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In this paper, a special sequence of controlled branching processes is considered. Using a martingale problem approach, we prove that under some mild conditions the limit of such processes is a kind of continuous branching process with dependent immigration constructed in Li (2019). The conditions are given in terms of probability generating functions and can be realized. [ABSTRACT FROM AUTHOR]

We study the N → ∞ limit of the normalized largest component in some systems of N diffusive particles with mean-field interaction. By applying a universal time change, the interaction in noises is transferred to the drift terms, and the asymptotic behavior of the maximum becomes well-understood due to existing results in the literature. We expect that the normalized maximum in the original setting has the same limiting distribution as that of i.i.d copies of a solution to the corresponding McKean–Vlasov SDE and we present some results and numerical simulations that support this conjecture. [ABSTRACT FROM AUTHOR]

We consider the class of quasi-infinitely divisible distributions. These distributions have appeared before in the theory of decompositions of probability laws, and nowadays they have various applications in theory of stochastic processes, physics, and insurance mathematics. The characteristic functions of quasi-infinitely divisible distributions admit Lévy type representation with real drift, nonnegative Gaussian variance, and "signed Lévy measure". Lindner et al. (2018) have recently done the first detailed analysis of these distributions based on such representations. The most complete results were established by them for the quasi-infinitely divisible distributions on the integers. In particular, the authors have obtained a criterion of weak convergence for distributions from this class in terms of parameters of their Lévy type representations. In the present short paper we complement this result by similar criteria of relative and stochastic compactness for quasi-infinitely divisible distributions with partial weak limits from this class. We also show that if a general sequence of distributions on the integers is relatively compact with quasi-infinitely divisible partial weak limits, then all distributions of the sequence are quasi-infinitely divisible except a finite number. [ABSTRACT FROM AUTHOR]

For large systems of Brownian particles interacting through their ranks introduced in (Banner et al., 2005), the empirical cumulative distribution function satisfies a porous medium PDE. However, when we introduce a common noise, the limit is no longer deterministic. Instead, we show that this limit is a solution of a stochastic PDE related to this porous medium PDE. This stochastic PDE is somewhat similar to the equations developed for conservation laws with rough stochastic fluxes (Lions et al., 2013). [ABSTRACT FROM AUTHOR]

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]

A law of large numbers for the empirical distribution of parameters of a one-layer artificial neural network with sparse connectivity is derived for a simultaneously increasing number of both, neurons and training iterations of the stochastic gradient descent. [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]

It is proved that the h -transform of the killed process for one-dimensional diffusions with respect to the scale function is weakly continuous with respect to the starting point. A similar result is obtained for its bridges. [ABSTRACT FROM AUTHOR]

We consider distributional limit of the Pearson chi-square statistic when the number of classes m n increases with the sample size n and n / m n → λ . Under mild moment conditions, the limit is Gaussian for λ = ∞ , Poisson for finite λ > 0 , and degenerate for λ = 0 . [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]

When a failure time observation is right censored, it is not observed beyond a random right threshold. When a possibly right censored observation is further susceptible to a random left threshold, a twice censored observation results. Patilea and Rolin have proposed product limit estimators for survival functions from twice censored data. Simultaneous confidence bands (SCBs) for survival functions from twice censored data are constructed in a way that mimics the approach that Hollander, McKeague and Yang pursued for random right censoring. Their nonparametric likelihood ratio function is adjusted, providing the basis for constructing the SCBs. The critical value needed for the SCBs is obtained using the bootstrap, for which asymptotic justification is provided. The SCBs align nicely as "neighborhoods" of the Patilea–Rolin nonparametric survival function estimator, in much the same way the likelihood ratio SCBs under random censoring are the "neighborhoods" of the Kaplan–Meier estimator. A simulation study supports the effectiveness of the proposed method. An illustration is given using synthetic data. [ABSTRACT FROM AUTHOR]

We find a random matrix to study a stochastic heat equation (SHE), and in doing so, we propose a method to discretize stochastic partial differential equations. Moreover, the convergence result helps to corroborate that standard partitions in the deterministic problem can also be considered in the stochastic case. In our study, we focus on the stochastic Schrödinger operator associated to the SHE, and prove a weak convergence of the random matrix to the stochastic operator. We do this by defining properly the space where the operators act, and by constructing a proper projection using the matrix. [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]

This paper presents a discrete-time equity derivatives pricing model with default risk in a no-arbitrage framework. Using the equity-credit reduced form approach where default intensity mainly depends on the firm’s equity value, we deduce the Arrow–Debreu state prices and the explicit pricing result in discrete time after embedding default risk in the pricing model. We prove that the discrete-time defaultable equity derivatives pricing has convergence stability, and it converges weakly to the continuous-time pricing results. [ABSTRACT FROM AUTHOR]

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about non-central moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps. [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]

We study linear eigenvalue statistics of band Hankel matrices with Brownian motion entries. We prove that the centred, normalized linear eigenvalue statistics of band Hankel matrices obey a central limit theorem (CLT) type result. We also discuss the convergence of linear eigenvalue statistics of band Hankel matrices with independent entries for odd power monomials. Our method is based on trace formula, moment method and some results of process convergence. [ABSTRACT FROM AUTHOR]

Abstract: We discuss inference on the Lévy measure in case of noisy observations. An extension of the pre-averaging method allows for a consistent estimation of the associated spectral function. The asymptotic behaviour of the novel estimator is the same as without noise. [Copyright &y& Elsevier]

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]

In this note we proved that weak limits, of sums of positive, independent and identically distributed random variables which are re-normalized by a non-linear shrinking transform max (0 , x − r) , are either degenerate or (some) compound Poisson distributions. [ABSTRACT FROM AUTHOR]

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: A lifetime with survival function , mean residual life function (MRL) , and finite mean is said to be new better than used in expectation (NBUE) if for all . We propose a new estimator for , based on a natural estimator of defined under the NBUE restriction. This is much simpler to implement than the only other restricted estimator in the literature. We also derive some asymptotic properties of the MRL of and extend our results to the censored case. [Copyright &y& Elsevier]

Abstract: In this paper we give an approximation theorem for Rosenblatt processes with , using martingale differences. [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: Multivariate measures of association are considered which, in the bivariate case, coincide with the population version of Spearman''s rho. For these measures, nonparametric estimators are introduced via the empirical copula. Their asymptotic normality is established under rather weak assumptions concerning the copula. The asymptotic variances are explicitly calculated for some copulas of simple structure. For general copulas, a nonparametric bootstrap is established. [Copyright &y& Elsevier]

Abstract: Let , be a triangular array of independent elliptical random vectors in . In this paper we investigate the asymptotic behaviour of the multivariate maxima of this triangular array. Generalising previous results for the bivariate set-up, we show that the normalised maxima of this elliptical triangular array is attracted by the multivariate Hüsler–Reiss distribution function provided that the components of the triangular array become asymptotically dependent with a specific rate, and further the random radius pertaining to the elliptical random vectors is in the Gumbel max-domain of attraction. [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: In this paper, we consider the Bickel–Rosenblatt test for a class of diffusion processes that covers the Ornstein–Uhlenbeck process. Using the discrete sampling scheme, we calculate residuals and construct the residual based Bickel–Rosenblatt test. We show that the test statistic is asymptotically normal under regularity conditions. The result is applicable to the test for the existence of jumps in diffusion models. [Copyright &y& Elsevier]

Abstract: In this paper we consider bivariate triangular arrays given in terms of linear transformations of asymptotically spherical bivariate random vectors. We show under certain restrictions that the componentwise maxima of such arrays is attracted by a bivariate max-stable distribution function with three parameters. This new class of max-stable distributions includes the bivariate max-stable Hüsler–Reiss distribution function for a special choice of parameters. [Copyright &y& Elsevier]

Abstract: We establish fluctuation limit theorems of immigration superprocesses with small branching rates. The weak convergence of the processes on a Sobolev space is established, which improves the result of Gorostiza and Li (High density fluctuations of immigration branching particle systems. In: Gorostiza, L.G., Ivanoff, B.G. (Eds.), Stochastic Models, (Ottawa, Ontario, 1998), CMS Conference Proceedings 2000, Series vol. 26, American Mathematical Society, Providence, RI, pp. 159–171). The limiting processes are infinite-dimensional Ornstein–Uhlenbeck type processes. [Copyright &y& Elsevier]

Abstract: A limit theorem in the sense of the weak convergence of probability measures on the complex plane for the Estermann zeta-function is proved. [Copyright &y& Elsevier]

Abstract: Let be a triangular array of independent bivariate elliptical random vectors. Hüsler and Reiss (1989. Statist. Probab. Lett. 7, 283–286) show that for the particular case that the array is Gaussian, the maxima of this array is in the max-domain of attraction of Hüsler–Reiss distribution function, provided that an asymptotic condition holds for the correlation . In this paper we obtain a similar result for the more general case of elliptical triangular arrays. [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 study the asymptotic distribution of the generally normalized maximum under finite mixture models. In one of the two theorems presented, we obtain a necessary and sufficient condition for this weak convergence, as well as the limit forms. The second theorem gives sufficient conditions for this convergence when the components of the mixture have different general normalizations. Examples are given to illustrate the applications of the two theorems. [Copyright &y& Elsevier]

We generalize a result of Sottinen (Finance Stochastics 5 (2001) 343) by proving an approximation theorem for the fractional Brownian motion, with H>1/2, using martingale-differences. [Copyright &y& Elsevier]

This work deals with asymptotic properties of the [αm]th-order statistic of a type-II progressively censored sample of size m. Such an order statistic, indexed by α∈[0,1], is called the quantile process. Our main results concern the normalized version of the quantile process for which a weak convergence result is obtained. This result is applied in order to construct non-parametric estimators of quantiles. Monte-Carlo simulations illustrate the behavior of the estimators for limited sample size. [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]

Sufficient conditions for convergence of monotone moments of weakly convergent random variables, concerning the rate of convergence, are given. They are often more convenient than the necessary and sufficient uniform integrability condition. Some asymptotic evaluations for inverse moments are presented. [Copyright &y& Elsevier]

We give a result of approximation in law, in the space of the continuous functions on [0,1]2, of Gaussian two-parameter processes that can be represented in law as a certain Wiener-type integral. The approximations are constructed from a Poisson process in the plane. [Copyright &y& Elsevier]

Let X1,…,Xn be a sequence of i.i.d. random variables with common distribution P on the real line. Assuming that P has a smooth density, we construct a histogram based estimator Pn,H and establish weak convergence of the empirical process √ of n(Pn,H−P)(f)f∈F under sharp conditions. If F is a class of indicators of sets, then the conditions imposed are necessary and sufficient. [Copyright &y& Elsevier]