On the Boundedness of a Recurrence Sequence in a Banach Space

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space B: where |y _n} and | _n } are sequences bounded in B, and A _k, k 1, are linear bounded operators. We prove that if, for any > 0, the condition is satisfied, then the sequence |x _n} is bounded for arbitrary bounded sequences |y _n} and | _n } if and only if the operator has the continuous inverse for every z C, |z| 1.     

Schur rings and cyclic association schemes of class three

LetG be a group of finite order andD ₀ = {e},D ₁,...,D _d be a partition ofG. Suppose{d ^–1|d D _i } =D _{i, i {0, 1,..., d}}, for eachi {0, 1,..., d}; and for alli, j where . Then the subalgebra spanned by is called a Schur ring overG. It is known that such a partitionD ₀,D ₁,...,D _d can be used to construct an association scheme of classd. In this paper, we obtain a complete classification for the case whenG is cyclic andd = 3. The result corresponds to a complete classification of cyclic association schemes of class three.     

Fourier coefficients of functions from the classesb andc. Parseval equality for the classc or for the Fourier-Stieltjes series

We study restrictions that should be imposed on the numbers sequences {_n} and {_n} in order to guarantee that the series cosnx and sinnx do not belong to the classesB orC for any {a _n } and {b _n } such thata _{n n} ,b _{n n} ,n=1, 2,.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1455–1460, October, 1993.     

Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

For a sequence of constants {a _n,n1}, an array of rowwise independent and stochastically dominated random elements { V _nj, j1, n1} in a real separable Rademacher type p (1p2) Banach space, and a sequence of positive integer-valued random variables {T _n, n1}, a general weak law of large numbers of the form is established where {c _nj, j1, n1}, _n , b _n are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {V _nj, j1, n1}. Illustrative examples include one wherein the strong law of large numbers fails.     

A limit theorem for renewal sequences with an application to local time

Summary Let 0<h() and {S _n} a renewal process. We find conditions under which ast wherem=ES ₁, (t) = min (nS _n>t}. We apply these results to obtain sample path representation of local time at a point for a Markov process.Research supported in part by a grant from the National Science Foundation, USA     

Bounded universal functions for sequences of holomorphic self-maps of the disk

We give several characterizations of those sequences of holomorphic self-maps {φ _n } _n≥1 of the unit disk for which there exists a function F in the unit ball of H ^∞ such that the orbit {F∘φ _n :n∈ℕ} is locally uniformly dense in . Such a function F is said to be a -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ _n . As a consequence we will see that if φ _n is the nth iterate of a map φ of into , then {φ _n } _n≥1 admits a -universal function if and only if φ is a parabolic or hyperbolic automorphism of . We show that whenever there exists a -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.     

Théorème de division avec des conditions au bord

Let be an open subset of ⁿ and be a subalgebra of the algebra of analytic functions on . We suppose that satisfies some weak conditions of noetherianity such that we can construct a finite stratification for each ideal of . We also suppose that satifies global £ojasiewicz's inequalities. We prove the following: Let andf C on flat on ; if for eacha the Taylor's serie off ata, T _a f, is in the ideal generated byT _a f ₁,...,T _a f _p in the ring of formal power series, then there exist ₁,..., _p ,C on flat on such that . This result extends the classic Hormander's theorem of division (for a polynomial) or the £ojasiewicz-Malgrange theorem in the local analytic case.Reherches menées dans le cadre du Programme d'Appui à la Recherche Scientifique (PARS MI 33)     

A remark on almost sure convergence of weighted sums

Summary As a generalization of a theorem of Chow [1] it is shown by an elementary method that for i.i.d. r.v.'s X ₁,...,X _n, with expectation zero and finite p-th absolute moment (p2) the weighted sums converge to zero a.s.     

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.     

On Ozeki's inequality for power sums

Let p (0, 1) be a real number and let n 2 be an even integer. We determine the largest value c _n(p) such that the inequality

A Symmetric Function Resolution of the Number of Permutations With Respect to Block-Stable Elements

We consider a question raised by Suhov and Voice from quantum information theory and quantum computing. An element of a partition of {1, ..., n} is said to be block-stable for if it is not moved to another block under the action of π. The problem concerns the determination of the generating series for elements of with respect to the number of block-stable elements of a canonical partition of a finite n-set, with block sizes k₁, ..., k_r, in terms of the moment (power) sums p_q(k₁, ..., k_r). We also consider the limit subject to the condition that exists for q = 1, 2,.... Received January 31, 2006     

Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Let X _n1 ^* , ... X _nn ^* be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M _n ^* = \max\{X _n1 ^* , ... X _nn ^* }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M _n ^* and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let _n, _n N_n, , and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case _n , _n/_n > 0, as n .     

On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesáro (C, ) means ( = (₁,...,_n),_i , ₁ > –1, ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them -convergence (1) of (C, ) means (₁ > 0, ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them -convergence cannot be replaced by ordinary convergence.     

Asymptotic approximation of crossing probabilities of random sequences

Summary Let {X _i , i1} be a random sequence and {u _ni ,1in, n1} be an array of boundary values. We consider the asymptotic approximation of the probability P _n =P{X _i u _ni ,1in} by . We give sufficient conditions on X _i such that P _n–P _n ^* 0 as n. This generalizes the situation considered in extreme-value theory where the boundary is constant in i. The general theory is applied in particular to Gaussian cases.     

Some propositions related to a dilation theorem of W. Benz

Summary This paper begins with another proof of a theorem of W. Benz [2] concerning dilations in normed linear spaces. Our proof motivates several questions which are addressed thereafter. For instance it is shown that, ifI is an open interval in ,: I ⁿ , is continuously differentiable and there exista ₁,...,a _n I such that {(a ₁,...,(a _n )} is linearly independent, then {(t): t I} contains a Hamel basis for ⁿ over .     

A sharper stability bound of Fourier frames

Given a real sequence {_n}_n. Suppose that is a frame for L²[–, ] with bounds A, B. The problem is to find a positive constant L such that for any real sequence {_n}_n with ¦_n –_n¦ <L, is also a frame for L²[–, ]. Balan [1] obtained arcsin . This value is a good stability bound of Fourier frames because it covers Kadec's 1/4-theorem and is better than (see Duffin and Schaefer [3]). In this paper, a sharper estimate is given.     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .     

Wold decomposition,prediction and parameterization of stationary processes with infinite variance

Summary A discrete time stochastic process {_t} is said to be a p-stationary process (1<p2)if , for all integers n1, t ₁,...t _n,h and scalars b ₁,...b _n.The class of p-stationary processes includes the class of second-order weakly stationary stochastic processes, harmonizable stable processes of order (1<2), and p ^thorder strictly stationary processes. For any nondeterministic process in this class a finite Wold decomposition (moving average representation) and a finite predictive decomposition (autoregressive representation) are given without alluding to any notion of covariance or spectrum. These decompositions produce two unique (interrelated) sequences of scalar which are used as parameters of the process {_t}. It is shown that the finite Wold and predictive decomposition are all that one needs in developing a Kolmogorov-Wiener type prediction theory for such processes.     

On outstanding values in a sequence of random variables

Summary A sequence {X _n} of independent and identically distributed (i.i.d.) random variables is considered. Outstanding values in the sequence are those that strictly exceed values preceding them. Let L _n be the index of the n-th outstanding value. Limit theorems are given for the sequences and {L _n} and { _n} where _n=L_n–L_n–1. A characterization of the exponential distribution in terms of the sequence is also given.     

On One Counterexample in Convex Approximation

We prove the existence of a function fcontinuous and convex on [–1, 1] and such that, for any sequence {p _n} _{n= 1} of algebraic polynomials p _nof degree nconvex on [–1, 1], the following relation is true: , where ₄(t, f) is the fourth modulus of continuity of the function fand . We generalize this result to q-convex functions.