On spaces of weak* to weak continuous compact operators

김주명 대한수학회 2013 대한수학회보 Vol.50 No.1

This paper is concerned with the space Kw * (X*, Y ) of weak* to weak continuous compact operators from the dual space X* of a Banach space X to a Banach space Y . We show that if X* or Y* has the Radon-Nikod´ym property, C is a convex subset of Kw* (X*, Y ) with 0 ∈C and T is a bounded linear operator from X* into Y , then T ∈ C τc if and only if [수식] , where τc is the topology of uniform convergence on each compact subset of X, moreover, if T 2 Kw* (X*, Y ), here C need not to contain 0, then T ∈ Cτc if and only if T ∈ C in the topology of the operator norm. Some properties of Kw* (X* , Y ) are presented.

ON SPACES OF WEAK^* TO WEAK CONTINUOUS COMPACT OPERATORS

Kim, Ju Myung Korean Mathematical Society 2013 대한수학회보 Vol.50 No.1

This paper is concerned with the space $\mathcal{K}_{w^*}(X^*,Y)$ of $weak^*$ to weak continuous compact operators from the dual space $X^*$ of a Banach space X to a Banach space Y. We show that if $X^*$ or $Y^*$ has the Radon-Nikod$\acute{y}$m property, $\mathcal{C}$ is a convex subset of $\mathcal{K}_{w^*}(X^*,Y)$ with $0{\in}\mathcal{C}$ and T is a bounded linear operator from $X^*$ into Y, then $T{\in}\bar{\mathcal{C}}^{{\tau}_{\mathcal{c}}}$ if and only if $T{\in}\bar{\{S{\in}\mathcal{C}:{\parallel}S{\parallel}{\leq}{\parallel}T{\parallel}\}}^{{\tau}_{\mathcal{c}}}$, where ${\tau}_{\mathcal{c}}$ is the topology of uniform convergence on each compact subset of X, moreover, if $T{\in}\mathcal{K}_{w^*}(X^*, Y)$, here $\mathcal{C}$ need not to contain 0, then $T{\in}\bar{\mathcal{C}}^{{\tau}_{\mathcal{c}}}$ if and only if $T{\in}\bar{\mathcal{C}}$ in the topology of the operator norm. Some properties of $\mathcal{K}_{w^*}(X^*,Y)$ are presented.

BOUNDED CONVERGENCE THEOREMS

Niemiec, Piotr Korean Mathematical Society 2017 대한수학회지 Vol.54 No.1

There are presented certain results on extending continuous linear operators defined on spaces of E-valued continuous functions (defined on a compact Hausdorff space X) to linear operators defined on spaces of E-valued measurable functions in a way such that uniformly bounded sequences of functions that converge pointwise in the weak (or norm) topology of E are sent to sequences that converge in the weak, norm or weak* topology of the target space. As an application, a new description of uniform closures of convex subsets of C(X, E) is given. Also new and strong results on integral representations of continuous linear operators defined on C(X, E) are presented. A new classes of vector measures are introduced and various bounded convergence theorems for them are proved.

BOUNDED CONVERGENCE THEOREMS

Piotr Niemiec 대한수학회 2017 대한수학회지 Vol.54 No.1

There are presented certain results on extending continuous linear operators defined on spaces of $E$-valued continuous functions (defined on a compact Hausdorff space $X$) to linear operators defined on spaces of $E$-valued measurable functions in a way such that uniformly bounded sequences of functions that converge pointwise in the weak (or norm) topology of $E$ are sent to sequences that converge in the weak, norm or weak* topology of the target space. As an application, a new description of uniform closures of convex subsets of $C(X,E)$ is given. Also new and strong results on integral representations of continuous linear operators defined on $C(X,E)$ are presented. A new classes of vector measures are introduced and various bounded convergence theorems for them are proved.

CONTINUATION THEOREMS OF THE EXTREMES UNDER POWER NORMALIZATION

Barakat, H.M.,Nigm, E.M.,El-Adll, M.E. 한국전산응용수학회 2002 Journal of applied mathematics & informatics Vol.10 No.1

In this paper an important stability property of the extremes under power normalizations is discussed. It is proved that the restricted convergence of the Power normalized extremes on an arbitrary nondegenerate interval implies the weak convergence. Moreover, this implication, in an important practical situation, is obtained when the sample size is considered as a random variable distributed geometrically with mean n.

Weak convergence for stationary bootstrap empirical processes of associated sequences

황은주 대한수학회 2021 대한수학회지 Vol.58 No.1

In this work the stationary bootstrap of Politis and Romano \cite{PR1994a} is applied to the empirical distribution function of stationary and associated random variables. A weak convergence theorem for the stationary bootstrap empirical processes of associated sequences is established with its limiting to a Gaussian process almost surely, conditionally on the stationary observations. The weak convergence result is proved by means of a random central limit theorem on geometrically distributed random block size of the stationary bootstrap procedure. As its statistical applications, stationary bootstrap quantiles and stationary bootstrap mean residual life process are discussed. Our results extend the existing ones of Peligrad \cite{P1998} who dealt with the weak convergence of non-random blockwise empirical processes of associated sequences as well as of Shao and Yu \cite{SY1996} who obtained the weak convergence of the mean residual life process in reliability theory as an application of the association.

On Weak Convergence of Some Rescaled Transition Probabilities of a Higher Order Stationary Markov Chain

Yun, Seok-Hoon The Korean Statistical Society 1996 Journal of the Korean Statistical Society Vol.25 No.3

In this paper we consider weak convergence of some rescaled transi-tion probabilities of a real-valued, k-th order (k $\geq$ 1) stationary Markov chain. Under the assumption that the joint distribution of K + 1 consecutive variables belongs to the domain of attraction of a multivariate extreme value distribution, the paper gives a sufficient condition for the weak convergence and characterizes the limiting distribution via the multivariate extreme value distribution.

WEAK CONVERGENCE OF MANN ITERATIVE SEQUENCE FOR NONEXPANSIVE MAPPINGS IN PROBABILISTIC HILBERT SPACES

Su, Yongfu,Wang, Xiuzhen,Gao, Junyu The Youngnam Mathematical Society Korea 2008 East Asian mathematical journal Vol.24 No.1

The purpose of this paper is to establish the weak convergence theorem of Mann iterative sequence for nonexpansive mappings in probabilistic Hilbert spaces. In order to establish the weak convergence theorem, a new method was presented in this paper, that is method of mathematical expectation.

CONVERGENCE PROPERTIES OF THE PARTIAL SUMS FOR SEQUENCES OF END RANDOM VARIABLES

Wu, Yongfeng,Guan, Mei Korean Mathematical Society 2012 대한수학회지 Vol.49 No.6

The convergence properties of extended negatively dependent sequences under some conditions of uniform integrability are studied. Some sufficient conditions of the weak law of large numbers, the $p$-mean convergence and the complete convergence for extended negatively dependent sequences are obtained, which extend and enrich the known results in the literature.

Convergence rates of spectral distributions of large dimensional quaternion sample covariance matrices

Huiqin Li,Zhidong Bai 한국통계학회 2015 Journal of the Korean Statistical Society Vol.44 No.1

In this paper, we study the convergence rates of empirical spectral distributions of largedimensional quaternion sample covariance matrices. Assume that the entries of Xn (p×n)are independent quaternion random variables with means zero, variances 1 and uniformlybounded sixth moments. Denote Sn = 1nXnX∗n. Using Bai’s inequality, we prove that theexpected empirical spectral distribution (ESD) converges to the limiting Marčenko–Pasturdistribution with the ratio of dimension to sample size yp = p/n at a rate of O n−1/2a−3/4n  when an > n−2/5 or O n−1/5when an ≤ n−2/5, where an = (1 − √yp)2. Moreover, therates for both the convergence in probability and the almost sure convergence are alsoestablished. The weak convergence rate of the ESD is O n−2/5a−1/2n when an > n−2/5 orO n−1/5when an ≤ n−2/5. The strong convergence rate of the ESD is O n−2/5+ηa−1/2n  when an > κn−2/5 or O n−1/5when an ≤ κn−2/5 for any η > 0 where κ is a positiveconstant.