Narrow Results

In this paper, we prove strong convergence of $q$-Gaussians with respect to a parameter $q$, which implies the spectrum of any self-adjoint non-commutative polynomial in $q$-Gaussians is continuously deformed with respect to $q$. With Bożejko’s Haagerup-type inequality, we follow Brannan’s approach to proving the asymptotic strong convergence of the free orthogonal quantum group.

In this paper, we study the split equality fixed point problem and propose a new iterative algorithm with a self-adaptive stepsize that does not need the prior information of the operator norms and is calculated easily. The L-Lipschitz and quasi-pseudo-contractive mappings are chosen as the operators in the algorithm since they have a wider range of applications. Moreover, we prove that the sequence generated by the algorithm strongly converges to the solution of the problem. Finally, we check the feasibility and effectiveness of the algorithm by comparing with other algorithms.

Motivated by weak convergence results in the paper of Takahashi & Yoshida (2005), we show strong convergence for an accelerated Euler–Maruyama scheme applied to perturbed stochastic differential equations. The Milstein scheme with the same acceleration is also discussed as an extended result. The theoretical results can be applied to analyze the multi-level Monte Carlo method originally developed by M.B. Giles. Several numerical experiments for the stochastic alpha-beta-rho (SABR) model of stochastic volatility are presented in order to confirm the efficiency of the schemes.

In this note, we discuss strong convergence of exponential integrator scheme based on spatial and time discretization for a class of neutral stochastic partial differential equations driven by α-stable processes.

Using an inequality due to Ricard and Xu, we give a different proof of Paul Skoufranis’s recent result showing that the strong convergence of possibly non-commutative random variables $X^{(k)}\to X$ is stable under reduced free product with a fixed non-commutative random variable $Y$. In fact we obtain a more general fact: assuming that the families $X^{(k)}=\{X_i^{(k)}\}$ and $Y^{(k)}=\{Y_j^{(k)}\}$ are ∗-free as well as their limits (in moments) $X=\{X_i\}$ and $Y=\{Y_j\}$, the strong convergences $X^{(k)}\to X$ and $Y^{(k)}\to Y$ imply that of $\{X^{(k)},Y^{(k)}\}$ to $\{X,Y\}$. Phrased in more striking language: the reduced free product is “continuous” with respect to strong convergence. The analogue for weak convergence (i.e. convergence of all moments) is obvious. Our approach extends to the amalgamated free product, left open by Skoufranis.

Convergence properties of multidimensional continued fraction algorithms introduced by V. Baladi and A. Nogueira are studied. The paper contains an arithmetic proof of almost everywhere exponentially strong convergence of some two-dimensional Markovian random algorithms and dynamically defined ones. A special three-dimensional deterministic case is also discussed.

This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor–corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the proposed symmetric predictor–corrector Euler methods.

We consider the strong solution of the 2D Navier–Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the space-time rate of convergence is the “optimal” one, namely, $\eta \in [0,1/2)$ in time and 1 in space. Let us mention that the coefficient $\eta$ is equal to the time regularity of the solution with values in ${𝕃}^2$. Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Unlike previous results, our main new idea is the use of a discrete Grönwall lemma for the error estimate without any localization.

Using compactly supported wavelets, Giné and Nickl [Uniform limit theorems for wavelet density estimators, Ann. Probab.37(4) (2009) 1605–1646] obtain the optimal strong $L^{\infty }(ℝ)$ convergence rates of wavelet estimators for a fixed noise-free density function. They also study the same problem by spline wavelets [Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections, Bernoulli16(4) (2010) 1137–1163]. This paper considers the strong $L^p(ℝ)(1\le p\le \infty )$ convergence of wavelet deconvolution density estimators. We first show the strong $L^p$ consistency of our wavelet estimator, when the Fourier transform of the noise density has no zeros. Then strong $L^p$ convergence rates are provided, when the noises are severely and moderately ill-posed. In particular, for moderately ill-posed noises and $p=\infty$, our convergence rate is close to Giné and Nickl’s.

This paper discusses the uniformly strong convergence of multivariate density estimation with moderately ill-posed noise over a bounded set. We provide a convergence rate over Besov spaces by using a compactly supported wavelet. When the model degenerates to one-dimensional noise-free case, the convergence rate coincides with that of Giné and Nickl’s (Ann. Probab., 2009 or Bernoulli, 2010). Our result can also be considered as an extension of Masry’s theorem (Stoch. Process. Appl., 1997) to some extent.

The output signal-to-interference (SIR) of conventional matched filter receiver in random environment is considered. When the number of users and the spreading factors tend to infinity with their ratio fixed, the convergence of SIR is showed to be with probability one under finite fourth moment of the spreading sequences. The asymptotic distribution of the SIR is also obtained.

This paper is concerned with strong convergence and almost sure convergence for neutral stochastic differential delay equations under non-globally Lipschitz continuous coefficients. Convergence rates of $𝜃$-EM schemes are given for these equations driven by Brownian motion and pure jumps, respectively, where the drift terms satisfy locally one-sided Lipschitz conditions, and diffusion coefficients obey locally Lipschitz conditions, and the corresponding coefficients are highly nonlinear with respect to the delay terms.

In this paper, we introduce four new iterative schemes by modifying the CQ-method with Ishikawa and $S$-iterations. The strong convergence theorems are given by the CQ-projection method with our modified iterations for obtaining a common fixed point of two $G$-nonexpansive mappings in a Hilbert space with a directed graph. Finally, to compare the rate of convergence and support our main theorems, we give some numerical experiments.

In this paper, we introduce a general iterative algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for a relaxed cocoercive mapping in a Hilbert space. Then, we prove that the iterative sequence converges strongly to a common element of the three sets. The results obtained in this paper extend and improve the several recent results in this area.

In this paper, we employ the idea of exclusion of a condition given by Shahzad and Zegeye [on Mann and Ishikawa iteration schemes for multivalued maps in Banach spaces, Nonlinear Anal. Theory Methods Appl.71(3–4) (2009) 838–844] to our iterative process which converges much faster than Mann, Ishikawa and Agarwal's iterative processes.

In this paper, we study the strong and Δ-convergence result for Thianwan's iteration procedure in the sense introduced by Laokul and Panyanak [Approximating fixed points of nonexpansive mappings in CAT(0) spaces, Int. J. Math. Anal.3(27) (2009) 1305–1315]. Our results are the extensions and generalizations of several recent results from the current literature.

We introduce an iterative process which converges strongly to the minimum-norm fixed point of Lipschitzian pseudocontractive mapping. As a consequence, convergence result to the minimum-norm zero of monotone mappings is proved. In addition, applications to convexly constrained linear inverse problems and convex minimization problems are included. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

In this paper, assuming an appropriate condition, we prove that $N$-generalized hybrid mappings are demiclosed in Hilbert spaces. Using this fact, we prove a weak convergence theorem of Ishikawa type for these nonlinear mappings. Also, a strong convergence theorem of Halpern–Ishikawa type and a nonlinear ergodic theorem for $N$-generalized hybrid mappings have been proven in Hilbert spaces.

In this paper, we introduce a new four-step iterative process and name it as RK-iterative process and prove some $\mathrm{\Delta }$-convergence and strong convergence results using this process for generalized nonexpansive mappings in CAT(0) spaces. Our results generalize the corresponding results of many authors in different directions.

Let $X$ be a complete CAT(0) space, $K\ne \varnothing$ be a closed and convex subset of $X$ and $T:K\to C(K)$ be a multivalued nonexpansive mapping. We prove that the sequence of Ishikawa iteration converges to an endpoint of $T$. This improves, extends and unifies some recently announced results of the current literature.