Weak Convergence Research Articles

Abstract We consider the estimation of the marginal expected shortfall $${\mathbb {E}}\left( X_h | Y_0>U_Y(1/p)\right) $$ E X h | Y 0 > U Y ( 1 / p ) at extreme levels, when $$((X_t, Y_t))_{t\in {\mathbb {Z}}}$$ ( ( X t , Y t ) ) t ∈ Z is a strictly stationary $$\beta -$$ β - mixing time series with marginal distributions of Pareto-type, $$U_Y$$ U Y is the tail quantile function associated to $$Y_t$$ Y t , h is a positive integer and $$p\in (0, 1)$$ p ∈ ( 0 , 1 ) is such that $$p\rightarrow 0$$ p → 0 . We propose an estimator for this risk measure based on a Weissman-type construction. First, in case of a non-negative time series, we establish the weak convergence of our estimator by using empirical processes arguments combined with the cluster method of Drees and Rootzén (2010). Then, we extend our result to the case of real-valued time series by using the decomposition of the original time series into the positive and negative parts, and we also propose a bootstrap procedure. The performance of our estimator is illustrated on a simulation experiment. Finally, the method is applied on river flow data.

The weak convergence for a measure related to a class of conformally invariant fully nonlinear operator

On weak convergence of Gaussian conditional distributions

Ergodicity and weak convergence of transition probabilities for the 2D primitive equations with multiplicative noise

In this paper, we establish a Freidlin-Wentzell type large deviation principle uniformly with respect to initial conditions in bounded subsets, that do not necessarily belongs to compact sets, of an infinite dimensional Banach space for two-dimensional primitive equations driven by multiplicative noise when the noise converges to zero. The proof is based on the weak convergence approach obtained by [Salins, Budhiraja, Dupuis; Trans. Amer. Math. Soc., 2019] and anisotropic estimates.

Abstract We reconsider the foundations of expected utility without assuming the linearity of the independence axiom. We consider a decision-maker who cancels out common outcomes when comparing a pair of lotteries with the same probability tree. We show that if the decision-maker is consistent with first-order stochastic dominance or topological continuity in weak convergence, then the decision-maker is an expected utility maximizer. First, this offers a simple method to differentiate behavior between prospect theory, canceling out common outcomes, and cumulative prospect theory, satisfying first-order stochastic dominance. Second, this offers a novel method to test technical continuity assumptions based on their behavioral content.

Abstract In this paper, we extend a criterion of Sodin on the convergence of graph spectral measures to regular graphs of growing degree. As a result, we show that for a sequence of random ‐regular graphs with vertices, if and tends to infinity, the normalized spectral measure converges almost surely in ‐Wasserstein distance to the semicircle distribution for any . This strengthens a result of Dumitriu and Pal. Many of the results are also extended to unitary‐colored regular graphs. For example, we give a short proof of the weak convergence to the Kesten–McKay distribution for the normalized spectral measures of random ‐lifts. This result is derived by generalizing a formula of Friedman involving Chebyshev polynomials and non‐backtracking walks.

This paper addresses the optimal control problem for a stochastic evolution system perturbed by a Markov-modulated Poisson process within a diffusion approximation framework. The considered system captures complex dynamics involving continuous evolution and discrete, state-dependent jumps, enabling the modeling of systems with regime-switching behavior or infrequent but significant events. The control function is constructed by minimizing a quality criterion through a stochastic optimization procedure. To analyze the asymptotic behavior of the system as the perturbation parameter vanishes, we derive the generator of the process and solve a corresponding singular perturbation problem. This allows us to prove the weak convergence of the stochastic system to a diffusion process. Furthermore, we establish sufficient conditions under which the control strategy converges almost surely to an optimal control. The obtained result makes it possible to study the rate of convergence of evolution under the optimal control for problems with a Markov-modulated Poisson perturbation.

In this paper, we propose a new general and stable fixed-point approach to compute the resolvents of the composition of a set-valued maximal monotone operator with a linear bounded mapping. Weak, strong and linear convergence of the proposed algorithms are obtained. Advantages of our method over the existing approaches are also thoroughly analyzed.

The split equality problem has gained significant recognition due to its widespread applicability across various applied mathematical fields. In this present paper, a hyperplane projection algorithm is introduced for solving the split equality problem in Hilbert spaces. This algorithm integrates the hyperplane projection technique with the gradient projection method, utilising Polyak's step sizes for efficient convergence. The weak convergence of our proposed algorithm is demonstrated as well as its relaxed version. Under mild conditions, the strong and linear convergence of the algorithms is established. Numerical experiments conducted on signal recovery problems reveal that our algorithm accelerates the convergence rate and outperforms some existing algorithms.

Ptychographic phase retrieval faces challenges of weak convergence, sensitivity to noise, and overlap ratio. Recognizing that the inherent structure of frequency information significantly impacts reconstruction, we propose phase retrieval via iterative spatial-frequency masking for ptychography (PRISM), a novel, to the best of our knowledge, reconstruction framework that decomposes reconstruction into a frequency-progressive optimization process. PRISM demonstrates improved convergence performance, overlap and noise robustness, and reconstruction quality. Both simulation and experimental results confirm PRISM's superior performance even under challenging imaging conditions with low overlap ratios and high noise levels.

Estimators of the conditional tail moment risk measure based on extreme Kaplan–Meier integral constructions are proposed. The situation when observations are heavy-tailed and subject to right-censoring is considered, which arises often in non-life insurance. Weak convergence is established for both standard and bias-reduced versions of the estimator, and the finite-sample performance is studied through simulations. A real data application to a theft guarantee from a Danish non-life insurer is considered.

Mapping the psychopathic brain: Divergent neuroimaging findings converge onto a common brain network.

This paper studies the large deviation principle (LDP) of a class of Hilfer fractional stochastic McKean–Vlasov differential equations with multiplicative noise. Firstly, by making use of the Laplace transform and its inverse transform, the solution of the equation is derived. Secondly, considering the equivalence between the LDP and the Laplace principle (LP), the weak convergence method is employed to prove that the equation satisfies the LDP. Finally, through specific example, it is elaborated how to utilize the LDP to analyze the behavioral characteristics of the system under small noise perturbation.

This article presents a general framework for checking the adequacy of the Cox proportional hazards model with interval-censored data, which arise when the event of interest is known only to occur over a random time interval. Specifically, we construct certain stochastic processes that are informative about various aspects of the model, that is, the functional forms of covariates, the exponential link function and the proportional hazards assumption. We establish their weak convergence to zero-mean Gaussian processes under the assumed model through empirical process theory. We then approximate the limiting distributions by Monte Carlo simulation and develop graphical and numerical procedures to check model assumptions and improve goodness of fit. We evaluate the performance of the proposed methods through extensive simulation studies and provide an application to the Atherosclerosis Risk in Communities Study. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Integrability of weak mixed first-order derivatives and convergence rates of scrambled digital nets

This paper derives the stochastic homogenization for two dimensional Navier–Stokes equations with random coefficients. By means of weak convergence method and Stratonovich–Khasminskii averaging principle approach, the solution of two dimensional Navier–Stokes equations with random coefficients converges in distribution to the solution of two dimensional Navier–Stokes equations with constant coefficients.

In this article, we introduce and rigorously analyze the concept of difference \(\lambda\)-weak convergence for sequences defined by an Orlicz function. This notion generalizes the classical weak convergence by incorporating a \(\lambda\)-density framework and an Orlicz function, providing a more flexible tool for analyzing convergence behavior in sequence spaces. We systematically investigate the algebraic and topological properties of these newly defined sequence spaces, establishing that they form linear and normed spaces under suitable conditions. Our results include demonstrating the convexity of these spaces and identifying several important inclusion relationships among them, such as strict inclusions between spaces involving different orders of difference operators and Orlicz functions satisfying the \(\Delta_{2}\)-condition.

Abstract We present a proof showing that the weak error of a system of $n$ interacting stochastic particles approximating the solution of the McKean–Vlasov equation is $\mathcal{O}({n^{-1}})$. Our proof is based on the Kolmogorov backward equation for the particle system and bounds on the derivatives of its solution which we derive more generally using the variations of the stochastic particle system. The convergence rate is verified by numerical experiments which also indicate that the assumptions made here and in the literature can be relaxed.

Abstract Convergence in distribution of fuzzy random variables can be studied via the notion of weak convergence of random elements in general metric spaces, with the $$d_p$$ metrics being the most common. Our main objective is to prove new properties of this type of convergence with respect to the endograph metric and as a consequence show that this definition of convergence is suitable while working with this particular metric. We study some topological properties of the space of fuzzy sets with this metric and use them to prove a new version of the Skorokhod representation theorem for fuzzy random variables. Next, we study some continuous operations, such as the sum and product by a scalar, and state some results regarding convergence in distribution of fuzzy random variables with respect to the endograph metric. In conclusion, convergence in distribution of fuzzy random variables in the endograph metric is preserved by the usual operations, making this approach to convergence in distribution feasible.