Sum Theorem Research Articles

On multiparameter Chacon’s type ergodic theorems

The purpose of this paper is to deal with generalizations of ratio ergodic theorems due to R.V. Chacon, G. Baxter, and K. Jacobs. We prove two weighted generalizations of the Chacon ergodic theorem and the Jacobs random ergodic theorem. L. Sucheston has formulated a general principle yielding simultaneous proofs of many almost everywhere multiparameter convergence theorems. This principle will allow us to derive multiparameter Chacon’s type ergodic theorems for positive linear contractions on L1. The advantage is that we can inquire further into the problem of improving the multiparameter Chacon-Ornstein ergodic theorem due to Frangos and Sucheston. A multiparameter generalization of the Dunford-Schwartz ergodic theorem is also obtained. In addition, our consideration comes to the a.e. convergence for sectorially restricted averages in the commutative case, as in the Brunel-Dunford-Schwartz theorem. Moreover, we establish two Chacon’s type nonlinear ergodic theorems for the nonlinear sums of affine operators on L1.

A generalization of Krasnosel’skii fixed point theorem for sums of compact and contractible maps with application

Abstract The existence of a fixed point for the sum of a generalized contraction and a compact map on a closed convex bounded set is proved. The result is applied to a kind of nonlinear integral equations.

Base dimension-like functions of the type ind

In Iliadis (2005) [4] base dimension-like functions of the type ind were introduced. These functions were studied only with respect to the property of universality. Here, we study these functions with respect to other standard properties of dimension theory. We present theorems concerning subspace theorems, partition theorems, sum theorems, and product theorems. Finally, we give some open questions concerning these functions.

Functional central limit theorems for sums of nearly nonstationary processes*

We study some Holderian functional central limit theorems for the polygonal partial-sum processes built on a first-order autoregressive process yn,k = φnyn,k−1 + ek with ϕn converging to 1 and i.i.d. centered square-integrable innovations. In the case where ϕn = eγ/n with a negative constant γ, we prove that the limiting process is an integrated Ornstein–Uhlenbeck one. In the case where ϕn = 1− γn/n, with γn tending to infinity slower than n, the convergence to Brownian motion is established in Holder space in terms of the rate of γn and the integrability of the eks.

Some generalizations of Aczél, Bellman's inequalities and related power sums

In this paper, we establish some functional generalizations and refinements of Aczel's inequality and of Bellman's inequality. We also establish several mean value theorems for the related power sums.

On the random functional central limit theorems with almost sure convergence for subsequences

Abstract In this paper we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related ‘logarithmic’ limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems for subsequences.

Principle of conditioning revisited

Abstract Principle of Conditioning is a well-known heuristic rule which allows constructing limit theorems for sums of dependent random variables from existing limit theorems for independent summands. In the paper we state a general limit theorem on converegnce to stable laws, which is valid for stationary sequences and provides a link between the Principle of Conditioning and ergodic theorems.

A strong approximation theorem for positively dependent Gaussian sequences and its applications

Under the polynomial decay rates and finite second moment, a strong approximation theorem for partial sums of linear positive quadrant dependent Gaussian sequences is derived. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem are obtained with minimal conditions.

Strong summability theorems for H p (Δ d )

We prove that certain means of the (C,α,…,α)-means (α=1/p−1) of the d-dimensional trigonometric Fourier series are uniformly bounded operators from the Hardy space Hp to Hp (1≦p≦2). As a consequence we obtain strong summability theorems concerning (C,α,…,α)-means.

A Complete Convergence Theorem for Row Sums from Arrays of Rowwise Independent Random Elements in Rademacher Type p Banach Spaces

We extend in several directions a complete convergence theorem for row sums from an array of rowwise independent random variables obtained by Sung, Volodin, and Hu [8] to an array of rowwise independent random elements taking values in a real separable Rademacher type p Banach space. An example is presented which illustrates that our result extends the Sung, Volodin, and Hu result even for the random variable case.

Goodness-of-fit testing: the thresholding approach

The classical Pearson's chi-square test for goodness-of-fit has found extensive applications in areas such as contingency tables and, recently, multiple testing. Mann and Wald [(1942), ‘On the Choice of the Number of Class Intervals in the Application of the Chi Square Test’, The Annals of Mathematical Statistics, 13, 306–317] were the first to establish the power advantages of letting the number n bin of bins tend to infinity with n, and found n bin=n 2/5 to be the optimal rate. For a corresponding development in the area of contingency tables, see Holst [(1972), ‘Asymptotic Normality and Efficiency for Certain Goodness-of-Fit Tests’, Biometrika, 59, 137–145], Morris [(1975), ‘Central Limit Theorems for Multinomial Sums’, The Annals of Statistics, 3, 165–188], and Koehler and Larntz [(1980), ‘An Empirical Investigation of Goodness-of-Fit Statistics for Sparse Multinomials’, Journal of the American Statistical Association, 75, 336–344]. In this paper, we consider the use of thresholding methods to further improve on the power of Pearson's chi-square test. An alternative statistic, based on the cell averages, is also studied. The Fourier or wavelet transformation is used to ensure power enhancement in both high- and low-signal-to-noise ratio alternatives. Simulations suggest that application of order thresholding (Kim, M.H., and Akritas, M.G. (2010), ‘Order Thresholding’, The Annals of Statistics, 38, 2314–2350) achieves accurate type I error rates, and competitive power.

Generalizations of some zero sum theorems

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law

Let X, X1, X2,... be a sequence of independent and identically distributed random variables in the domain of attraction of a normal distribution. A universal result in almost sure limit theorem for the self-normalized partial sums S n /V n is established, where . Mathematical Scientific Classification: 60F15.

Approximating class approach for empirical processes of dependent sequences indexed by functions

We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class ${\mathcal{F}}$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_i))_{i\geq0}$. We introduce a new bracketing number to measure the size of the index class ${\mathcal{F}}$ which fits this setting. Our results apply to the empirical process of data $(X_i)_{i\geq0}$ satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron-Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, for example, to ergodic torus automorphisms.

Almost sure central limit theorem for partial sums of Markov chain

The authors prove an almost sure central limit theorem for partial sums based on an irreducible and positive recurrent Markov chain using logarithmic means, which realizes the extension of the almost sure central limit theorem for partial sums from an i.i.d. sequence of random variables to a Markov chain.

An Extended Limit Theorem for Correlated Lognormal Sums

Recent work has reported a lognormal limit theorem for sums of identically distributed equicorrelated lognormal random variables. An extended lognormal limit theorem for sums of nonidentically distributed correlated lognormal random variables having a particular correlation structure is derived.

A strong invariance principle for nonconventional sums

In Kifer and Varadhan (Nonconventional limit theorems in discrete and continuous time via martingales, 2010) we obtained a functional central limit theorem (known also as a weak invariance principle) for sums of the form \({\sum_{n=1}^{[Nt]} F\big(X(n), X(2n), .\, .\, .\, .\, X(kn), X(q_{k+1}(n)), X(q_{k+2}(n)), .\, .\, .\, , X(q_\ell(n))\big)}\) (normalized by \({1/\sqrt N}\)) where X(n), n ≥ 0 is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties and qi, i > k are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when qi’s are polynomials of growing degrees. This paper deals with strong invariance principles (known also as strong approximation theorems) for such sums which provide their uniform in time almost sure approximation by processes built out of Brownian motions with error terms growing slower than \({\sqrt N}\) . This yields, in particular, an invariance principle in the law of iterated algorithm for the above sums. Among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems as well, as to some questions in metric number theory and they can be considered as a natural follow up of a series of papers dealing with nonconventional ergodic averages.

A note on a sum theorem for dimension [formula omitted]-Ind

Main results are:1.Let Y be a closed subspace of a hereditarily normal X such that K-IndY⩽n and K-Ind(X∖Y)⩽n. Then K-IndX⩽n.2.Let X be a perfectly normal space. Then a finite sum theorem for dimension K-Ind holds in X if and only if K-Ind is monotonic in X.We denote by K a non-empty set of finite complete simplicial complexes.

Cauchy's Continuum

Cauchy's sum theorem of 1821 has been the subject of rival interpretations ever since Robinson proposed a novel reading in the 1960s. Some claim that Cauchy modified the hypothesis of his theorem in 1853 by introducing uniform convergence, whose traditional formulation requires a pair of independent variables. Meanwhile, Cauchy's hypothesis is formulated in terms of a single variable x, rather than a pair of variables, and requires the error term rn = rn(x) to go to zero at all values of x, including the infinitesimal value generated by explicitly specified by Cauchy. If one wishes to understand Cauchy's modification/clarification of the hypothesis of the sum theorem in 1853, one has to jettison the automatic translation-to-limits.

ESSENTIAL STATE SURFACES FOR KNOTS AND LINKS

AbstractWe study a canonical spanning surface obtained from a knot or link diagram, depending on a given Kauffman state. We give a sufficient condition for the surface to be essential. By using the essential surface, we can deduce the triviality and splittability of a knot or link from its diagrams. This has been done on the extended knot or link class that includes all semiadequate, homogeneous knots and links, and most algebraic knots and links. In order to prove the main theorem, we extend Gabai’s Murasugi sum theorem to the case of nonorientable spanning surfaces.