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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]

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 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]

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]

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]

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]

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]

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]

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]

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]

Different aspects of mean inactivity time (MIT) function such as properties, bounds and limiting behaviour are studied. We proceed to prove an important characterization theorem and also explore weak convergence issues. Finally, connections between MIT and reversed hazard rate functions are examined. [ABSTRACT FROM AUTHOR]

In this paper we study the sojourn time on the positive half-line up to time t of a drifted Brownian motion with starting point u and subject to the condition that min 0 ≤ z ≤ l B (z) > v , with u > v. This process is a drifted Brownian meander up to time l and then evolves as a free Brownian motion. We also consider the sojourn time of a bridge-type process, where we add the additional condition to return to the initial level at the end of the time interval. We analyze the weak limit of the occupation functional as u ↓ v. We obtain explicit distributional results when the barrier is placed at the zero level, and also in the special case when the drift is null. [ABSTRACT FROM AUTHOR]

Consider a non-explosive positive Feller process with no negative jumps. It is shown in this note that when infinity is an entrance boundary, in the sense that the entrance times of the process remain bounded when the initial value tends to infinity, the process admits a Feller extension on the compactified state space [ 0 , ∞ ]. Moreover, when started from infinity, the extended Markov process on [ 0 , ∞ ] leaves infinity instantaneously and stays finite, almost-surely. Arguments are adapted from a proof given by Kallenberg (2002) for diffusions. We also show that the process started from x converges weakly towards that started from infinity in the Skorokhod space, when x goes to infinity. [ABSTRACT FROM AUTHOR]