- 學位論文
p類型Banach空間中逐行獨立的隨機元素陣列的完全收斂性質之研究
Complete Convergence of Sums of Row-wise Independ-ent Random Elements in Type p Banach Space
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摘要
本文主要考慮在p類型可分離的Banach空間上的逐行獨立的隨機元素陣列的完全收斂性質。在第一章中，我們給出了隨機元素的基本定義，介紹了p類型可分離的Banach空間及其基本性質，引入了完全收斂性質的基本概念及其拓展，並敘述和證明了幾個必要的不等式。在第二章中，我們從Kolmolgorov三級數定理獲得啟發，給出了在p類型可分離的Banach空間上的隨機元素陣列的完全收斂性定理。 在第三章中，我們給出了一些特殊形態的隨機元素陣列完全收斂性的判別法，並研究了加權的隨機元素陣列的完全收斂性。在第四章中，我們構造了四個重要的例證，以解釋說明上述工作的意義。在第五章中，我們總結了全文，並提供了一些改進的方向。
並列摘要
In this thesis, we investigate the complete convergence for row sums of arrays of row-wise independent random elements taking values in type p separable Banach spaces. In Chapter 1, we introduce basic definitions and properties of random elements taking val-ue in separable Banach spaces and type p separable Banach spaces. And we introduce the basic concept of complete convergence of random elements and their improvement and describe and prove several necessary inequalities. In Chapter 2, being inspired by the Kolmogorov three-series theorem, we obtain two complete convergence theorems for arrays of random elements taking values in type p separable Banach spaces. In Chap-ter 3, We provide several criteria of complete convergences for some particular cases of random element arrays. We also investigate complete convergences of row-wise weighted sums for arrays of random elements. In Chapter 4, we construct four illustra-tive examples to explain those works in previous Chapters. In chapter 5, we summarize and provide some possible directions for study in the future.
參考文獻
[1] Hsu, P.L., and Robbins, H. (1947). Complete convergence and the law of large
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[3] Mourier, E.(1953). Eléments aleatories dan un espace de Banach. Ann. Inst. Henri Poincaré. 13:159-244.
[4] Hoffmann-Jørgensen, J., and Pisier, G. (1976). The law of large numbers and the central limit theorem in Banach spaces. Ann. Prob. 4:587-599.