Selection principle for pointwise bounded sequences of functions

For a number ? > 0 and a real function f on an interval [a, b], denote by N(?, f, [a, b]) the least upper bound of the set of indices n for which there is a family of disjoint intervals [a _i , b _i ], i = 1, …, n, on [a, b] such that |f(a _i ) ? f(b _i )| > ? for any i = 1, …, n (sup Ø = 0). The following theorem is proved: if {f _j } is a pointwise bounded sequence of real functions on the interval [a, b] such that n(?) ≡ lim sup _j→∞ N(?, f _j , [a, b]) < ∞ for any ? > 0, then the sequence {f _j } contains a subsequence which converges, everywhere on [a, b], to some function f such that N(?, f, [a, b]) ≤ n(?) for any ? > 0. It is proved that the main condition in this theorem related to the upper limit is necessary for any uniformly convergent sequence {f _j } and is “almost” necessary for any everywhere convergent sequence of measurable functions, and many pointwise selection principles generalizing Helly’s classical theorem are consequences of our theorem. Examples are presented which illustrate the sharpness of the theorem.     

Approximation by antiderivatives

Let Ω ?C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ ^(?n)} ofn-fold antiderivatives (i.e., we haveφ ⁽⁰⁾(z)∶=φ(z) andφ ^(?n)(z)=dφ ^(?n?1)(z)/dz for alln ∈ N₀ and z ∈ Ω) such that the following properties hold:

1. For any compact setB ?Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n _k} such that {φ^?nk} converges tof uniformly onB.
2. For any open setU ?Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m _k} such that {φ^?mk} converges tof compactly onU.
3. For any measurable setE ?Ω and any functionf that is measurable onE, there exists a sequence {p _k} such that {φ ^(-Pk)} converges tof almost everywhere onE.

     

Almost continuity on generalized topological spaces

For any unbounded sequence {n _k } of positive real numbers, there exists a permutation {n _σ(k)} such that the discrepancies of {n _σ(k) x} obey the law of the iterated logarithm exactly in the same way as the uniform i.i.d. sequence {U _k }.     

Some limit theorems for negatively associated random variables

Let {X _n , n ≥ 1} be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the L ^p -convergence theorem and Marcinkiewicz–Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables.     

On a property related to convergence in probability and some applications to branching processes

It is shown that any real-valued sequence of random variables {X_n} converging in probability to a non-degenerate, not necessarily a.s. finite limit X possesses the following property: for any c with P(X? (c ? δ, c + δ)) > 0 for all δ > 0, there exists a sequence {c_n} with lim_n→∞ c_n = c such that for any ε > 0, lim_n→∞ P(Xδ (c ? ε, c + ε) |X_n = c_n) = 1. This property is applied to various types of branching processes where $\text{X}{}_{\mathrm{n}}\text{=}\text{Z}{}_{\mathrm{n}}\text{C}{}_{\mathrm{n}}$ or $\text{X}{}_{\mathrm{n}}\text{=}\text{U(Z}{}_{\mathrm{n}}\text{)}\text{C}{}_{\mathrm{n}}\text{{C}{}_{\mathrm{n}}\text{}}$ being a sequence of constants or random variables and U a slowly varying function. If {Z_n} is a supercritical branching process in varying or random environment, X is shown to have a continuous and strictly increasing distribution function on (0, ∞). Characterizations of the tail of the liniting distribution of the finite mean and the infinite mean supercritical Galton-Watson processes are also obtained.     

Long-range dependence of stationary processes in single-server queues

The stationary processes of waiting times {W _n } _{n = 1,2,…} in a GI/G/1 queue and queue sizes at successive departure epochs {Q _n}_{n = 1,2,…} in an M/G/1 queue are long-range dependent when 3 < κ _S < 4, where κ _S is the moment index of the independent identically distributed (i.i.d.) sequence of service times. When the tail of the service time is regularly varying at infinity the stationary long-range dependent process {W _n } has Hurst index ½(5?κ _S ), i.e.

${\rm sup} \left\{h : {\rm lim sup}_{n\to\infty}\, \frac{{\rm var}(W_1+\cdots+W_n)}{n^{2h}} = \infty \right\} = \frac{5-\kappa_S} {2}\,.$

If this assumption does not hold but the sequence of serial correlation coefficients {ρ _n } of the stationary process {W _n } behaves asymptotically as cn ^?α for some finite positive c and α ? (0,1), where α = κ _S ? 3, then {W _n } has Hurst index ½(5?κ _S ). If this condition also holds for the sequence of serial correlation coefficients {r _n } of the stationary process {Q _n } then it also has Hurst index ½(5κ _S )

     

Functions generating normal Toeplitz matrices

To any complex function there corresponds a Fourier series, which is often associated with a sequence {T _n} of Toeplitz n × n matrices. Functions whose Fourier series generate sequences of normal Toeplitz matrices are classified, and a procedure for constructing Fourier series for which the sequence {T _n} contains an infinite subsequence of normal matrices is described.     

A hereditarily indecomposable Banach space with rich spreading model structure

We present a reflexive Banach space \(\mathfrak{X}_{usm}\) which is Hereditarily Indecomposable and satisfies the following properties. In every subspace Y of \(\mathfrak{X}_{usm}\) there exists a weakly null normalized sequence {y _n } _n , such that every subsymmetric sequence {z _n } _n is isomorphically generated as a spreading model of a subsequence of {y _n } _n . Also, in every block subspace Y of \(\mathfrak{X}_{usm}\) there exists a seminormalized block sequence {z _n } and \(T:\mathfrak{X}_{usm} \to \mathfrak{X}_{usm}\) an isomorphism such that for every n ∈ ?, T(z _2n?1) = z _2n . Thus the space is an example of an HI space which is not tight by range in a strong sense.     

On the connection between weak and strong convergencies in Cm

In this work wome connections are pursued between weak and strong convergence in the spaces C^m (m-times continuously differentiable functions on Rⁿ). Let f_n, f?C^{m + 1}, where n = 1, 2,…, and m is a nonnegative integer. Suppose that the sequence {f_n} converges to f relative to the weak topology of C^{m + 1}. It is shown that this implies the convergence of {f_n} to f with respect to the strong topology of C^m. Several corollaries to this theorem are established; among them is a sufficient condition for uniform convergence. A stronger result is shown to exist when the sequence constitutes an output sequence of a linear weakly continuous operator.     

Remark on a theorem of Aharonov and Walsh

The following theorem is proved: there is a functionf(z) analytic in |z|<1 and having the natural boundary |z|=1 such that for an infinite sequence of rational functions of degreen, r _n(z)=P_n(z)/q_n(z), the inequality 1 $$\left| {f(z) - r_n (z)} \right|< \varepsilon _n $$ holds in the closed unit circle |z|≦1. Here? ₁,? ₂,...,? _n is any sequence of positive numbers, tending to zero asn approaches infinity. This theorem is a refinement of a theorem of Aharonov and Walsh, who showed the existence of anf(z) satisfying (*) in |z|≦1 (with an infinite sequence {r _n(z)}) but having the natural boundary |z|=3.     

On Müntz rational approximation

One of the main results of the present paper shows that, for any sequence of real numbers {λ_n} with infinitely many distinct elements, the monotone rational combinations of {X^λn} always form a dense set in the uniform norm in the subspace of monotone functions fromC _{[0, 1]}.     

Limit Theory for Moderate Deviations from a Unit Root Under Innovations with a Possibly Infinite Variance

An asymptotic theory was given by Phillips and Magdalinos (J Econom 136(1):115–130, 2007) for autoregressive time series Y _t ?=?ρY _t?1?+?u _t , t?=?1,...,n, with ρ?=?ρ _n ?=?1?+?c/k _n , under (2?+?δ)-order moment condition for the innovations u _t , where δ?>?0 when c?<?0 and δ?=?0 when c?>?0, {u _t } is a sequence of independent and identically distributed random variables, and (k _n ) _n?∈?? is a deterministic sequence increasing to infinity at a rate slower than n. In the present paper, we established similar results when the truncated second moment of the innovations $l(x)=\textsf{E} [u_1^2I\{|u_1|\le x\}]$ is a slowly varying function at ∞, which may tend to infinity as x?→?∞. More interestingly, we proposed a new pivotal for the coefficient ρ in case c?<?0, and formally proved that it has an asymptotically standard normal distribution and is nuisance parameter free. Our numerical simulation results show that the distribution of this pivotal approximates the standard normal distribution well under normal innovations.     

Systems of functions which are complete in measure

In this note it is proved that if a complete orthonormal system {? _n} in L₂[0, 1] contains a subsystem {? _nk} of a lacunary order p>2, then for some bounded measurable function h(x) the system {h(x)? _n(x)}_n≠_nk is complete in L₂[0, 1].     

Perturbed sampling formulas and local reconstruction in shift invariant spaces

Let V? be a closed subspace of L₂(R) generated from the integer shifts of a continuous function ? with a certain decay at infinity and a non-vanishing property for the function Φ^†(γ)=_∑n∈Z?(n)e−inγ on [−π,π]. In this paper we determine a positive number δ? so that the set {n+δn}_n∈Z is a set of stable sampling for the space V? for any selection of the elements δn within the ranges ±δ?. We demonstrate the resulting sampling formula (called perturbation formula) for functions f∈V? and also we establish a finite reconstruction formula approximating f on bounded intervals. We compute the corresponding error and we provide estimates for the jitter error as well.     

Universal meromorphic approximation on Vitushkin sets

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ? ?, there exists a function ?, meromorphic on ?, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ? such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ℕ such that converges to f(z) uniformly on K. A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λ _n }={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3]. Original Russian Text ? W.Luh, T.Meyrath, M.Niess, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 6, pp. 66–75.     

The Shannon-McMillan-Breiman theorem for a class of amenable groups

We prove the SMB theorem for amenable groups that possess Følner sets {A _n } with the property that for some constantM, and all,n, |A _n ^?1 A _n | ≦M· |A _n |.     

Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space

Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}_n?0⊂[1,+∞), limn→∞kn=1 such that F(T)≠∅. Let {αn}_n?0⊂[0,1] be such that _∑n?0αn=∞, and _∑n?0αn(kn−1)<∞. Suppose {xn}_n?0 is iteratively defined by xn+1=(1−αn)xn+αnTnxn, n?0, and suppose there exists a strictly increasing continuous function , ?(0)=0 such that 〈Tnx−x^∗,j(x−x^∗)〉?kn‖x−x^∗‖2−?(‖x−x^∗‖), ∀x∈K. It is proved that {xn}_n?0 converges strongly to x^∗∈F(T). It is also proved that the sequence of iteration {xn} defined by xn+1=anxn+bnTnxn+cnun, n?0 (where {un}_n?0 is a bounded sequence in K and {an}_n?0, {bn}_n?0, {cn}_n?0 are sequences in [0,1] satisfying appropriate conditions) converges strongly to a fixed point of T.     

A Characterization of L-orthogonal Polynomials from?Three Term Recurrence Relations

We consider the sequence of polynomials {Q _n } satisfying the L-orthogonality ?[z ^?n+m Q _n (z)]=0, 0??m??n?1, with respect to a linear functional ? for which the moments ?[t ⁿ ]=?? _n are all complex. Under certain restriction on the moment functional these polynomials also satisfy a three term recurrence relation. We consider three special classes of such moment functionals and characterize them in terms of the coefficients of the associated three term recurrence relations. Relations between the polynomials {Q _n } associated with two of these special classes of moment functionals are also given. Examples are provided to justify this characterization.     

Normal family and the sequence of omitted functions

Let {fn} be a sequence of meromorphic functions on a plane domain D, whose zeros and poles have multiplicity at least 3. Let {hn} be a sequence of meromorphic functions on D, whose poles are multiple, such that {hn} converges locally uniformly in the spherical metric to a function h which is meromorphic and zero-free on D.If fn≠hn, then {fn} is normal on D.     

Distribution of roots of quasianalytic functions

For functions of certain quasianalytic classes C{m_n} on (?∞, ∞) we determine a function ξ (x), depending on {m_n}, which is such that a sequence {x_k} is a sequence of the roots off(x) ε C{m_n} if and only if for somea $$\int_a^\infty {\tfrac{{dn(x)}}{{\xi (x - a}}< \infty ,} $$ where n(x) is a distribution function of the sequence {x_k}.