由於適應相依強度具有強大的靈活性，擴充負相依結構被廣泛地使用在高維度統計應用及風險管理應用上。因為許多數學家及統計學家對於相依隨機變數的特別關注，這份研究的目標是系統性地探討擴充負相依隨機變數的基本機率性質和研究各種擴充負相依隨機變數的極限定理。
我們建立了擴充負相依隨機變數的Borel-Cantelli引理以及各種不同的機率不等式和矩不等式。機率不等式包含Bernstein 型不等式和Hoeffding型不等式，而矩不等式包含指數型不等式和Rosenthal型不等式。我們也建構一個擴充負相依隨機變數的基本最大值不等式，且經由這個不等式我們得到了Hàjek-Rényi型不等式和Kolmogorov型不等式。
Kolmogorov型三級數定理也被推廣至擴充負相依隨機變數。擴充負相依隨機變數的Kolmogorov-Chung型和Marcinkiewicz-Zygumund型強大數法則也被得到。 基於擴充負相依隨機變數的Borel-Cantelli引理和Rosenthal型不等式，我們使用子數列方法給出了強大數法則的充份和必要條件。
使用擴充負相依隨機變數的機率不等式和矩不等式，我們給出列擴充負相依隨機變數陣列的完全收斂性和完全矩收斂性。此外，我們估計擴充負相依且相同分布隨機變數在完全收斂性和完全矩收斂性上的精準漸進行為。

Due to its great flexibility of adjusting dependence strength, the extended negatively dependent structure is wildly used in high-dimensional statistical applications and risk management applications. Since many mathematicians and statisticians pay special attention to dependent random variables, the aim of this study is to systematically explore the fundamental probability property and investigate the various limiting theorems for extended negatively random variables.
We establish the Borel-Cantelli lemma and several different probability inequalities and moment inequalities for extended negatively dependent random variables. The probability inequalities include Bernstein type inequality and Hoeffding type inequality, and the moment inequalities contain exponential type inequality and Rosenthal type inequality. We also construct a fundamental maximal inequality for extended negatively random variables, and through this theorem we obtain Hàjek-Rényi type inequality and Kolmogorov type inequality.
The Kolmogorov type three-series theorem is generalized to extended negatively random variables. The Kolmogorov-Chung type and the Marcinkiewicz-Zygumund type strong law of large numbers is obtained for extended negatively dependent random variables. Based on the Borel-Cantelli lemma and the Rosenthal type inequality for extended negatively dependent random variables, we use the method of subsequence to provide the necessary and sufficient condition for the strong law of large numbers.
Using the probability inequality and moment inequality for extended negatively dependent random variables, we present the complete convergence and complete moment convergence theorems for array of rowwise extended negatively dependent random variables. Furthermore, we estimate the precise asymptotic in complete convergence and complete moment convergence for extended negatively dependent and identical distributed random variables.

Abstract i
Acknowledgements ii
1 Introduction 1
2 Extended negatively dependent random variables 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Probabilistic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Probability inequalities for partial sums . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Moment inequalities for partial sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Maximal inequalities for partial sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 The law of large numbers for extended negatively dependent random variables 32
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 The weak law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 The strong law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Complete convergence theorems for extended negatively dependent random
variables 79
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Complete convergence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 More results on complete convergence for weighted sums . . . . . . . . . . . . . . . . 94
4.4 Complete convergence theorems for other kind of weighted sums . . . . . . . . . . . 111
5 Complete moment convergence theorems for extended negatively dependent
random variables 126
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2 Complete moment convergence theorems . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3 More results on complete moment convergence theorems . . . . . . . . . . . . . . . . 145
6 Precise asymptotic of complete convergence and complete moment convergence
for extended negatively dependent random variables 159
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2 Precise asymptotic of complete convergence . . . . . . . . . . . . . . . . . . . . . . . 161
6.3 Precise asymptotic of complete moment convergence . . . . . . . . . . . . . . . . . . 172
Bibliography 195

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