Continuity of Usual Operations and Variational Convergences

Given convergent sequences of functions (f _n ) and (g _n ), we look for conditions ensuring that the sequences (f _n +g _n ), (max(f _n ,g _n )) and (f _n g _n ) converge, being the infimal convolution. The convergences we use are variational convergences. This study is motivated by applications to Hamilton–Jacobi equations.     

Weak statistical convergence and weak filter convergence for unbounded sequences

For every weakly statistically convergent sequence (xn) with increasing norms in a Hilbert space we prove that . This estimate is sharp. We study analogous problem for some other types of weak filter convergence, in particular for the Erdös-Ulam filters, analytical P-filters and Fσ filters. We present also a refinement of the recent Aron-Garcia-Maestre result on weakly dense sequences that tend to infinity in norm.     

Some results related with statistical convergence and Berezin symbols

The concept of discrete statistical Abel convergence is introduced. In terms of Berezin symbols we present necessary and sufficient condition under which a series with bounded sequence {an}_n?0 of complex numbers is discrete statistically Abel convergent. By using concept of statistical convergence we also give slight strengthening of a result of Gokhberg and Krein on compact operators.     

Stability of the geometric Ekeland variational principle: Convex case

In geometric terms, the Ekeland variational principle says that a lower-bounded proper lower-semicontinuous functionf defined on a Banach spaceX has a point (x ₀,f(x ₀)) in its graph that is maximal in the epigraph off with respect to the cone order determined by the convex coneK _λ = {(x, α) ∈X × ?:λ ∥x∥ ≤ ? α}, where λ is a fixed positive scalar. In this case, we write (x ₀,f(x ₀))∈λ-extf. Here, we investigate the following question: if (x ₀,f(x ₀))∈λ-extf, wheref is a convex function, and if 〈f _n 〉 is a sequence of convex functions convergent tof in some sense, can (x ₀,f(x ₀)) be recovered as a limit of a sequence of points taken from λ-extf _n ? The convergence notions that we consider are the bounded Hausdorff convergence, Mosco convergence, and slice convergence, a new convergence notion that agrees with the Mosco convergence in the reflexive setting, but which, unlike the Mosco convergence, behaves well without reflexivity.     

Pointwise -convergence and -convergence in measure of sequences of functions

Let be an ideal. We say that a sequence of real numbers is -convergent to if for every neighborhood U of y the set of n's satisfying y_nU is in . Basing upon this notion we define pointwise -convergence and -convergence in measure of sequences of measurable functions defined on a measure space with finite measure. We discuss the relationship between these two convergences. In particular we show that for a wide class of ideals including Erdős–Ulam ideals and summable ideals the pointwise -convergence implies the -convergence in measure. We also present examples of very regular ideals such that this implication does not hold.     

On convergence of moment generating functions

Mukherjea et al. [Mukherjea, A., Rao, M., Suen, S., 2006. A note on moment generating functions. Statist. Probab. Lett. 76, 1185-1189] proved that if a sequence of moment generating functions M_n(t) converges pointwise to a moment generating function M(t) for all t in some open interval of the real line, not necessarily containing the origin, then the distribution functions F_n (corresponding to M_n) converge weakly to the distribution function F (corresponding to M). In this note, we improve this result and obtain conditions of the convergence which seem to be sharp: F_n converge weakly to F if M_n(t_k) converge to M(t_k), k=1,2,…, for some sequence {t₁,t₂,…} having the minimal and the maximal points. A similar result holds for characteristic functions.     

Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C with a fixed point, for some 0?κ<1. Given an initial guess x₀∈C and given also a real sequence {αn} in (0,1). The Mann's algorithm generates a sequence {xn} by the formula: xn+1=αnxn+(1−αn)Txn, n?0. It is proved that if the control sequence {αn} is chosen so that κ<αn<1 and , then {xn} converges weakly to a fixed point of T. However this convergence is in general not strong. We then modify Mann's algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372-379] from nonexpansive mappings to strict pseudo-contractions.     

Strong representation of weak convergence

Skorokhod's representation theorem states that if on a Polish space,there is a weakly convergent sequence of probability measures μnw→μ0,as n →∞,then there exist a probability space(Ω,F,P) and a sequence of random elements Xnsuch that Xn→ X almost surely and Xnhas the distribution function μn,n = 0,1,2,... We shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces Sn,a sequence of probability measures μnand a sequence of measurable mappings n such that μnn-1w→μ0,then there exist a probability space(Ω,F,P) and Sn-valued random elements Xndefined on Ω,with distribution μnand such that n(Xn) → X0 almost surely. In addition,we present several applications of our result including some results in random matrix theory,while the original Skorokhod representation theorem is not applicable.     

Über Folgen,die BezÜglich Eines Mittels Rekurrent Sind

Let X be a convex subset of a finite-dimensional real vector space. A function M: X ^k → X is called a strict mean value, if M(x₁,…, x_k) lies in the convex hull of x₁,…, x_k), but does not coincide with one of its vertices. A sequence (x_n)_{n∈ ?} in X is called M-recursive if x_n+k = M(x_n, x_n+1,…, x_n+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05.     

Pointwise properties of convergence in probability

If a sequence of random variables X_n converges to X in probability we know little about the pointwise behavior X_n(ω). In this note we show that if X_n converges to X quickly enough (for example, like n^?α for α > 0) then, for almost all ω, X_n(ω) converges to X(ω) outside a set of density zero.     

Moments of Two-Variable Functions and the Uniqueness of Graph Limits

For a symmetric bounded measurable function W on [0, 1]² and a simple graph F, the homomorphism density $t(F,W) = \int _{[0,1]^{V (F)}} \prod_ {i j\in E(F)} W(x_i, x_j)dx .$ can be thought of as a “moment” of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. The main motivation for this result comes from the theory of convergent graph sequences. A sequence (G _n ) of dense graphs is said to be convergent if the probability, t(F, G _n ), that a random map from V(F) into V(G _n ) is a homomorphism converges for every simple graph F. The limiting density can be expressed as t(F, W) for a symmetric bounded measurable function W on [0, 1]². Our results imply in particular that the limit of a convergent graph sequence is unique up to measure preserving transformation.     

On Almost Convergent and Statistically Convergent Subsequences

There are two well-known non-matrix summability methods which we will consider, namely “almost convergence” and “statistical convergence”. The results presented in this paper will be of two types, dealing with Lebesgue measure and Baire category. Establishing a one-to-one correspondence between the interval (0; 1] and the collection of all subsequences of a given sequence s = (s _n), we will examine the measure and category of the set of all almost convergent subsequences of (s _n). Similar questions for statistical and lacunary statistical convergence are considered. Results on rearrangements of sequences are also presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Functional convergence of stochastic integrals with application to statistical inference

Assuming that {(U_n,V_n)} is a sequence of càdlàg processes converging in distribution to (U,V) in the Skorohod topology, conditions are given under which {?f_n(β,u,v)dU_ndV_n} converges weakly to ?f(β,x,y)dUdV in the space C(R), where f_n(β,u,v) is a sequence of “smooth” functions converging to f(β,u,v). Integrals of this form arise as the objective function for inference about a parameter β in a stochastic model. Convergence of these integrals play a key role in describing the asymptotics of the estimator of β which optimizes the objective function. We illustrate this with a moving average process.     

A bounded sequence of normal functionals has a subsequence which is nearly weakly convergent

Let (φ_n) be a norm bounded sequence in the pre-dual of a von Neumann algebra M. In general it is not true that this sequence has a weakly convergent subsequence. But given a normal state ψ, then, for any ε>0, there exists a projection e such that ψ(1−e)?ε and the restriction of (φ_n) to eMe has a subsequence which converges weakly to a normal functional on eMe.     

Diophantine equations with products of consecutive terms in Lucas sequences

In this paper, we show that if (u_n)_n?1 is a Lucas sequence, then the Diophantine equation in integers n?1, k?1, m?2 and y with |y|>1 has only finitely many solutions. We also determine all such solutions when (u_n)_n?1 is the sequence of Fibonacci numbers and when u_n=(xⁿ-1)/(x-1) for all n?1 with some integer x>1.     

Increasing the convergence rate of series

This paper deals with the question of obtaining from the sequence {s_n} of partial sums of a convergent series s a new sequence {t_n} which converges to the same limit s as s_n, but more rapidly. When the general term u_n of the series s possesses certain types of expansion involving inverse powers of n, it is shown how t_n is obtained by adding a fixed number of terms to s_n. When the series s is convergent, these terms tend to zero as n tends to infinity, but they are such as to make t_n much more rapidly convergent to s—in fact we can make the convergence rate as great as we wish. Explicit general formulas are obtained for a wide range of important series.     

Stability of Doob—Meyer Decomposition Under Extended Convergence

In what follows, we consider the relation between Aldous‘s extended convergence and weak convergence of filtrations. We prove that, for a sequence (X^n) of Ft^n )-special semimartingales, with canonical decomposition X^n =M^n A^n, if the extended convergence (X^n,F.^n)→(X,T. ) holds with a quasi-left continuous (Ft)-special semimartingale X = M A, then, under an additional assumption of uniform integrability,we get the convergence in probability under the Skorokhod topology: M^n↑P→M and A^n↑P→ A.     

General theorems on rates of convergence in distribution of random variables I. General limit theorems

Let (X_n)_n?$\text{N}$ be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions F_Xn, and S_n = Σ_i=1ⁿX_i. The authors present limit theorems together with convergence rates for the normalized sums ?(n)S_n, where ?: $\text{N}$ → $\text{R}$⁺, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝_$\text{R}$f(x) d[F_?(n)Sn(x) ? F_X(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.     

A Trotter-Kato type result for a second order difference inclusion in a Hilbert space

A Trotter-Kato type result is proved for a class of second order difference inclusions in a real Hilbert space. The equation contains a nonhomogeneous term f and is governed by a nonlinear operator A, which is supposed to be maximal monotone and strongly monotone. The associated boundary conditions are also of monotone type. One shows that, if An is a sequence of operators which converges to A in the sense of resolvent and fn converges to f in a weighted l²-space, then under additional hypotheses, the sequence of the solutions of the difference inclusion associated to An and fn is uniformly convergent to the solution of the original problem.