrast with finite and one-sided infinite systems, two-sided infinite systems can have one- or two-parameter family of stationary gap distributions, or the gap process weakly converging to zero as time goes to infinity. [ABSTRACT FROM AUTHOR]

For each n ≥ 1, let $\{ X_{in}, \quad i \geqslant 1 \}$ be independent copies of a nonnegative continuous stochastic process X = ( X( s)) indexed by a compact metric space S. We are interested in the process of partial maxima $\tilde M_{n}(t,s) =\max \{ X_{in}(s), 1 \leqslant i\leqslant [nt] \},\quad t\geq 0,\ s\in S,$ where the brackets [ ⋅ ] denote the integer part. Under a regular variation condition on the sequence of processes X, we prove that the partial maxima process $\tilde M_{n}$ weakly converges to a superextremal process $\tilde M$ as $n\to \infty $. We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko. [ABSTRACT FROM AUTHOR]

This paper concerns the macroscopic behavior of solutions to parabolic equations with large, highly oscillatory, random potential. When the correlation function of the random potential satisfies a specific integrability condition, we show that the random solution converges, as the correlation length of the medium tends to zero, to the deterministic solution of a homogenized equation in dimension d ≥ 3. Our derivation is based on a Feynman-Kac probabilistic representation and the Kipnis-Varadhan method applied to weak convergence of Brownian motions in random sceneries. For sufficiently mixing coefficients, we also provide an optimal rate of convergence to the homogenized limit using a quantitative martingale central limit theorem. As soon as the above integrability condition fails, the solution is expected to remain stochastic in the limit of a vanishing correlation length. For a large class of potentials given as functionals of Gaussian fields, we show the convergence of solutions to stochastic partial differential equations (SPDE) with multiplicative noise. The Feynman-Kac representation and the corresponding weak convergence of Brownian motions in random sceneries allows us to explain the transition from deterministic to stochastic limits as a function of the correlation function of the random potential. [ABSTRACT FROM AUTHOR]

We consider a risk-reserve process for an insurance company where premium income and the claim sum process are modeled as a renewal reward processes. Moreover, dividends are paid out according to a barrier rule. The aim of the article is to establish a diffusion approximation of this model and to compute ruin probabilities (in finite and in infinite time) and other relevant statistics approximately using the limiting diffusion process. We also demonstrate that, under special circumstances, there exists a stationary distribution for the limiting diffusion. [ABSTRACT FROM PUBLISHER]

We study the problem of the approximation in law of the Rosenblatt sheet. We prove the convergence in law of two families of process to the Rosenblatt sheet: the first one is constructed from a Poisson process in the plane and the second one is based on random walks. [ABSTRACT FROM AUTHOR]

We propose a new adequacy test and a graphical evaluation tool for nonlinear dynamic models. The proposed techniques can be applied in any set-up where parametric conditional distribution of the data is specified and, in particular, to models involving conditional volatility, conditional higher moments, conditional quantiles, asymmetry, Value at Risk models, duration models, diffusion models, etc. Compared to other tests, the new test properly controls the nonlinear dynamic behaviour in conditional distribution and does not rely on smoothing techniques that require a choice of several tuning parameters. The test is based on a new kind of multivariate empirical process of contemporaneous and lagged probability integral transforms. We establish weak convergence of the process under parameter uncertainty and local alternatives. We justify a parametric bootstrap approximation that accounts for parameter estimation effects often ignored in practice. Monte Carlo experiments show that the test has good finite-sample size and power properties. Using the new test and graphical tools, we check the adequacy of various popular heteroscedastic models for stock exchange index data. [ABSTRACT FROM AUTHOR]

We prove two functional limit theorems for empirical multiparameter second moment functions (generalizing Ripley's K-function) obtained from a homogeneous Poisson point field observed in an unboundedly expanding convex sampling window W in ℝ. The cases of known and unknown (estimated) intensity lead to distinct Gaussian limits and require quite different proofs. Further we determine the limit distributions of the maximal deviation and the integrated squared distance between empirical and true multiparameter second moment function. These results give rise to construct goodness-of-fit tests for checking the hypothesis that a given point pattern is completely spatially random (CSR), that is, a realization of a homogeneous Poisson process. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove several preservation properties of stochastic comparisons based on the discrete mean residual life ordering d-MRL under the reliability operations of convolutions, mixtures. Finally we introduce a discrete renewal process application. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic efficiency of the delayed rejection strategy. In particular, the efficiency of the delayed rejection Metropolis-Hastings algorithm is compared to that of the regular Metropolis algorithm. To allow for a fair comparison, the study is carried under optimal mixing conditions for each of these algorithms. After introducing optimal scaling results for the delayed rejection (DR) algorithm, we outline the fact that the second proposal after the first rejection is discarded, with a probability tending to 1 as the dimension of the target density increases. To overcome this drawback, a modification of the delayed rejection algorithm is proposed, in which the direction of the different proposals is fixed once for all, and the Metropolis-Hastings accept-reject mechanism is used to select a proper scaling along the search direction. It is shown that this strategy significantly outperforms the original DR and Metropolis algorithms, especially when the dimension becomes large. We include numerical studies to validate these conclusions. [ABSTRACT FROM AUTHOR]

Let { X( t), t ≥ 0} be a standard (zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. In this paper, we consider the maxima M( T) = max{ X( t), ∀ t ∈ [0, T]} with random index T, where T/ T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M( T) exists under some additional conditions related to the correlation function r(·). [ABSTRACT FROM AUTHOR]

The subject of the paper is to find existence conditions of weak solutions to multivalued stochastic differential equations with discontinuous coefficients. First we prove that a non-exploding solution exists when the drift coefficientbsatisfies linear growth and the diffusion coefficient σ is uniformly elliptic. On this basis, we continue to obtain a solution (up to the explosion time) in the weak sense under certain local integrability, improving the result of Rozkosz and Słomiński. [ABSTRACT FROM AUTHOR]

Abstract: Detection of changes in health care performance, financial markets, and industrial processes have recently gained momentum due to the increased availability of complex data in real-time. As a consequence, there has been a growing demand in developing statistically rigorous methodologies for change-point detection in various types of data. In many practical situations, the data being monitored for the purpose of detecting changes are autocorrelated binary time series. We propose a new statistical procedure based on the partial likelihood score process for the retrospective detection of change in the coefficients of a logistic regression model with AR(p)-type autocorrelations. We carry out some Monte Carlo experiments to evaluate the power of the detection procedure as well as its probability of false alarm (type I error). We illustrate the utility using data on 30-day mortality rates after cardiac surgery and to data on IBM share transactions. [Copyright &y& Elsevier]

Abstract: In this work we develop an efficient lattice approach for option pricing with two underlying assets whose prices are governed by regime-switching models. Jump amplitudes are specified in a way such that the lattice achieves complete node recombination along each asset variable and grows quadratically as the number of time steps increases. Jump probabilities are obtained by solving a related quadratic programming problem. The weak convergence of the discrete lattice approximations to the continuous-time regime-switching diffusion processes is established. The lattice is employed to price both European and American options written on the maximum and minimum of two assets in different regimes. Numerical results are provided and compared for the European options with a Monte-Carlo simulation approach. [Copyright &y& Elsevier]

Abstract: We wish to test the null hypothesis if the means of N panels remain the same during the observation period of length T. A quasi-likelihood argument leads to self-normalized statistics whose limit distribution under the null hypothesis is double exponential. The main results are derived assuming that the each panel is based on independent observations and then extended to linear processes. The proofs are based on an approximation of the sum of squared CUSUM processes using the Skorokhod embedding scheme. A simulation study illustrates that our results can be used in case of small and moderate N and T. We apply our results to detect change in the “corruption index”. [Copyright &y& Elsevier]

We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node, where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight ϑ > 0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For ϑ = 1, the resulting tree is the well- studied random recursive tree. We analyze the height, internal path length, and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n → ∞. Mainly expectations, variances, and asymptotic distributions of these parameters are derived. [ABSTRACT FROM AUTHOR]

Abstract: Given a random variable regular enough in the sense of the Malliavin calculus, we are able to measure the distance between its law and any probability measure with a density function which is continuous, bounded, strictly positive on an interval in the real line and admits finite variance. The bounds are given in terms of the Malliavin derivative of . Our approach is based on the theory of Itô diffusions and the stochastic calculus of variations. Several examples are considered in order to illustrate our general results. [Copyright &y& Elsevier]

Abstract: Let be a -statistic based on a symmetric kernel and -mixing samples . Under some suitable conditions, the weak convergence of the self-normalized -statistics is obtained, which extends a previous result for independent cases. [Copyright &y& Elsevier]

In this paper we study a general class of population genetic models where the total population is divided into a number of subpopulations or types. Migration between subpopulations is fast. Extending the results of Nordborg and Krone (2002) and Sagitov and Jagers (2005), we prove, as the total population size N tends to ∞, weak convergence of the joint ancestry of a given sample of haploid individuals in the Skorokhod topology towards Kingman's coalescent with a constant change of time scale c. Our framework includes age-structured models, geographically structured models, and combinations thereof. We also allow each individual to have offspring in several subpopulations, with general dependency structures between the number of offspring of various types. As a byproduct, explicit expressions lbr the coalescent effective population size N/c are obtained. [ABSTRACT FROM AUTHOR]

Hidden regular variation requires regular variation on 𝔼 = [0, ∞] d  \ {(0, 0,…, 0)} and another regular variation on the sub-cone , where 𝕃 i is the ith axis. We extend this concept to sub-cones of 𝔼(2) as well. We suggest a procedure for detecting hidden regular variation, and when it exists, propose a method of estimating the limit measure exploiting its semi-parametric structure. We give an example where hidden regular variation yields improved estimates of probabilities of risk sets. [ABSTRACT FROM AUTHOR]

This paper presents some results on convergence in distribution of random variables. We define concepts as the weak convergence of probabilities, the function of Kolmogorov, the real Brownian motion, and finally, we refer to the Invariance Principle as one of the notable results in the intervening notions outlined above. [ABSTRACT FROM AUTHOR]

In this paper it is proved that the weak convergence on a compact interval to a continuous, strictly increasing limit function, of the distribution function of a record value, implies weak convergence throughout . [ABSTRACT FROM AUTHOR]

We obtain a limit theorem for L-functions of elliptic curves twisted by Dirichlet characters with increasing modulus. [ABSTRACT FROM AUTHOR]

We prove limit theorems in the sense of the weak convergence of probability measures on the complex plane for the periodic Hurwitz zeta function. Three cases (transcendental, rational and algebraic irrational) of the parameter are considered. [ABSTRACT FROM AUTHOR]

We prove a two-dimensional limit theorem for Leach zetafunctions with transcendental and algebraic irrational parameters. [ABSTRACT FROM AUTHOR]

Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let Xϵ be the sum of jumps not exceeding ϵ, and let μ(ϵ) = E[Xϵ(l)]. We study the question of weak convergence of Xϵ/μ(ϵ) as ϵ ↓ 0, in terms of the limit behavior of μ(ϵ)/ϵ. The most interesting case reduces to the weak convergence of Xϵ/ϵ to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001). [ABSTRACT FROM AUTHOR]

For the two-stage quadratic stochastic program where the second-stage problem is a general mixed-integer quadratic program with a random linear term in the objective function and random right-hand sides in constraints, we study continuity properties of the second-stage optimal value as a function of both the first-stage policy and the random parameter vector. We also present sufficient conditions for lower or upper semicontinuity, continuity, and Lipschitz continuity of the second-stage problem's optimal value function and the upper semicontinuity of the optimal solution set mapping with respect to the first-stage variables and/or the random parameter vector. These results then enable us to establish conclusions on the stability of optimal value and optimal solutions when the underlying probability distribution is perturbed with respect to the weak convergence of probability measures. [ABSTRACT FROM AUTHOR]

Abstract: There are many ways to measure the dispersion of a random variable. One such method uses the concept of peakedness. If the random variable is symmetric about a point , then Birnbaum [Z.W. Birnbaum, On random variables with comparable peakedness, The Annals of Mathematical Statistics 19 (1948) 76–81] defined the function , as the peakedness of . If two random variables, and , are symmetric about the points and , respectively, then is said to be less peaked than , denoted by , if for all , i.e.,  is stochastically larger than . For normal distributions this is equivalent to variance ordering. Peakedness ordering can be generalized to the case where and are arbitrary points. However, in this paper we study the comparison of dispersions in two continuous random variables, symmetric about their respective medians, using the peakedness concept where normality, and even moment assumptions are not necessary. We provide estimators of the distribution functions under the restriction of symmetry and peakedness ordering, show that they are consistent, derive the weak convergence of the estimators, compare them with the empirical estimators, and provide formulas for statistical inferences. An example is given to illustrate the theoretical results. [Copyright &y& Elsevier]