Random Variable Column To Complete The Convergence Properties

This thesis consists of four chapters:In chapter one, we give some classic conclusions, which inspired me.In other chapters we will give some results, which are generation of this conclusions.In chapter two, we discuss complete convergence for arrays of rowwise independent random variables. Frist we give a version of Hoffmann-Jφrgensen inequality for independent, but not necessarily symmetric r.v. Then, using the inequality we proved our result, from which the main result of Hu (2003), Gan (2005) and Sung (2005) and Chen (2004) proved can be-derived .The main results in this chapter are: Theorem 2.1 Let {X_nk, 1≤k≤m_n, n≥1} be an array of rowwise independent random variable, and {m_n, n≥1} be a sequence of positive integers such that lim(?) m_n=∞,{c_n, n≥1} be a sequence of positive constants, suppose that: then for allε＞0,Theorem 2.2 Let {X_nk, 1≤k≤m_n, n≥1} be an array of rowwise independent random variable, and {m_s, n≥1} be a sequence of positive integers such that lim(?) m_n=∞,{c_n, n≥1} be a sequence of positive constants, suppose that: then for allε＞0, Using Theorem2.2 we can get a conclusion, which is a generation of the main result of Hu (1989) (cf.also Gut (1992), Theorem2.1). Victor (2006) proved the situation when 1≤q＜2. We proved the conclusion also right for 0＜q＜2. The conclusion is: Corollary 2.7 Let {X_nk, 1≤k≤m_n, n≥1} be an array of rowwise independent mean zero random variable which ara stochastically dominated in the Cesàro sense by a random variable X, 0＜q＜2, p≥-1, if E|X|^q(p+2)＜∞, then for allε＞0,In chapter three, we discuss the arrays of random variables which are NA sequences or p-mixing sequence in each row, we also give some results in this chapter: Theorem 3.1 Let {X_nk, 1≤k≤m_n, n≥1} be an array of rowwise negatively as sociated random variables, and {m_n, n≥1} be a sequence of positive integers such that lim(?) m_n=∞, {c_n, n≥1} be a sequence of positive constants, suppose that: We can see this theorem is generation of theorem2.1 for pj＞1. Theorem 3.2 Let{X_nk, 1≤k≤m_n, n≥1} be an array of rowwise negatively associated random variables, r＞1, suppose that: then for allε＞0, Theorem 3.4 Let {X_nk, 1≤k, n≥1} be an array of r.v., which are p-mixing sequences in each row, p≤2,αp＞1, r＞1, b_n of positive integers, suppose that: then for allε＞0, In chapter four, we mainly discuss complete convergence for arrays of rowwise independent B-valued r.v., by suing of the moment inequality for B-valued r.v. Theorem 4.1 Let {X_nk, 1≤k≤m_n, n≥1} be an array of rowwise independent random elements taking values in a real separable Banach space of type p (1≤p≤2), and {m_n, n≥1} be a sequence of positive integers such that lim(?) m_n=∞, {c_n, n≥1} be a sequence of positive constants, suppose that: then for allε＞0, Theorem 4.2 Let {X_nk, 1≤k≤m_n n＞1} be an array of rowwise independent random elements taking values in a real separable Banach space of type q (1≤q≤2), 0＜r＜q＜α, rp＞1, suppose that: then for allε＞0,...