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We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged…
This paper discusses possible approaches to the escape rate in infinite lattices of weakly coupled maps with uniformly expanding repeller. It is proved that computed-via-volume rates of spatially periodic approximations grow linearly with…
This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on $N$-dimensional ${\rm RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. Moreover, we obtain almost…
Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these…
By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative H\'enon map, both of which exhibit the strong homoclinic chaos.…
We introduce a weak asymptotic version of nonlinear contraction, termed \emph{asymptotic pointwise contraction}. For a mapping on a metric space, this notion requires the existence of a sequence of functions that dominate the distances…
In this paper, we prove first that the iterates of a mean nonexpansive map defined on a weakly compact, convex set converge weakly to a fixed point in the presence of Opial's property and asymptotic regularity at a point. Next, we prove the…
In this monograph, we prove an asymptotic approximation for integrals of probability densities over sets in finite dimensional euclidean space, which are far away from the origin (asymptotic sets). We use this approximation to investigate…
In this paper, we propose a general framework for the asymptotic analysis of node-based verification-based algorithms. In our analysis we tend the signal length $n$ to infinity. We also let the number of non-zero elements of the signal $k$…
A common method for deriving non-parametric tests is to reformulate a parametric test in terms of sample ranks. Despite being distribution free (even in finite samples), the resulting tests often display remarkable asymptotic power…
Approximate Bayesian computation allows for statistical analysis in models with intractable likelihoods. In this paper we consider the asymptotic behaviour of the posterior distribution obtained by this method. We give general results on…
In this paper, we compute quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration in W-hyperbolic spaces. This iteration is an extension to a nonlinear setting of the modified Mann iteration defined recently by Bot, Csetnek…
Fixed points for uniformly local asymptotic nonexpansive maps are discussed in this article. An approximate fixed point sequence for such a map over a uniformly convex Banach space is derived. At the end, we study the unique fixed point for…
In this paper, we present a simple analysis of {\bf fast rates} with {\it high probability} of {\bf empirical minimization} for {\it stochastic composite optimization} over a finite-dimensional bounded convex set with exponential concave…
This is the third in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this…
In this paper we present a systematic study of regular sequences of quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular sequences of operators. We show that the type of…
This paper has two main goals: (a) establish several statistical properties---consistency, asymptotic distributions, and convergence rates---of stationary solutions and values of a class of coupled nonconvex and nonsmoothempirical risk…
Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, presented here for the first time, we introduce and study the class of {\it enriched nonexpansive mappings} in Hilbert spaces. In order to…
In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large $N$ limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and…
In this paper, we study inference for high-dimensional data characterized by small sample sizes relative to the dimension of the data. In particular, we provide an infinite-dimensional framework to study statistical models that involve…