Note on Strong Law of Large Number under Sub-linear Expectation

황교신 영남수학회 2020 East Asian mathematical journal Vol.36 No.1

The classical limit theorems like strong law of large numbers, central limit theorems and law of iterated logarithms are fundamental the- ories in probability and statistics. These limit theorems are proved under additivity of probabilities and expectations. In this paper, we investigate strong law of large numbers under sub-linear expectation which generalize the classical ones. We give strong law of large numbers under sub-linear expectation with respect to the partial sums and some conditions sim- ilar to Petrov’s. It is an extension of the classical Chung type strong law of large numbers of Jardas et al.’s result. As an application, we ob- tain Chung’s strong law of large number and Marcinkiewicz’s strong law of large number for independent and identically distributed random vari- ables under the sub-linear expectation. Here the sub-linear expectation and its related capacity are not additive.

A supplement to precise asymptotics in the law of the iterated logarithm for self-normalized sums

황교신 대한수학회 2008 대한수학회지 Vol.45 No.6

Let X,X1,X2, . . . be i.i.d. random variables with zero means, variance one, and set [수식] Gut and Spataru [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If [수식] is replaced by [수식] in their results, the new one is called the moment version of precise asymptotics in the law of the iterated logarithm. We establish such a result for self-normalized sums, when X belongs to the domain of attraction of the normal law. Let X,X1,X2, . . . be i.i.d. random variables with zero means, variance one, and set [수식] Gut and Spataru [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If [수식] is replaced by [수식] in their results, the new one is called the moment version of precise asymptotics in the law of the iterated logarithm. We establish such a result for self-normalized sums, when X belongs to the domain of attraction of the normal law.

Strassen's functional LIL for $d$-dimensional self-similar Gaussian process in H\"older norm

황교신,Zhengyan Lin 대한수학회 2005 대한수학회지 Vol.42 No.5

In this paper, based on largedeviation probabilities on Gaussian random vectors, we obtainStrassen's functional LIL for d-dimensional self-similarGaussian process in H"older norm via estimating large deviationprobabilities for d-dimensional self-similar Gaussian process in H\"older norm.

ALMOST SURE CONVERGENCE OF WEIGHTED SUMS FOR WIDELY NEGATIVE DEPENDENT RANDOM VARIABLES UNDER SUB-LINEAR EXPECTATIONS

황교신 장전수학회 2022 Advanced Studies in Contemporary Mathematics Vol.32 No.4

The sub-linear expectation space is a nonlinear expectation space having advantages of modeling the uncertainty of probability and statistics. In this paper we study the almost sure convergence for weighted sums of widely negative dependent random variables in the sub-linear expectation spaces. An almost sure convergence theorem is obtained for weighted sums of widely negative dependent random variables under sub-linear expectations. Our results extend and generalize the corresponding ones of Hu and Wu(2021) to widely negative dependent random variables under sub-linear expectation.

A note on functional limit theorem for the increments of FBM in sup-norm

황교신 영남수학회 2008 East Asian mathematical journal Vol.24 No.3

In this paper, using large deviation results for Gaussian processes, we establish some functional limit theorems for increments of a fractional Brownian motion in the usual sup-norm via estimating large deviation probabilities for increments of a fractional Brownian motion.

A NOTE ON STRONG CONVERGENCE OF SUMS OF INDEPENDENT RANDOM VARIABLES UNDER SUB-LINEAR EXPECTATION

황교신 장전수학회 2020 Advanced Studies in Contemporary Mathematics Vol.30 No.3

The sub-linear expectation space is a nonlinear expecta- tion space having advantages of modeling the uncertainty of probability and statistics. Strong convergence for non-additive probabilities or non- linear expectation are challenging issues which have raised progressive interest recently. In the sub-linear expectation space, we use capac- ity and sub-linear expectation to replace probability and expectation of classical probability theory. Recently Zhang has proved very important theorems in the sub-linear expectation space which can be looked upon as being extensions of Kolmogorov's three series theorem in classical probability theory. Zhang and Lin obtained Marcinkiewicz's strong law of large numbers for sums of independent random variables under sub- linear expectation space. In addition, they have given a theorem about strong convergence of a random series for independent random variables under sub-linear expectation. In this paper we investigate the strong convergence for sums of inde- pendent random variables under sub-linear expectation and gives also almost surely convergence of sums of independent random variables in capacity. They are extensions of strong convergence of a random series and almost surely convergence of sums for independent random variables in the framework of sub-linear expectation. By using Zhang's result, the proof rests on the methods of proof due to Petrov's result concerning the almost sure convergence of series of independent random variable. As an application Marcinkiewicz's strong law of large number for nonlinear expectation are obtained.

조선시대 전통직물에 나타난 용무늬의 조형적 특성

황진영(주저자) ( Jin Young Hwang ),강혜승(교신저자) ( Hae Seung Kang ) 커뮤니케이션디자인학회 2016 커뮤니케이션 디자인학연구 Vol.55 No.-

본 연구는 조선시대 전통직물에 나타난 용무늬를 시기별로 구분하여 변화되는 조형적 특성을 파악하는 것을 목적으로 한다. 이를 위해 조선전기, 중기, 후기로 1차 분류하고, 조형적 특성으로 실루엣과 구성, 형태의 세 가지를 2차 분류기준으로 설정하였다. 전통직물에 나타난 용무늬의 조형적 변화는 다음과 같다. 첫째, 실루엣은 꼬임이 적은 ∪형을 시작으로 조선후기의 꼬임이 많거나 두 마리의 용을 배치한 ○형의 쌍용문으로 변화되며 화려한 실루엣의 유행이 변화되는 과정을 확인 하였다. 둘째, 구성은 네 가지 중 용+화문, 용+구름이 전 시기에 나타났으며, 그 중 용+화문이 가장 많이 사용되었다. 셋째, 형태를 살펴보면 조선전기는 세부표현은 하지 않고 외형라인으로 표현하고 몸통의 비율은 4등신으로 머리보다 몸체를 더욱 두껍게 표현하였다. 조선중기는 초기의 특징과 함께 조금 더 자세한 표현을 한 양식으로 두 가지가 보여 진다. 몸통의 비율이6등신으로 생동감 넘치고, 몸통의 두께는 머리와 같은 비율로 나타내었다. 조선후기에는 몸통의 비율이 8등신으로 가장 길게 표현 하였으며 세부표현 또한 모두 자세히 나타내었다. 이 연구를 통해 전통복식관련 콘텐츠와 디자인분야에서 다방면으로 응용할 수 있는 기초자료가 되기를 기대한다. The objectives of this research is to categorize dragon patterns in traditional woven fabrics of Joseon dynasty by period, to understand the changing formative characteristics. To do this, categorization standards were set up, first to be categorized by early, middle and later periods of Joseon dynasty, and secondly by three formative characteristics - silhouette, structure and shape. The summary of the results are as follows. First, we were able to observe a change of trends, beginning with a u-shape with small number of contortions, to many contortions and o-shaped placement of two dragons for a double dragon pattern and decorative silhouettes in the later periods of Joseon dynasty. Secondly, out of the four structures, dragons+flowers and dragons+clouds were used in earlier periods, and out of those dragons and flowers were most often used. Thirdly, in the earlier periods of Joseon, detailed expressions were not used, only expressing with an outward form, and the body was as long as four heads and was drawn much thicker than the head. From the middle period of Joseon era, two styles were found that used more detailed expressions. The length of the body is as long as six heads, the image is very lively and the thickness of the body is the same as the head. In the later period of Joseon era, the length of the body is as long as eight heads, the longest of all periods, and the minor expressions are all very detailed. It is hoped that this research can become a basis resource that can be diversely utilized in traditional costume related contents and design fields.

定常增分을 갖는 가우스 過程

박순규,황교신 진주산업대학교 산업과학기술연구소 1996 산업과학기술연구소보 Vol.- No.3

Let {X(t):0≤t≤∞} be a almost surely continuous Gaussian Process with E(X(t)}=0 and stationary increments E{X(t)-X(s)}=σ^2(|t-s|)=|t-s|^2r, 0< r ≤1/2. Let a_T(0<T<∞) be a nondecreasing function of T with 0<a_T≤T and T/a_T nondecreasing. Suppose that lim(logT-loga_T)/log log T-r (o<r≤∞).In this paper we obtain liminf _(r→∞) sup_(0≤s≤a_r)sup_(0≤t≤r-d) |X(t+s)-X(t)|/βγα(αγ) =liminf_(r→∞) where β_r=(s{log(T/a_r)+log log T})^1/2 The above results are proved via using some moduli of continuity for Gaussian Processers.

Moduli of continuity for the increment of a two-parameter fractional Levy Brownian motion

Park, Soon-kyu,Hwang, Kyo-Shin 진주산업대학교 1999 산업과학기술연구소보 Vol.- No.6

We establish moduli of continuity for the increment of a two-parameter fractional LevyBrownian motion which is a general form of a two-parameter Levy Brownian motion via estimating bounds of large deviation probabilitices on suprema of the two-parameter fractional Levy Brownian motion.

가우스 과정의 증분에 관한 소고

박순규,황교신 진주산업대학교 1997 산업과학기술연구소보 Vol.- No.4

Book-shore type limit theorems for Wiener process are extended to those for Gaussian process. In particular, in this paper, we show that superior and inferior limits for the increments of the Gaussian process are differently obtained under mild conditions.