Conditioned rates of convergence in the CLT for sums and maximum sums

An interesting recent result of Landers and Roggé (1977, Ann. Probability5, 595–600) is investigated further. Rates of convergence in the conditioned central limit theorem are developed for partial sums and maximum partial sums, with positive mean and zero mean separately, of sequences of independent identically distributed random variables. As corollaries we obtain a conditioned central limit theorem for maximum partial sums both for positive and zero mean cases.     

A unifying Radon-Nikodym theorem for vector measures

It is a known fact that certain derivation bases from martingales with a directed index set. On the other hand it is also true that the strong convergence of certain abstract martingales is a consequence of the Radon-Nikodym theory for vector measures (cf. Uhl, J. J., Jr., Trans. Amer. Math. Soc.145 1969, 271–285). Many other connections and applications of the latter theory with multidimensional problems in stochastic processes and representation theory are known (cf. Dinculeanu, N., Studia Math.25 1965, 181–205; Dinculeanu, N., and Foias, C., Canad. J. Math.13 1961, 529–556; Rao, M. M., Ann. Mat. pura et applicata76 1967, 107–132; Rybakov, V. I., Izv. Vys?. U?ebn. Zaved. Matematika19 1968, 92–101; Rybakov, V. I., Dokl. Akad. Nauk SSSR180 1968, 620–623). Starting from various vantage points, many authors have proposed several hypotheses for establishing abstract Radon-Nikodym theorems. In view of the great interest and importance of this problem in the areas mentioned above, it is natural to obtain a unifying result with a general enough hypothesis to deduce the various forms of the Radon-Nikodym theorem for vector measures. This should illuminate the Radon-Nikodym theory for vector measures and stimulate further work in abstract martingale problems. In this paper the first problem is attacked, leaving the martingale part and other applications for another study.The main result (Theorem 7 of Section 2) provides the desired unification and from if the Dunford-Pettis theorem, the Phillips theorem and several others are obtained. As martingale-type arguments are constantly present, a careful reader may note the easy translation of the hypothesis to the martingale convergence problem but we treat only the Radon-Nikodym problem using the language of measure theory and linear analysis.     

L 1 bounds for some martingale central limit theorems

The aim of this paper is to extend the results in [E. Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann. Probab., 10(3):672–688, 1982] and [J.C. Mourrat, On the rate of convergence in the martingale central limit theorem, Bernoulli, 19(2):633–645, 2013] to the L1-distance between distributions of normalized partial sums for martingale-difference sequences and the standard normal distribution.     

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L ² observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.     

多维随机向量序列加权和的中心极限定理

本文研究了多维随机向量序列加权和的渐近行为.利用Lindeberg中心极限定理的基本思想,得到了多维随机向量序列加权和的中心极限定理及其收敛速度,为Lindeberg中心极限定理的推广.     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

Detecting abrupt changes of the long-range dependence or the self-similarity of a Gaussian process

In this Note, an estimator of m instants (m is known) of abrupt changes of the parameter of long-range dependence or self-similarity is proved to satisfy a limit theorem with an explicit convergence rate for a sample of a Gaussian process. In each estimated zone where the parameter is supposed not to change, a central limit theorem is established for the parameter's (of long-range dependence, self-similarity) estimator and a goodness-of-fit test is also built. To cite this article: J.-M. Bardet, I. Kammoun, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A Multivariate CLT for Local Dependence withn log nRate and Applications to Multivariate Graph Related Statistics

This paper concerns the rate of convergence in the central limit theorem for certain local dependence structures. The main goal of the paper is to obtain estimates of the rate in the multidimensional case. Certain one-dimensional results are also improved by using some more flexible characteristics of dependence. Assuming the summands are bounded, we obtain rates close to those for independent variables. As an application we study the rate of the normal approximation of certain graph related statistics which arise in testing equality of several multivariate distributions     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

Noneuclidean harmonic analysis,the central limit theorem,and long transmission lines with random inhomogeneities

This is an expository paper. The derivation of the ordinary central limit theorem using the Fourier transform on the real line is reviewed. Harmonic analysis on the Poincaré-Lobatchevsky upper half plane H is sketched. The Fourier inversion formula on H reduces to that for the classical integral transform of F. G. Mehler (1881, Math. Ann.18, 161–194) and V. A. Fock (1943, Compt. Rend. Acad. Sci. URSS Dokl N. S.39, 253–256), for example. This result is then used to solve the heat equation on H, producing a non-Euclidean analogue of the density function for the Gaussian or normal distribution on H. The non-Euclidean central limit theorem for rotation invariant distributions on H with an application to the statistics of long transmission lines is also discussed.     

Determinants of Period Matrices and an Application to Selberg's Multidimensional Beta Integral

In work on critical values of linear functions and hyperplane arrangements, A. Varchenko (Izv. Akad. Nauk SSSR Ser. Mat.53 (1989), 1206–1235; 54 (1990), 146–158) defined certain period matrices whose entries are Euler-type integrals representing hypergeometric functions of several variables and derived remarkable closed-form expressions for the determinants of those matrices. In this article, we present elementary proofs of some of Varchenko's determinant formulas. By the same method, we obtain proofs of variations of Varchenko's determinants. As an application, we deduce new proofs of the multidimensional beta integrals of Selberg and of Aomoto. Further, we obtain a new proof of a determinant formula of A. Varchenko (Funct. Anal. Appl.25 (1999), 304–305) in which the entries are multidimensional Selberg-type integrals.     

Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces

In this paper, we continue to discuss the properties of iterates generated by a strict pseudo- contraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.： Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336-349 （2007）]. In order to get a strong convergence theorem, we modify the normal Mann＇s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.： Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51- 60 （2005）] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.     

Central Limit Theorems for Supercritical Superprocesses with Immigration

In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem extends and generalizes the results obtained by Ren et al. (Stoch Process Appl 125:428–457, 2015). We first give laws of large numbers for supercritical superprocesses with immigration since there are few convergence results on immigration superprocesses, then based on these results, we establish the central limit theorem.     

Central limit theorems for empirical andU-processes of stationary mixing sequences

This paper gives sufficient conditions for the weak convergence to Gaussian processes of empirical processes andU-processes from stationary β mixing sequences indexed byV-C subgraph classes of functions. If the envelope function of theV-C subgraph class is inL _p for some 2<p<∞, we obtain a uniform central limit theorem for the empirical process under the β mixing condition $$k^{p/(p - 2)} (\log k)^{2(p - 1)/(p - 2)} \beta _k \to 0 as k \to \infty $$ In the case that the functions in theV-C subgraph class are uniformly bounded, we obtain uniform central limit theorems for the empirical process and theU-process, provided that the decay rate of the β mixing coefficient satisfies β _k =O(k ^?r ) for somer>1. These conditions are almost minimal.     

Joint functional convergence of partial sums and maxima for linear processes<Superscript>*</Superscript>

For linear processes with independent identically distributed innovations that are regularly varying with tail index α ∈ (0, 2), we study the functional convergence of the joint partial-sum and partial-maxima processes. We derive a functional limit theorem under certain assumptions on the coefficients of the linear processes, which enable the functional convergence in the space of ?²-valued càdlàg functions on [0, 1] with the Skorokhod weak M₂ topology.We also obtain a joint convergence in the M₂ topology on the first coordinate and in theM₁ topology on the second coordinate.     

Étude asymptotique locale de l'estimateur par ondelettesA law of the iterated logarithm for wavelet density estimator

We state a pointwise central limit theorem for the linear wavelet density estimator in a more general setting than the result of Wu [12]. Furthermore, we also give a pointwise law of the iterated logarithm for this density estimator. Our proof of the law of the iterated logarithm uses the results of Mason [9] on the asymptotic behavior of the tail empirical process. To cite this article: A. Massiani, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 553–556.     

On Complete Convergence for Arrays of Random Elements

Abstract

A complete convergence theorem for arrays of rowwise independent random variables was obtained by Kruglov, Volodin, and Hu (Statistics and Probability Letters 2006, 76:1631–1640). In this article, we extend the result to a Banach space without any additional conditions. No assumptions are made concerning the geometry of the underlying Banach space.     

General theorems on rates of convergence in distribution of random variables II. Applications to the stable limit laws and weak law of large numbers

This part is concerned with the applications of the general limit theorems with rates of Part I, achieved by specializing the limiting r.v. X. This leads to new convergence theorems with higher order rates in the one- and multi-dimensional case for the stable limit law, for the central limit theorem, and the weak law of large numbers.     

Central Limit Theorem for Nonlinear Semi-Markov Reward Processes

Abstract

In this work, we obtain a central limit theorem for reward processes defined on a finite state space semi-Markov process, when reward functions assumed to have general forms and are not of constant rates. Martingale theory is the main tool which have been used for establishing the convergence of scaled and shifted reward process to a zero mean Brownian motion. The striking point in this article is considering general forms for the reward functions which are realistic in applications. The conditions needed for these results are existence of variances for sojourn times in each state and second order integrability of reward functions with respect to sojourn times distributions.     

A universal result in almost sure central limit theory

The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561–574; Schatte, Math. Nachr. 137 (1988) 249–256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular, a.s. central limit theorems and various related ‘logarithmic’ limit theorems have been obtained for several classes of independent and dependent random variables. In this paper we extend this theory and show that not only the central limit theorem, but every weak limit theorem for independent random variables, subject to minor technical conditions, has an analogous almost sure version. For many classical limit theorems this involves logarithmic averaging, as in the case of the CLT, but we need radically different averaging processes for ‘more sensitive’ limit theorems. Several examples of such a.s. limit theorems are discussed.