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Abstract In this note, we restudy the general halfspace depth trimmed means and establish the weak convergence of their sample versions, which extends the result of Massé (2009) for dimensions one and two to any dimension. The asymptotic distribution of the Donoho (1982) halfspace depth trimmed mean is obtained as a special case and concretized for elliptically symmetric distributions. Highlights • We establish the weak convergence of general halfspace depth trimmed means. • The result of Massé (2009) for dimensions one and two is extended to any dimension. • The asymptotic distribution of Donoho (1982) trimmed mean is obtained as a special case. • The asymptotic distribution is concretized for elliptically symmetric distributions. [ABSTRACT FROM AUTHOR]

Weak convergence and related issues as well as reliability and moment bounds are studied for a non-monotonic analogue of the NBUFR class of life distributions. Connections of this new family with existing non-monotonic ageing classes are explored. Certain properties of the NBUFR family, hitherto unknown, are obtained as special cases. [ABSTRACT FROM AUTHOR]

We obtain the asymptotic distribution of the linear rank statistics under weak dependence. We consider a sequence of strong mixing random vectors with unequal dimensions and show the asymptotic normality of the rank statistics based on overall ranking. [ABSTRACT FROM AUTHOR]

The class of limit distribution functions of the joint upper record values, as well as the joint of lower record values, is fully characterized. Sufficient conditions for the weak convergence are obtained. As an application of this result, the sufficient conditions for the weak convergence of the record quasi-range, record quasi-midrange, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit distribution functions of these statistics are derived. [ABSTRACT FROM AUTHOR]

Two-step estimation of ordinary differential equation (ODE) is investigated when the transformation parameter is unknown. First we build the transformation parameter estimator by profile M-estimation. Then with a new consistency result of M-estimator, two-step estimator is proved to be consistent. Numerical studies validate the effectiveness of the proposed estimator. [ABSTRACT FROM AUTHOR]

The weak convergence of posterior distributions conditional on maximum pseudo-likelihood estimates (MPLE) is studied and exploited to justify the use of MPLE as summary statistics in approximate Bayesian computation (ABC). Our study could be generalized by replacing the pseudo-likelihood by other estimating functions (e.g. quasi-likelihoods and contrasts). [ABSTRACT FROM AUTHOR]

We establish an invariance principle for the fractional Brownian sheet, starting from discrete random fields constructed from two-parameter strong martingales. This is an approximation in law of the fractional Brownian sheet in Skorohord space in the plane. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the weak convergence of a sequence of Markov-modulated diffusion processes when the modulating Markov chain is ergodic and rapidly switching. We prove, in particular, its tightness property based on Aldous’ tightness criterion. [Copyright &y& Elsevier]

Abstract: We consider the number of survivors in a broad class of fair leader election algorithms after a number of election rounds. We give sufficient conditions for the number of survivors to converge to a product of independent identically distributed random variables. The number of terms in the product is determined by the round number considered. Each individual term in the product is a limit of a scaled random variable associated with the splitting protocol. The proof is established via convergence (to 0) of the first-order Wasserstein distance from the product limit. In a broader context, the paper is a case study of a class of stochastic recursive equations. We give two illustrative examples, one with binomial splitting protocol (for which we show that a normalized version is asymptotically Gaussian) and one with uniform splitting protocol. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the weak convergence of functional stochastic differential equations with variable delays driven by Wiener processes and jump processes, respectively. Moreover, an example is established to demonstrate the theory derived. [Copyright &y& Elsevier]

Abstract: We consider a Markov chain obtained by random iterations of Lipschitz maps chosen with a probability depending on the current position . We assume this system has a property of “contraction on average”, that is for some . In the present note, we study the weak convergence of the empirical process associated to this Markov chain. [Copyright &y& Elsevier]

Abstract: This note focuses on estimating the mean residual life function with left-truncated and right-censored data. We show that the proposed estimator converges weakly to a Gaussian process. The performances of the estimator and its pointwise confidence intervals are illustrated through simulation studies. [Copyright &y& Elsevier]

Abstract: We establish asymptotic normality for sums of triangular arrays of near integrated linear processes with martingale difference innovations. The results are obtained under minimal conditions. We also obtain weak convergence of the corresponding partial sum processes. The results are applicable to near unit root settings in statistical and econometric applications. [Copyright &y& Elsevier]

Abstract: In this paper we prove the invariance of some causality relationships between flows of information (represented by filtrations) under some types of convergence. We consider a statistical concept of causality which is based on Granger’s definition of causality, but instead of time series we focus on continuous time processes. [Copyright &y& Elsevier]

Abstract: We study the problem of parameter estimation for the continuous state branching processes with immigration, observed at discrete time points. The weighted conditional least square estimators (WCLSEs) are used for the drift parameters. Under the proper moment conditions, asymptotic distributions of the WCLSEs are obtained in the supercritical, sub- or critical cases. [Copyright &y& Elsevier]

Abstract: Mukherjea et al. [Mukherjea, A., Rao, M., Suen, S., 2006. A note on moment generating functions. Statist. Probab. Lett. 76, 1185–1189] proved that if a sequence of moment generating functions converges pointwise to a moment generating function for all in some open interval of the real line, not necessarily containing the origin, then the distribution functions (corresponding to ) converge weakly to the distribution function (corresponding to ). In this note, we improve this result and obtain conditions of the convergence which seem to be sharp: converge weakly to if converge to , , for some sequence having the minimal and the maximal points. A similar result holds for characteristic functions. [Copyright &y& Elsevier]

Abstract: and provided the limiting distribution for linear placement statistics under null hypotheses only when one of the sample sizes goes to infinity. In this paper we prove the asymptotic normality and the weak convergence of the linear placement statistics of and when the sample sizes of each group go to infinity simultaneously. [ABSTRACT FROM AUTHOR]

Abstract: Let be a regular metric as defined below for the space. Even when () is not a separable and complete metric space we show (i) that the usual conditions on a sequence of probability measures in () ensures its weak convergence and (ii) that Prohorov''s theorem in () can be derived as a consequence of our results. [Copyright &y& Elsevier]

Abstract: A new approach to the study of asymptotic behavior of truncated sumsis proposed, where is an increasing sequence of sets and are i.i.d. random vectors from the domain of attraction of a stable law. This approach is based on the investigation of weak convergence (more strong than usual) for empirical point processes associated with the vectors [Copyright &y& Elsevier]

Abstract: We consider a sequence of stochastic processes on converging weakly to X and call it polynomially convergent, if for continuous functionals F of polynomial growth. We present a sufficient moment conditions on for polynomial convergence and provide several examples, e.g. discrete excursions and depth first path associated to Galton–Watson trees. This concept leads to a new approach to moments of functionals of rooted trees such as height and path length. [Copyright &y& Elsevier]

In this paper we deal with the Skorohod representation of a given system of probability measures. More precisely, we give conditions for the existence of a Skorohod representation (X,(Xn)) with the following additional property: for each real number p⩾1 and each real random variable Z in Lp, the conditional expectation E[Z|Xn] converges in Lp to the conditional expectation E[Z|X]. [Copyright &y& Elsevier]

We consider a modification of the classical bootstrap procedure for censored observations by choosing a resample size m which is possibly different from the original sample size n. In the situation of quantile estimation we establish weak convergence of the bootstrap error process and show that modified bootstrapping leads to improved consistency rates for the maximum error. [Copyright &y& Elsevier]