• Journal of the Korean Mathematical Society
• /
• v.32 no.4
• /
• pp.679-688
• /
• 1995

Let $(B, \left\$\mid$ \right\$\mid$)$ be a real separable Banach space. Let $(\Omega, F, P)$ denote a probability space. A random elements in B is a function from $\Omega$ into B which is $F$-measurable with respect to the Borel $\sigma$-field $B$(B) in B.

• Journal of the Korean Statistical Society
• /
• v.22 no.1
• /
• pp.83-91
• /
• 1993

Random elements in $L^1(R)$ and some properties of $L^1(R)$ space are investigated with application to kernel density estimators. A weak law of large numbers for compact uniformly integrable random elements is introduced for further application.

• Journal of the Korean Mathematical Society
• /
• v.44 no.2
• /
• pp.467-476
• /
• 2007

A complete convergence theorem for arrays of rowwise independent random variables was proved by Sung, Volodin, and Hu [14]. In this paper, we extend this theorem to the Banach space without any geometric assumptions on the underlying Banach space. Our theorem also improves some known results from the literature.

• Journal of the Korean Statistical Society
• /
• v.23 no.1
• /
• pp.39-51
• /
• 1994

Some exponential moment inequalities for sub-gaussian random variables are studied in this paper. These inequalities are used to obtain laws of large numbers for random variable and random elements in $L^1(R)$.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.2
• /
• pp.369-383
• /
• 2010

We obtain a result on complete convergence of weighted sums for arrays of rowwise independent Banach space valued random elements. No assumptions are given on the geometry of the underlying Banach space. The result generalizes the main results of Ahmed et al. [1], Chen et al. [2], and Volodin et al. [14].

• Journal of the Korean Statistical Society
• /
• v.26 no.2
• /
• pp.231-243
• /
• 1997

In this paper we investigate weak convergence of the intergral processes whose index set is the non-compact infinite time interval. Our first goal is to develop the empirical central limit theorem as random elements of [0, .infty.) for an integral process which is constructed from iid variables. In developing the weak convergence as random elements of D[0, .infty.), we will use a result of Ossiander(4) whose proof heavily depends on the total boundedness of the index set. Our next goal is to establish the empirical central limit theorem for the Kaplan-Meier integral process as random elements of D[0, .infty.). In achieving the the goal, we will use the above iid result, a representation of State(6) on the Kaplan-Meier integral, and a lemma on the uniform order of convergence. The first result, in some sense, generalizes the result of empirical central limit therem of Pollard(5) where the process is regarded as random elements of D[-.infty., .infty.] and the sample paths of limiting Gaussian process may jump. The second result generalizes the first result to random censorship model. The later also generalizes one dimensional central limit theorem of Stute(6) to a process version. These results may be used in the nonparametric statistical inference.

• The Journal of the Korea Contents Association
• /
• v.6 no.5
• /
• pp.29-34
• /
• 2006

For the almost certainly convergent series $S_n=\sum_{i=1}^nV-i$ of independent random elements in Banach spaces, by investigating tail series laws of large numbers, the rate of convergence of the series $S_n$ to a random variable s is studied in this paper. More specifically, by studying the duality between the limiting behavior of the tail series $T_n=S-S_{n-1}=\sum_{i=n}^{\infty}V-i$ of random variables and that of Banach space valued random elements, an alternative way of proving a result of the previous work, which establishes the equivalence between the tail series weak law of large numbers and a limit law, is provided in a Banach space setting.

• Journal of the Korean Mathematical Society
• /
• v.49 no.5
• /
• pp.1053-1064
• /
• 2012

For a double array of random elements $\{X_{mn};m{\geq}1,n{\geq}1\}$ in a $p$-uniformly smooth Banach space, $\{b_{mn};m{\geq}1,n{\geq}1\}$ is an array of positive numbers, convergence of double random series ${\sum}^{\infty}_{m=1}{\sum}^{\infty}_{n=1}X_{mn}$, ${\sum}^{\infty}_{m=1}{\sum}^{\infty}_{n=1}b^{-1}_{mn}X_{mn}$ and strong law of large numbers $$b^{-1}_{mn}\sum^m_{i=1}\sum^n_{j=1}X_{ij}{\rightarrow}0$$ as $$m{\wedge}n{\rightarrow}{\infty}$$ are established.

• Journal of the Korean Institute of Telematics and Electronics C
• /
• v.34C no.8
• /
• pp.79-87
• /
• 1997

For the representation of a complex object, the object is decomposed into several parts, and it is described by these decomposed parts and their relations. In genral, the parts can be the primitive elements that can not be decomposed further, or can be decomposed into their subparts. Therefore, the hierarchical description method is very natural and it si represented by a hierarchical attributed graph whose vertieces represent either primitive elements or graphs. This graphs also have verties which contain primitive elements or graphs. When some uncertainty exists in the hierarchical description of a complex object either due to noise or minor deformation, a probabilistic description of the object ensemble is necessary. For this purpose, in this paper, we formally define the hierarchical attributed random graph which is extention of the hierarchical random graph, and erive the equations for the entropy calculation of the hierarchical attributed random graph, and derive the equations for the entropy calculation of the hierarchical attributed random graph. Finally, we propose the application areas to use these concepts.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.5
• /
• pp.1199-1210
• /
• 2019

In this paper, we consider a class of inhomogeneous random intersection graphs by assigning random weight to each vertex and two vertices are adjacent if they choose some common elements. In the inhomogeneous random intersection graph model, vertices with larger weights are more likely to acquire many elements. We show the Poisson convergence of the number of induced copies of a fixed subgraph as the number of vertices n and the number of elements m, scaling as $m={\lfloor}{\beta}n^{\alpha}{\rfloor}$ (${\alpha},{\beta}>0$), tend to infinity.