Convergence of series of translations

For a mean zero norm one sequence (f _n )L ₂[0, 1], the sequence (f _n {nx+y}) is an orthonormal sequence inL ₂([0, 1]²); so if , then converges for a.e. (x, y)[0, 1]² and has a maximal function inL ₂([0, 1]²). But for a mean zerofL ₂[0, 1], it is harder to give necessary and sufficient conditions for theL ₂-norm convergence or a.e. convergence of . Ifc _n 0 and , then this series will not converge inL ₂-norm on a denseG subset of the mean zero functions inL ₂[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c _n | = 0(n ^–) for >1/2, then converges a.e. and unconditionally inL ₂[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL _p[0, 1] with 1p<. Finally, if (f _n )L [0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x _n )[0, 1]. Moreover, iffL [0, 1] is mean zero and , then for a.e. (x _n )[0, 1], converges for a.e.y and in allL _p [0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.     

The precise asymptotics of the complete convergence for moving average processes of <Emphasis Type="Italic">m</Emphasis>-dependent <Emphasis Type="Italic">B</Emphasis>-valued elements

Let {εt;t ∈ Z} be a sequence of m-dependent B-valued random elements with mean zeros and finite second moment. {a3;j ∈ Z} is a sequence of real numbers satisfying ∑j=-∞^∞｜aj｜ 〈 ∞. Define a moving average process Xt = ∑j=-∞^∞aj＋tEj,t ≥ 1, and Sn = ∑t=1^n Xt,n ≥ 1. In this article, by using the weak convergence theorem of { Sn/√ n _〉 1}, we study the precise asymptotics of the complete convergence for the sequence {Xt; t ∈ N}.     

Realizing a singular first fundamental form as a nonimmersed surface in Euclidean 3-space

We describe local properties of the first fundamental form for a C surface in Euclidean 3-space which fails to immerse on a small set D⁰. We then show that (subject to the regularity assumption KdA 0, on D⁰) given such a locally defined analytic tensor on an open subset of the plane there exists an analytic mapping into whose first fundamental form is the given tensor. We then describe several global consequences of the condition KdA 0 on D⁰.     

Differentiability of Inverse Operators and Limit Theorems for Inverse Functions

Let H be a map from a set SR ^d to R ^d . For tR ^d let _H (t) denote the distance from t to the set H(S). Consider sequences {s _n} _n1 in S such that . Any limit point of any such sequence (finite or infinite) is considered as a possible value of the inverse H ^–1(t). Any map defined in such a way will be called an SC-inverse (a selected closest inverse) to H. In the paper we study differentiability of the nonlinear operator at H=G, where G is a one-to-one map from S onto a set TR ^d with good analytic properties (specifically, a diffeomorphism). We establish compact differentiability of this operator tangentially to continuous functions and introduce a family of norms such that it is Fréchet differentiable with respect to them. We also obtain optimal bounds for the remainder of the differentiation, extending to the multivariate case recent results of Dudley. These differentiability results are applied to random maps , which could be statistical estimators of an unknown map G. For a function J on R ^d , let (J) _T be its restriction to T. It is shown that for a diffeomorphism G and for an increasing sequence of positive numbers {a _n } _n1 weak convergence of the sequence {a _n (G _n – G)} _n1 (locally in S) is equivalent to weak convergence of the sequence (locally in T) along with the convergence of the sequence to 0 in probability (locally uniformly in S). The equivalence holds for all SC-inverses and all double SC-inverses and it extends to the multivariate case a theorem of Vervaat. Moreover, each of these equivalent statements implies a kind of Taylor expansion of the SC-inverse at G (locally uniformly in T) where inv(A) denotes the inverse of a nonsingular linear transformation A in R ^d . Such limit theorems for functional inverses can be used to study asymptotic behavior of statistical estimators defined implicitly (as solutions of equations involving the empirical distribution P _n ). We show how to apply this approach to get asymptotic normality of M-estimators in the multivariate case under minimal assumptions. We consider an extension of the quantile function to the multivariate case related to M-parameters of a distribution P in R ^d (an M-quantile function)and use limit theorems for functional inverses to study limit behavior of the empirical M-quantile process. We also show how to use these theorems to study asymptotics of regression quantiles.     

Sharp Conditions for the CLT of Linear Processes in a Hilbert Space

In this paper we study the behavior of sums of a linear process associated to a strictly stationary sequence with values in a real separable Hilbert space and are linear operators from H to H. One of the results is that satisfies the CLT provided are i.i.d. centered having finite second moments and . We shall provide an example which shows that the condition on the operators is essentially sharp. Extensions of this result are given for sequences of weak dependent random variables under minimal conditions.     

On Complete Arcs Arising from Plane Curves

We point out an interplay between -Frobenius non-classical plane curves and complete -arcs in . A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete -arcs with parameters and being a power of the characteristic. In addition, for q a square, new complete -arcs with either and or and are constructed by using certain reducible plane curves.     

Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series and whose coefficients satisfy the condition for , where the sequence can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If as , then the series is uniformly convergent. If for all , then the sequence of partial sums of this series is uniformly bounded. If the series is convergent for and as , then this series is uniformly convergent. If the sequence of partial sums of the series for is bounded and for all , then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence is lacunary. In the general case, they are not necessary.     

Rates of convergence for the stability of large order statistics

Forr1 and eachnr, letM _nr be therth largest ofX ₁,X ₂, ...,X _n , where {X _n ,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of for all >0 and some –1, where {a _n } is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0.     

Accelerating infinite products

Slowly convergent infinite products are considered, where is a sequence of numbers, or a sequence of linear operators. Using an asymptotic expansion for the “remainder” of the infinite product a method for convergence acceleration is suggested. The method is in the spirit of the d-transformation for series. It is very simple and efficient for some classes of sequences . For complicated sequences it involves the solution of some linear systems, but it is still effective. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Limit Result for U-Statistics of Binary Variables

Define , where is a symmetric U-type statistic, H _k() is the Hermite polynomial of degree k, and {X, X _n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that according as EX=0 or EX0, respectively.     

A Change-of-Variable Formula with Local Time on Curves

Let be a continuous semimartingale and let be a continuous function of bounded variation. Setting and suppose that a continuous function is given such that F is C^1,2 on and F is on . Then the following change-of-variable formula holds: where is the local time of X at the curve b given by and refers to the integration with respect to . A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.     

Tauberian conditions for almost convergence

In [Acta Math. 80(1948), 167–190], G. G. Lorentz characterized almost convergent sequences in (or in ) in terms of the concept of uniform convergence of the de la Vallée-Poussin means. In this paper, we present Tauberian results which relate almost convergence to norm convergence or to the (C, 1) convergence. Our results generalize Kronecker lemma. As a consequence, we prove that almost convergence and norm convergence are equivalent for the sequence of the partial sums of the Fourier series of (or ), where . We also show that our results can be used to derive Fatou’s theorem.      

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Let = (₁,...,_d) be a vector with positive components and let D be the corresponding mixed derivative (of order _j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that

Quadrature formulas obtained by variable transformation

Quadrature formulas suitable for evaluation of improper integrals such as are obtained by means of variable transformations =tanhu and =erfu, and subsequent use of trapezoidal quadrature rule. Error analysis is carried out by the method of contour integral, and the results are confirmed on several concrete examples. Similar formulas are also obtained to accelerate the convergence of infinite integrals by means of variable transformations =sinhu and =tanu.     

Nondiagonal Hermite–Sobolev Orthogonal Polynomials

In this work, we study algebraic and analytic properties for the polynomials { Q _n } _{n 0}, which are orthogonal with respect to the inner product where , R such that – ² > 0.     

Limit sets and strengths of convergence for sequences in the duals of thread-like Lie groups

We consider a properly converging sequence of non-characters in the dual space of a thread-like group and investigate the limit set and the strength with which the sequence converges to each of its limits. We show that, if (π _k ) is a properly convergent sequence of non-characters in , then there is a trade-off between the number of limits σ which are not characters, their degrees, and the strength of convergence i _σ to each of these limits (Theorem 3.2). This enables us to describe various possibilities for maximal limit sets consisting entirely of non-characters (Theorem 4.6). In Sect. 5, we show that if (π _k ) is a properly converging sequence of non-characters in and if the limit set contains a character then the intersection of the set of characters (which is homeomorphic to ) with the limit set has at most two components. In the case of two components, each is a half-plane. In Theorem 7.7, we show that if a sequence has a character as a cluster point then, by passing to a properly convergent subsequence and then a further subsequence, it is possible to find a real null sequence (c _k ) (with ) such that, for a in the Pedersen ideal of C ^*(G _N ), exists (not identically zero) and is given by a sum of integrals over .     

On Lower Tail Probabilities of Positive Random Sums

Let be a sequence of independent identically distributed positive random variables with O-regularly varying distribution F at 0. Given a sequence of positive numbers, we show that belongs to the Type I domain of attraction of extremes for minima, by means of relating the asymptotic behaviour of P{S < } as 0, to that of E{e^-S/}. Our contribution is that we dispense with the unnatural moment condition from the literature, that F has finite variance. This in turn permits a novel application to lower tails of -stable distributions on Hilbert space.AMS 2000 Subject Classification. Primary—60G50, 60G70, Secondary—60B12, 60E07, 60F05, 60G52Research supported by NFR Grant M 650-19981841/2000, and by M.R. Leadbetter     

Convergence of Discrete Snakes

The discrete snake is an arborescent structure built with the help of a conditioned Galton-Watson tree and random i.i.d. increments Y. In this paper, we show that if and , then the discrete snake converges weakly to the Brownian snake (this result was known under the hypothesis ). Moreover, if this condition fails, and the tails of Y are sufficiently regular, we show that the discrete snake converges weakly to an object that we name jumping snake. In both case, the limit of the occupation measure is shown to be the integrated super-Brownian excursion. The proofs rely on the convergence of the codings of discrete snake with the help of two processes, called tours.     

Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We obtain upper bounds in terms of Fourier coefficients for the best approximation by an angle and for norms in the metric of L _p for functions of two variables defined by trigonometric series with coefficients such that as l ₁ + l ₂ and

Asymptotic properties of type I elliptical random vectors

Let be an elliptical random vector with a non-singular square matrix and a spherical random vector in , and let be a sequence of vectors in such that . We assume in this paper that the associated random radius R _k =(S ₁ + S ₂ +...+S _k )^1/2 is almost surely positive, and it has distribution function in the Gumbel max-domain of attraction. Relying on extreme value theory we obtain an exact asymptotic expansion of the tail probability for converging as to a boundary point. Further we discuss density convergence under a suitable transformation. We apply our results to obtain an asymptotic approximation of the distribution of partial excess above a high threshold, and to derive a conditional limiting result. Further, we investigate the asymptotic behaviour of concomitants of order statistics, and the tail asymptotics of associated random radius for subvectors of .