• Communications of the Korean Mathematical Society
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• v.14 no.3
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• pp.581-595
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• 1999

In the present paper we provide a short of the uni-form CLT for the function-indexed martingale difference process under the uniformly integrable entropy by establishing a maximal inequality.

• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.39-51
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• 2010

In this paper we consider the uniform central limit theorem for a martingale-difference array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-difference arrays of process indexed by a class of measurable functions by a method as Ziegler [19] did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-difference and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler [19] and other results of independent problems. The results also generalizes those of Bae and Choi [3] to martingale-difference array of a function-indexed stochastic process. Finally, an application to classes of functions changing with n is given.

• Journal of the Korean Data and Information Science Society
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• v.28 no.6
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• pp.1565-1571
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• 2017

The standard logistic map is an iterative function, which forms a discrete-time dynamic system. The chaotic logistic map is a kind of ergodic map defined over the unit interval. In this paper we study the limiting behaviors on the several processes induced by the chaotic logistic map. We derive the law of large numbers for the process induced by the chaotic logistic map. We also derive the uniform law of large numbers for this process. When deriving the uniform law of large numbers, we study the role of bracketing of the indexed class of functions associated with the process. Then we apply the idea of DeHardt (1971) associated with the bracketing method to the process induced by the logistic map. We finally illustrate an application to Monte Carlo integration.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.221-231
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• 1995

Let $Z_1, Z_2, \ldots, Z_l$ be random set functions or intergrals. Then it is possible to discuss their products. In the case of random integrals, $Z_i$ is a random set function indexed y a family, $G_i$ say, of real valued functions g on $S_i$ for which the integrals $Z_i(g) = \smallint gdZ_i$ are well defined. If $g_i = \in g_i (i = 1, 2, \ldots, l) and g_1 \otimes \cdots \otimes g_l$ denotes the tensor product $g(s) = g_1(s_1)g_2(s_2) \cdots g_l(s_l) for s = (s_1, s_2, \ldots, s_l) and s_i \in S_i$, then we can defined $Z(g) = (Z_1 \times Z_2 \times \cdots \times Z_l)(g) = Z_1(g_1)Z_2(g_2) \cdots Z_l(g_l)$.

• Journal of the Korean Institute of Telematics and Electronics C
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• v.34C no.11
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• pp.95-103
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• 1997

A robust center estimation tecnique of n-fold engineering parts is presented, which use self-organizing neural networks with generating and merging learning for training neural units. To estimate the center of the n-fold engineering parts using neural networks, the segmented boundaries of the interested part are approximated to strainght lines, and the temporal estimated centers by thecosine theorem which formed between the approximaged straight line and the reference point, , are indexed as (.sigma.-.theta.) parameteric vecstors. Then the entries of parametric vectors are fed into self-organizing nerual network. Finally, the center of the n-fold part is extracted by mean of generating and merging learning of the neurons. To accelerate the learning process, neural network uses an adaptive learning rate function to the merging process and a self-adjusting activation to generating process. Simulation results show that the centers of n-fold engineering parts are effectively estimated by proposed technique, though not knowing the error distribution of estimated centers and having less information of boundaries.

• Journal of Korea Multimedia Society
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• v.10 no.10
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• pp.1328-1337
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• 2007

We describe a method for streaming specific section using the section information among peers on P2P network. As peers tend to extract only the part of contents in which they are interested, an indexed file is created based on the section information of multimedia contents and registered in the search server. If a peer requests the streaming of specific parts of the contents to other corresponding peer using the section information that the server returns, the peer streams only separated section of the multimedia contents according to the information. For this process, we implement a function that a peer's PC can play a role as a server, and the peer streams the separated content based on the section information that the server supplies.