Convergence in Distribution of Products of I.I.D. Nonnegative Matrices

Let (X _i) be a sequence of m × m i.i.d. stochastic matrices with distribution . Then ⁿ is the distribution of X _n X _n–1 ...X ₁. Simple sufficient conditions for the weak convergence of ( ⁿ ) are presented here. An extremely simple (and verifiable) necessary and sufficient condition is provided for m= 3. The method for m= 3 works for m> 3 even though calculations are more involved for higher values of m. We also discuss the purity of the limit distribution for m2.     

A computational comparison of the first nine members of a determinantal family of root-finding methods

For each natural number m greater than one, and each natural number k less than or equal to m, there exists a root-finding iteration function, B_m^(k) defined as the ratio of two determinants that depend on the first m−k derivatives of the given function. This infinite family is derived in Kalantari (J. Comput. Appl. Math. 126 (2000) 287–318) and its order of convergence is analyzed in Kalantari (BIT 39 (1999) 96–109). In this paper we give a computational study of the first nine root-finding methods. These include Newton, secant, and Halley methods. Our computational results with polynomials of degree up to 30 reveal that for small degree polynomials B_m^(k−1) is more efficient than B_m^(k), but as the degree increases, B_m^(k) becomes more efficient than B_m^(k−1). The most efficient of the nine methods is B₄⁽⁴⁾, having theoretical order of convergence equal to 1.927. Newton's method which is often viewed as the method of choice is in fact the least efficient method.     

Testing for the Mean of Random Curves: A Penalization Approach

Let X ₁,...,X _n be an i.i.d. sample of random curves, viewed as Hilbert space valued random elements, with mean curve m. An asymptotic test of m = m ₀ vs m ≠ m ₀ is proposed, when m ₀ is a fixed known function. The test statistics converges under very mild assumptions and relies on the pseudo-inversion of the covariance operator (leading to a non standard inverse problem). The power against local alternatives is investigated. In final form November 2004     

Fragmentation Processes with an Initial Mass Converging to Infinity

We consider a family of fragmentation processes where the rate at which a particle splits is proportional to a function of its mass. Let F ₁^(m)(t),F ₂^(m)(t),… denote the decreasing rearrangement of the masses present at time t in a such process, starting from an initial mass m. Let then m→∞. Under an assumption of regular variation type on the dynamics of the fragmentation, we prove that the sequence (F ₂^(m),F ₃^(m),…) converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly with the convergence of m−F ₁^(m) to a stable subordinator. A continuum random tree counterpart of this result is also given: the continuum random tree describing the genealogy of a self-similar fragmentation satisfying the required assumption and starting from a mass converging to ∞ will converge to a tree with a spine coding a fragmentation with immigration. Research supported in part by EPSRC GR/T26368.     

Convergence conditions for a restarted GMRES method augmented with eigenspaces

We consider the GMRES(m,k) method for the solution of linear systems Ax=b, i.e. the restarted GMRES with restart m where to the standard Krylov subspace of dimension m the other subspace of dimension k is added, resulting in an augmented Krylov subspace. This additional subspace approximates usually an A‐invariant subspace. The eigenspaces associated with the eigenvalues closest to zero are commonly used, as those are thought to hinder convergence the most. The behaviour of residual bounds is described for various situations which can arise during the GMRES(m,k) process. The obtained estimates for the norm of the residual vector suggest sufficient conditions for convergence of GMRES(m,k) and illustrate that these augmentation techniques can remove stagnation of GMRES(m) in many cases. All estimates are independent of the choice of an initial approximation. Conclusions and remarks assessing numerically the quality of proposed bounds conclude the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

Strongly almost convergent generalized difference sequences associated with multiplier sequences

Let Λ = (λ _k ) be a sequence of non-zero complex numbers. In this paper we introduce the strongly almost convergent generalized difference sequence spaces associated with multiplier sequences i.e. w ₀[A,Δ ^m ,Λ,p], w ₁[A,Λ ^m ,Λ,p], w _∞[A,Δ ^m ,Λ,p] and study their different properties. We also introduce Δ _Λ ^m -statistically convergent sequences and give some inclusion relations between w ₁[Δ ^m ,λ,p] convergence and Δ _Λ ^m -statistical convergence. Communicated by Pavel Kostyrko     

Preconditioners for block Toeplitz systems based on circulant preconditioners

We study the numerical solution of a block system T _m,n x=b by preconditioned conjugate gradient methods where T _m,n is an m×m block Toeplitz matrix with n×n Toeplitz blocks. These systems occur in a variety of applications, such as two-dimensional image processing and the discretization of two-dimensional partial differential equations. In this paper, we propose new preconditioners for block systems based on circulant preconditioners. From level-1 circulant preconditioner we construct our first preconditioner q ₁(T _m,n ) which is the sum of a block Toeplitz matrix with Toeplitz blocks and a sparse matrix with Toeplitz blocks. By setting selected entries of the inverse of level-2 circulant preconditioner to zero, we get our preconditioner q ₂(T _m,n ) which is a (band) block Toeplitz matrix with (band) Toeplitz blocks. Numerical results show that our preconditioners are more efficient than circulant preconditioners.     

On a Class of Generalized Lacunary Difference Sequence Spaces Defined by Orlicz Functions

In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞using m~(th)- difference.We study their properties like completeness,solidness,symmetricity.Also we obtain some inclusion relations involving the spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and[N_θ,M,Δ~m]_∞and the Cesàro summable and strongly Cesàro summable sequences.     

On the optimal convergence factor of the accelerated parameterized inexact Uzawa method with three parameters for augmented systems

Under the assumption that all eigenvalues of the preconditioned Schur complement are real, we present an analytical proof for obtaining the optimal convergence factor of the real accelerated parameterized inexact Uzawa (APIU) method when P=A. It is proved that the optimal convergence factor is the same as that of the generalized successive overrelaxation method, which was published at the same time, and that it can be attained only at the unique optimum point of parameters, regardless of whether m>n or m=n. In addition, we generalize the APIU method and analyze the relationship between the APIU method and 10 additional Uzawa‐like methods.     

Simplicial algorithm to solve the nonlinear complementarity problem onS n×R + m

In this paper, a simplicial algorithm is developed to solve the nonlinear complementarity problem onS ⁿ×R ₊ ^m . Furthermore, a condition for convergence is formulated. The triangulation which underlies the algorithm is a combination of the V-triangulation ofS ⁿ and the K-triangulation ofR ₊ ^m . Therefore, we will call it the VK-triangulation.The author wishes to thank Professor G. van der Laan for his valuable comments.     

Robust nonparametric estimators of monotone boundaries

This paper revisits some asymptotic properties of the robust nonparametric estimators of order-m and order-α quantile frontiers and proposes isotonized version of these estimators. Previous convergence properties of the order-m frontier are extended (from weak uniform convergence to complete uniform convergence). Complete uniform convergence of the order-m (and of the quantile order-α) nonparametric estimators to the boundary is also established, for an appropriate choice of m (and of α, respectively) as a function of the sample size. The new isotonized estimators share the asymptotic properties of the original ones and a simulated example shows, as expected, that these new versions are even more robust than the original estimators. The procedure is also illustrated through a real data set.     

Graphs of Non-Crossing Perfect Matchings

 Let P _n be a set of n=2m points that are the vertices of a convex polygon, and let ℳ _m be the graph having as vertices all the perfect matchings in the point set P _n whose edges are straight line segments and do not cross, and edges joining two perfect matchings M ₁ and M ₂ if M ₂=M ₁−(a,b)−(c,d)+(a,d)+(b,c) for some points a,b,c,d of P _n . We prove the following results about ℳ _m : its diameter is m−1; it is bipartite for every m; the connectivity is equal to m−1; it has no Hamilton path for m odd, m>3; and finally it has a Hamilton cycle for every m even, m≥4. Received: October 10, 2000 Final version received: January 17, 2002 RID="*" ID="*" Partially supported by Proyecto DGES-MEC-PB98-0933 Acknowledgments. We are grateful to the referees for comments that helped to improve the presentation of the paper.     

On some new type generalized difference sequence spaces

In this paper we introduce a new type of difference operator Δ _m ⁿ for fixed m, n ∈ ℕ. We define the sequence spaces ℓ_∞(Δ _m ⁿ ), c(Δ _m ⁿ ) and c ₀(Δ _m ⁿ ) and study some topological properties of these spaces. We obtain some inclusion relations involving these sequence spaces. These notions generalize many earlier existing notions on difference sequence spaces.      

Greedy algorithms in Banach spaces

We study efficiency of approximation and convergence of two greedy type algorithms in uniformly smooth Banach spaces. The Weak Chebyshev Greedy Algorithm (WCGA) is defined for an arbitrary dictionary D and provides nonlinear m-term approximation with regard to D. This algorithm is defined inductively with the mth step consisting of two basic substeps: (1) selection of an mth element _m ^c from D, and (2) constructing an m-term approximant G _m ^c . We include the name of Chebyshev in the name of this algorithm because at the substep (2) the approximant G _m ^c is chosen as the best approximant from Span( ₁ ^c ,..., _m ^c ). The term Weak Greedy Algorithm indicates that at each substep (1) we choose _m ^c as an element of D that satisfies some condition which is t _m -times weaker than the condition for _m ^c to be optimal (t _m =1). We got error estimates for Banach spaces with modulus of smoothness (u)u ^q , 1<q2. We proved that for any f from the closure of the convex hull of D the error of m-term approximation by WCGA is of order (1+t ₁ ^p ++t _m ^p )^–1/p , 1/p+1/q=1. Similar results are obtained for Weak Relaxed Greedy Algorithm (WRGA) and its modification. In this case an approximant G ^r _m is a convex linear combination of 0,₁ ^r ,..., ^r _m . We also proved some convergence results for WCGA and WRGA.     

Stability in vector-valued and set-valued optimization

In this paper, we discuss the stability of the sets of efficient points of vector-valued and set-valued optimization problems when the data (E _n,f _n) (resp. (E _n, F _n)) of the approximate problems converge to the data (E, f) (resp. (E, F)) of the original problem in the sense of Painleve-Kuratowski or Mosco. Our results improve and generalize those obtained by Attouch and Riahi in Section 5 in [1].     

The (n + 1)2 m -ray algorithm: a new simplicial algorithm for the variational inequality problem on ℝ + m ×S n

In this paper we propose a variable dimension simplicial algorithm for solving the variational inequality problem on the cross product of the nonnegative orthant ₊ ^m of them-dimensional Euclidean space ^m and then-dimensional unit simplexS ⁿ of ⁿ⁺¹. Starting from an arbitrary point (u, v) є ₊ ^m ×S ⁿ, the algorithm generates a piecewise linear path in ₊ ^m ×S ⁿ. The path is traced by making alternately linear programming pivot operations and replacement steps in an appropriate simplicial subdivision of ₊ ^m ×S ⁿ. The algorithm differs from the thus far known algorithm in the number of directions in which it may leave the starting point. More precisely, the algorithm has (n+1)2 ^m rays to leave the starting point whereas the existing algorithm hasn+m+1 rays. A convergence condition is presented and the accuracy estimation of an approximate solution generated is also given.     

On the Betti number of the image of a generic map

Let be a differentiable map of a closed m-dimensional manifold into an (m + k)-dimensional manifold with k > 0. We show, assuming that f is generic in a certain sense, that f is an embedding if and only if the (m - k + 1)-th Betti numbers with respect to the Čech homology of M and f(M) coincide, under a certain condition on the stable normal bundle of f. This generalizes the authors' previous result for immersions with normal crossings [BS1]. As a corollary, we obtain the converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results of [BR, BMS1, BMS2, Sae1] for immersions with normal crossings. Received: January 3, 1996     

Adaptive Semiparametric Estimation of the Memory Parameter

In Giraitis, Robinson, and Samarov (1997), we have shown that the optimal rate for memory parameter estimators in semiparametric long memory models with degree of “local smoothness” β is n^−r(β), r(β)=β/(2β+1), and that a log-periodogram regression estimator (a modified Geweke and Porter-Hudak (1983) estimator) with maximum frequency m=m(β)n^2r(β) is rate optimal. The question which we address in this paper is what is the best obtainable rate when β is unknown, so that estimators cannot depend on β. We obtain a lower bound for the asymptotic quadratic risk of any such adaptive estimator, which turns out to be larger than the optimal nonadaptive rate n^−r(β) by a logarithmic factor. We then consider a modified log-periodogram regression estimator based on tapered data and with a data-dependent maximum frequency m=m(β), which depends on an adaptively chosen estimator β of β, and show, using methods proposed by Lepskii (1990) in another context, that this estimator attains the lower bound up to a logarithmic factor. On one hand, this means that this estimator has nearly optimal rate among all adaptive (free from β) estimators, and, on the other hand, it shows near optimality of our data-dependent choice of the rate of the maximum frequency for the modified log-periodogram regression estimator. The proofs contain results which are also of independent interest: one result shows that data tapering gives a significant improvement in asymptotic properties of covariances of discrete Fourier transforms of long memory time series, while another gives an exponential inequality for the modified log-periodogram regression estimator.     

Pointwise convergence along a tangential curve for the fractional Schrödinger equation with 0 < m < 1

In this article, we study the pointwise convergence problem about solution to the fractional Schrödinger equation with 0 < m < 1 along a tangential curve and estimate the capacitary dimension of the divergence set. We extend the results of Cho and Shiraki (2021) for the case m > 1 to the case 0 < m < 1, which is sharp up to the endpoint.     

Strong limit theorems for quasi-orthogonal random fields

This is a systematic and unified treatment of a variety of seemingly different strong limit problems. The main emphasis is laid on the study of the a.s. behavior of the rectangular means ζ_mn = 1/(λ₁(m) λ₂(n)) Σ_i=1^m Σ_k=1ⁿ X_ik as either max{m, n} → ∞ or min{m, n} → ∞. Here {X_ik: i, k ≥ 1} is an orthogonal or merely quasi-orthogonal random field, whereas {λ₁(m): m ≥ 1} and {λ₂(n): n ≥ 1} are nondecreasing sequences of positive numbers subject to certain growth conditions. The method applied provides the rate of convergence, as well. The sufficient conditions obtained are shown to be the best possible in general. Results on double subsequences and 1-parameter limit theorems are also included.