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U-statistics represent a fundamental class of statistics arising from modeling quantities of interest defined by multi-subject responses. U-statistics generalize the empirical mean of a random variable X to sums over every m-tuple of distinct observations of X. Stute [182] introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: r(m)(φ, t) ≔ 피[φ(F1,..., Ym)|(X1,..., Xm) = t], for t ∈ Xm. In this paper, we are mainly interested in the study of the kNN conditional U-processes in a functional mixing data framework. More precisely, we investigate the weak convergence of the conditional empirical process indexed by a suitable class of functions and of the kNN conditional U-processes when the explicative variable is functional. We treat the uniform central limit theorem in both cases when the class of functions is bounded or unbounded satisfying some moment conditions. The second main contribution of this study is the establishment of a sharp almost complete Uniform consistency in the Number of Neighbors of the constructed estimator. Such a result allows the number of neighbors to vary within a complete range for which the estimator is consistent. Consequently, it represents an interesting guideline in practice to select the optimal bandwidth in nonparametric functional data analysis. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The theoretical results established in this paper are (or will be) key tools for further functional data analysis developments. Potential applications include the set indexed conditional U-statistics, Kendall rank correlation coefficient, the discrimination problems and the time series prediction from a continuous set of past values. [ABSTRACT FROM AUTHOR]

We consider infinitely wide multi-layer perceptrons (MLPs) which are limits of standard deep feed-forward neural networks. We assume that, for each layer, the weights of an MLP are initialized with independent and identically distributed (i.i.d.) samples from either a light-tailed (finite-variance) or a heavy-tailed distribution in the domain of attraction of a symmetric $\alpha$ -stable distribution, where $\alpha\in(0,2]$ may depend on the layer. For the bias terms of the layer, we assume i.i.d. initializations with a symmetric $\alpha$ -stable distribution having the same $\alpha$ parameter as that layer. Non-stable heavy-tailed weight distributions are important since they have been empirically seen to emerge in trained deep neural nets such as the ResNet and VGG series, and proven to naturally arise via stochastic gradient descent. The introduction of heavy-tailed weights broadens the class of priors in Bayesian neural networks. In this work we extend a recent result of Favaro, Fortini, and Peluchetti (2020) to show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit, under a suitable scaling, to a vector of i.i.d. random variables with symmetric $\alpha$ -stable distributions, $\alpha\in(0,2]$. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak convergence and convergence rate in the weak law of large numbers for weighted sums of a class of random variables satisfying the Rosenthal type inequality. The necessary and sufficient conditions for the convergence rates in the weak law of large numbers under some mild conditions are provided. Moreover, the main results that we established are applied to simple linear errors-in-variables regression models and nonparametric regression models based on a class of random errors. Finally, we present some numerical simulations to assess the finite sample performance of the theoretical results. [ABSTRACT FROM AUTHOR]

Each sequence f1, f2, ... of random variables satisfying limM→∞ (M supk∈N P(|fk| > M)) = 0 contains a subsequence fk1, fk2, ??? which, along with all its subsequences, satisfies the weak law of large numbers limN→∞ ((1/N) ∑n=1N fkn -- DN) = 0 in probability. Here, DN is a "corrector" random variable with values in [--N, N] for each N ∈ N; these correctors are all equal to zero if, in addition, lim infn→∞ E (fn² 1{|fn|≤M})) = 0 for every M ∈ (0, ∞). [ABSTRACT FROM AUTHOR]

We consider the asymptotic behavior of the infimal values and the statistical estimators of the solutions to a general stochastic optimization problem. We establish the epi-convergence of the performance criteria of approximate problems when the approximate probability laws, obtained by sampling the values of the random variable, converge weakly and tightly. Based on this key convergence, consistency properties of the infimal values and the estimators of the solutions to the approximate problems are obtained. Applying these results and properties of epi/hypo-convergence of bifunctions to Lagrangians of stochastic mathematical programs, we obtain the consistency of the saddle points of approximate Lagrangians and hence the consistency of the optimal values and the estimators of the solutions of approximate mathematical programs and their dual programs. [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]

In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains the Poisson and geometric summations as particular cases. We show that the weak limit of the FP summation, when properly normalized, is a mixture between the normal and Mittag-Leffler distributions, which we call by Normal-Mittag-Leffler (NML) law. A parameter estimation procedure for the NML distribution is developed and the associated asymptotic distribution is derived. Simulations are run to check the performance of the proposed estimators under finite samples. An empirical illustration on the daily log-returns of the Brazilian stock exchange index (IBOVESPA) shows that the NML distribution captures better the tails than some of its competitors. Related problems such as a mixed Poisson representation for the FP law and the weak convergence for the Conway-Maxwell-Poisson random sum are also addressed. [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]

Nonreversible Markov chain Monte Carlo methods often outperform their reversible counterparts in terms of asymptotic variance of ergodic averages and mixing properties. Lifting the state-space is a generic technique for constructing such samplers. The idea is to think of the random variables we want to generate as position variables and to associate to them direction variables so as to design Markov chains which do not have the diffusive behavior often exhibited by reversible schemes. In this article, we explore the benefits of using such ideas in the context of Bayesian model choice for nested models, a class of models for which the model indicator variable is an ordinal random variable. By lifting this model indicator variable, we obtain nonreversible jump algorithms, a nonreversible version of the popular reversible jump algorithms. This simple algorithmic modification provides samplers which can empirically outperform their reversible counterparts at no extra computational cost. The code to reproduce all experiments is available online1. [ABSTRACT FROM AUTHOR]

Can convergence of random variables, random vectors or, more generally, random elements of a metric space be studied using fuzzy integrals, in particular the seminormed integral? Among the positive results we present, it is shown that for large families of t-seminorms convergence in distribution and convergence in probability can be characterized via seminormed integrals. This suggests that, while fuzzy integrals were devised for working with non-additive fuzzy measures, some of their theoretical properties within the realm of probability measures can be comparable to those of the Lebesgue integral. [ABSTRACT FROM AUTHOR]

Let ξ 0 , ξ 1 , ... be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form f (z) = ∑ k = 0 ∞ ξ k c k z k , where c 0 , c 1 , ... is a real sequence such that c n 2 is regularly varying with index γ − 1 , where γ > 0. We prove that 𝔼 N [ 0 , 1 − 𝜖 ] ∼ γ 2 π | log 𝜖 | as 𝜀 ↓ 0 , where N [ 0 , r ] denotes the number of real zeroes of f in the interval [ 0 , r ]. [ABSTRACT FROM AUTHOR]

Let (X , d) be a locally compact separable ultrametric space. Given a measure m on X and a function C (B) defined on the set of all non-singleton balls B of X , we consider the hierarchical Laplacian L = L C . The operator L acts in ℒ 2 (X , m) , is essentially self-adjoint and has a purely point spectrum. Choosing a sequence { 𝜀 (B) } of i.i.d. random variables, we consider the perturbed function C (B , ω) and the perturbed hierarchical Laplacian L ω = L C (ω). Under certain conditions, the density of states 𝔪 exists and it is a continuous function. We choose a point λ such that 𝔪 (λ) > 0 and build a sequence of point processes defined by the eigenvalues of L ω located in the vicinity of λ. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity 𝔪 (λ). [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]

General properties of convergence in distribution for fuzzy random variables are studied as regards its interplay with the structure of the space of fuzzy sets. In particular, its behaviour with respect to taking tuples of fuzzy random variables, adding, multiplying by a scalar, taking the union, preserving inclusion ordering, and subsuming convergence in distribution of random sets is established. [ABSTRACT FROM AUTHOR]

In this paper, we study the limit distributions of upper and lower record values of a stationary Gaussian sequence under an equi-correlated set up. Moreover, the class of limit distribution functions (df's) of the joint upper (and the lower) record values of a stationary Gaussian sequence is fully characterized. As an application of this result, the sufficient conditions for the weak convergence of the record quasi-range, record quasi-mid-range, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit df's of these statistics are derived. [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]

In the definition of weak convergence of probability measures it is assumed that the limit is a probability measure as well. We get rid of this assumption and require that the limit merely needs to be a Choquet-capacity functional. In terms of random variables this means that the distributional limit no longer is a random point, but a random closed set, namely one uniquely determined by the Choquet capacity. For our extended notion of weak convergence there is a counterpart of the portmanteau theorem. Moreover, we demonstrate basic relations to the theory of random closed sets with emphasis on weak convergence in hyperspace topologies including two correspondence theorems. Finally, the approach carries over to sequences of Choquet capacities. [ABSTRACT FROM AUTHOR]

We provide conditions on dependent and on non-stationary random variables X n ensuring that the mantissa of the sequence of products ∏ 1 n X k is almost surely distributed following Benford’s law or converges in distribution to Benford’s law. This is achieved through proving new generalizations of Lévy’s and Robbins’s results on distribution modulo 1 of sums of independent random variables. [ABSTRACT FROM AUTHOR]

We offer an umbrella type result which extends weak convergence of classical empirical processes on the line to that of more general processes indexed by functions of bounded variation. This extension is not contingent on the type of dependence of the underlying sequence of random variables. As a consequence we establish weak convergence for stationary empirical processes indexed by general classes of functions under α -mixing conditions. [ABSTRACT FROM AUTHOR]

Under Markovian assumptions, we leverage a central limit theorem for the empirical measure in the test statistic of the composite hypothesis Hoeffding test so as to establish weak convergence results for the test statistic, and, thereby, derive a new estimator for the threshold needed by the test. We first show the advantages of our estimator over an existing estimator by conducting extensive numerical experiments. We find that our estimator controls better for false alarms while maintaining satisfactory detection probabilities. We then apply the Hoeffding test with our threshold estimator to detect anomalies in two distinct applications domains: One in communication networks and the other in transportation networks. The former application seeks to enhance cyber security and the latter aims at building smarter transportation systems in cities. [ABSTRACT FROM AUTHOR]

The extremal ratio has been used in several fields, most notably in industrial quality control, life testing, small-area variation analysis, and the classical heterogeneity of variance situation. In many biological, agricultural, military activity problems and in some quality control problems, it is almost impossible to have a fixed sample size, because some observations are always lost for various reasons. Therefore, the sample size itself is considered frequently to be an random variable (rv). Generalized order statistics (GOS) have been introduced as a unifying theme for several models of ascendingly ordered rvs. The concept of dual generalized order statistics (DGOS) is introduced to enable a common approach to descendingly ordered rvs. In this article, the limit dfs are obtained for the extremal ratio and product with random indices under non random normalization based on GOS and DGOS. Moreover, this article considers the conditions under which the cases of random and non random indices give the same asymptotic results. Some illustrative examples are obtained, which lend further support to our theoretical results. [ABSTRACT FROM AUTHOR]

The class of limit distribution functions of bivariate extreme, intermediate and central dual generalized order statistics from independent and identically distributed random variables with random sample size is fully characterized. Two cases are considered. The first case is when the random sample size is assumed to be independent of all basic random variables. The second case is when the interrelation of the random size and the basic random variables is not restricted. [ABSTRACT FROM AUTHOR]

In this paper, new weak second-order stochastic Runge-Kutta (SRK) methods for Itô stochastic differential equations (SDEs) with an m-dimensional Wiener process are introduced. Two new explicit SRK methods with weak order 2.0 are proposed. As the main innovation, the new explicit SRK methods have two advantages. First, only three evaluations of each diffusion coefficient are needed in per step. Second, the number of necessary random variables which have to be simulated is only $$m+2$$ for each step. Compared to well-known explicit SRK methods, these good properties can be used to reduce the computational effort. Our methods are compared with other well-known explicit weak second-order SRK methods in numerical experiments. And the numerical results show that the computational efficiency of our methods is better than other methods. [ABSTRACT FROM AUTHOR]

Numerical models have to be used in most applications to characterize approximately the random fields describing the spatial fluctuation of material properties and estimate extremes and other functionals of materials responses. These models, referred to as finite dimensional (FD) models, are deterministic functions of the spatial coordinate and finite sets of random variables. Conditions are established under which the distributions of extremes and other functionals of material responses can be estimated from samples of material responses to FD material models. An one-dimensional stochastic transport equation is used to illustrate the construction and FD models and assess the accuracy of the extreme material response based on these models. • A rigorous but practical method for estimating extremes of random material responses. • Finite dimensional (FD) models of random material microstructures. • Numerical examples illustrating method implementation and accuracy. [ABSTRACT FROM AUTHOR]

For a strictly stationary sequence of nonnegative regularly varying random variables ( X) we study functional weak convergence of partial maxima processes $M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]$ in the space D[0, 1] with the Skorohod J topology. Under the strong mixing condition, we give sufficient conditions for such convergence when clustering of large values do not occur. We apply this result to stochastic volatility processes. Further we give conditions under which the regular variation property is a necessary condition for J and M functional convergences in the case of weak dependence. We also prove that strong mixing implies the so-called Condition $\mathcal {A}(a_{n})$ with the time component. [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]

This paper is concerned with studying the dependence structure between two random variables Y1 and Y2 in the presence of a covariate X, which affects both marginal distributions but not the dependence structure. This is reflected in the property that the conditional copula of Y1 and Y2 given X, does not depend on the value of X. This latter independence often appears as a simplifying assumption in pair-copula constructions. We introduce a general estimator for the copula in this specific setting and establish its consistency. Moreover, we consider some special cases, such as parametric or nonparametric location-scale models for the effect of the covariate X on the marginals of Y

In this article, we first introduce multifrational Riemann-Liouville Brownian sheets. Then, we show a result of approximation in law of the multifractional Riemann-Liouville Brownian sheet. The construction of these approximations is based on a sequence of I.I.D random variables. [ABSTRACT FROM PUBLISHER]

The Mittag-Leffler process X = (Xt)t ≥0 is introduced. This Markov process has the property that its marginal random variables Xt are Mittag-Leffler distributed with parameter e-1, t ∈ [0, ∞), and the semigroup (Tt)t≥o of X satisfies Ttf (x) = E(f(xe-tXt)) for all x ≥ 0 and all bounded measurable functions f : [0, ∞) → ℝ. Further characteristics of the process X are derived, for example an explicit formula for the joint moments of its finite-dimensional distributions. The Mittag-Leffler process turns out to be Siegmund dual to Neveu's continuous-state branching process. The main result states that the block counting process of the Bolthausen-Sznitman n-coalescent, properly scaled, converges in the Skorohod topology to the Mittag-Leffler process X as the sample size n tends to infinity. We provide an equivalent version of this convergence result involving stable distributions. [ABSTRACT FROM AUTHOR]

In this study, a semi-Markovian random walk process (X(t)) with a discrete interference of chance is investigated. Here, it is assumed that the ζn, n = 1; 2; 3; ..., which describe the discrete interference of chance are independent and identically distributed random variables having restricted normal distribution with parameters (α; σ²). Under this assumption, the ergodicity of the process X(t) is proved. Moreover, the exact forms of the ergodic distribution and characteristic function are obtained. Then, weak convergence theorem for the ergodic distribution of the process Wa(t) ≡ X(t)/a is proved under additional condition that σ/a → 0 when a →∞. [ABSTRACT FROM AUTHOR]

Numerical solutions of stochastic problems involving random processes X (t) , which constitutes infinite families of random variables, require to represent these processes by finite dimensional (FD) models X d (t) , i.e., deterministic functions of time depending on finite numbers d of random variables. Most available FD models match the mean, correlation, and other global properties of X (t). They provide useful information to a broad range of problems, but cannot be used to estimate extremes or other sample properties of X (t). We develop FD models X d (t) for processes X (t) with continuous samples and establish conditions under which these models converge weakly to X (t) in the space of continuous functions as d → ∞. These theoretical results are illustrated by numerical examples which show that, under the conditions established in this study, samples and extremes of X (t) can be approximated by samples and extremes of X d (t) and that the discrepancy between samples and extremes of these processes decreases with d. [ABSTRACT FROM AUTHOR]

It is known that for a sequence of independent and identically distributed random variables ( X) the regular variation condition is equivalent to weak convergence of partial maxima $M_{n}= \max \{X_{1}, \ldots , X_{n}\}$, appropriately scaled. A functional version of this is known to be true as well, the limit process being an extremal process, and the convergence takes place in the space of càdlàg functions endowed with the Skorohod J topology. We first show that weak convergence of partial maxima M holds also for a class of weakly dependent sequences under the joint regular variation condition. Then using this result we obtain a corresponding functional version for the processes of partial maxima $M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]$, but with respect to the Skorohod M topology, which is weaker than the more usual J topology. We also show that the M convergence generally can not be replaced by the J convergence. Applications of our main results to moving maxima, squared GARCH and ARMAX processes are also given. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of stochastic Liénard equations with random switching. The motivation of our study stems from modeling of complex systems in which both continuous dynamics and discrete events are present. The continuous component is a solution of a stochastic Liénard equation and the discrete component is a Markov chain with a finite state space that is large. A distinct feature is that the processes under consideration are time inhomogeneous. Based on the idea of nearly decomposability and aggregation, the state space of the switching process can be viewed as “nearly decomposable” into l subspaces that are connected with weak interactions among the subspaces. Using the idea of aggregation, we lump the states in each subspace into a single state. Considering the pair of process (continuous state, discrete state), under suitable conditions, we derive a weak convergence result by means of martingale problem formulation. The significance of the limit process is that it is substantially simpler than that of the original system. Thus, it can be used in the approximation and computation work to reduce the computational complexity. [ABSTRACT FROM PUBLISHER]

In this paper, we extend the well-studied fractional Brownian motion of Riemann-Liouville type to the multivariate case, and the corresponding processes are called operator fractional Brownian motions of Riemann-Liouville type. We also provide two results on approximation to operator fractional Brownian motions of Riemann-Liouville type. The first approximation is based on a Poisson process, and the second one is based on a sequence of I.I.D. random variables. [ABSTRACT FROM AUTHOR]

It is well known that the standard independent, identically distributed (iid) bootstrap of the mean is inconsistent in a location model with infinite variance (α-stable) innovations. This occurs because the bootstrap distribution of a normalised sum of infinite variance random variables tends to a random distribution. Consistent bootstrap algorithms based on subsampling methods have been proposed but have the drawback that they deliver much wider confidence sets than those generated by the iid bootstrap owing to the fact that they eliminate the dependence of the bootstrap distribution on the sample extremes. In this paper we propose sufficient conditions that allow a simple modification of the bootstrap (Wu, 1986) to be consistent (in a conditional sense) yet to also reproduce the narrower confidence sets of the iid bootstrap. Numerical results demonstrate that our proposed bootstrap method works very well in practice delivering coverage rates very close to the nominal level and significantly narrower confidence sets than other consistent methods. [ABSTRACT FROM PUBLISHER]

Abstract: Motivated by problems in functional data analysis, in this paper we prove the weak convergence of normalized partial sums of dependent random functions exhibiting a Bernoulli shift structure. [Copyright &y& Elsevier]

Abstract: This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall''s tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall''s tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures. [Copyright &y& Elsevier]

Abstract: The peakedness of a random variable (RV) about a point is defined by . A RV is said to be less peaked about than a RV about , denoted by , if for all , i.e., is stochastically larger than . These generalize the original definitions of Birnbaum (1948) who considered the cases where and were symmetric about and , respectively. Statistical inferences about the distribution functions of continuous and under peakedness ordering in the symmetric case have been treated in the literature. Rojo et al. (2007) provided estimators of the distributions in the general case and analyzed their properties. We show that these estimators could have poor asymptotic properties relative to those of the empiricals. We provide improved estimators of the DFs, 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 also used to illustrate our theoretical results. [Copyright &y& Elsevier]

We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet Wα, β with Hurst parameter (α, β) ∈ (0, 1)2. When $0 \lt \alpha \leq 1-\frac{1}{2q}$ or $0 \lt \beta \leq 1-\frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order q ≥ 2, while for $1-\frac{1}{2q} \lt \alpha, \beta \lt 1$ we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time (1, 1). [ABSTRACT FROM AUTHOR]

In a wide subclass of generalized order statistics (gOs), which contains most of the known and important models of ordered random variables, weak convergence of lower extremes are developed. A recent result of extreme value theory of m −gOs (as well as the classical extreme value theory of ordinary order statistics) yields three types of limit distributions that are possible in case of linear normalization. In this paper a similar classification of limit distributions holds for extreme gOs, where the parameters γ, j = 1, . . . , n, are assumed to be pairwise different. Two illustrative examples are given to demonstrate the practical importance for some of the obtained results. [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]

The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process. In this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence. We will apply our results to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and solutions to stochastic recurrence equations. [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]

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically, it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable. [ABSTRACT FROM AUTHOR]

In survival analysis, the classical Koziol-Green random censorship model is commonly used to describe informative censoring. Hereby, it is assumed that the distribution of the censoring time is a power of the distribution of the survival time. In this article, we extend this model by assuming a general function between these distributions. We determine this function from a relationship between the observable random variables which is described by a copula family that depends on an unknown parameter θ. For this setting, we develop a semi-parametric estimator for the distribution of the survival time in which we propose a pseudo-likelihood estimator for the copula parameter θ. As results, we show first the consistency and asymptotic normality of the estimator for θ. Afterwards, we prove the weak convergence of the process associated to the semi-parametric distribution estimator. Furthermore, we investigate the finite sample performance of these estimators through a simulation study and finally apply it to a practical data set on survival with malignant melanoma. [ABSTRACT FROM AUTHOR]

By the Cramér method, the large deviation principle for a form of compound Poisson process Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is obtained, where N(t), t > 0, is a nonhomogeneous Poisson process with intensity λ (t) > 0, Xi, i ≥ 1, are i. i. d. nonnegative random variables independent of N(t), and h(t), t > 0, is a nonnegative monotone real function. Consequently, weak convergence for S(t) is also obtained. [ABSTRACT FROM AUTHOR]

We prove that the sequences generated by the Douglas-Rachford method converge weakly to zero of the sum of two maximal monotone operators using new tools introduced in recent works of Eckstein and the author. The assumption of maximal monotonicity of the sum is also removed, using a recent result of Bauschke. [ABSTRACT FROM AUTHOR]

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a 'distance' between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov-Smirnov, Cramér-von-Mises, and Anderson-Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555-581; 2010 © 2010 Statistical Society of Canada [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 present a notion of semi-self-decomposability for distributions with support in Z +. We show that discrete semi-self-decomposable distributions are infinitely divisible and are characterized by the absolute monotonicity of a specific function. The class of discrete semi-self-decomposable distributions is shown to contain the discrete semistable distributions and the discrete geometric semistable distributions. We identify a proper subclass of semi-self-decomposable distributions that arise as weak limits of subsequences of binomially thinned sums of independent Z +-valued random variables. Multiple semi-self-decomposability on Z + is also discussed. [ABSTRACT FROM AUTHOR]