一类强正相依随机变量列部分和次序统计量的矩不等式与收敛定理

INEQUALITIES OF MOMENT AND CONVERGENCE THEOREM OF ORDER STATISTICS OF PARTIAL SUMS FOR A CLASS OF STRONGLY POSITIVE DEPENDENT STOCHASTIC SEQUENCE

设（Ｘｎ； ｎ≥１）是一类由作者提出的强正相依（ＳＰＤ）随机序列[1] ，它的相依条件弱于由 Ｅｓａｒｙ等［２］所定义的相协性（Ａｓｓｏｃｉａｔｉｏｎ）；假设 ＥＸｎ＝０，令Ｓｎ＝ Ｘ1＋Ｘ２+… ＋Ｘn（n≥１），以Ｓ_（１，ｎ）≤…≤Ｓ_（ｋ，ｎ）≤…≤Ｓ_（ｎ，ｎ）表示Ｓ_１，…，Ｓ_ｎ对应的次序统计量．本文主要结果（１）导出 Ｓ_（ｋ，ｎ）（１≤ｋ≤ ｎ； ｎ≥１）的若干矩不等式，特别是 Ｄｏｏｂ不等式；（２）固定ｋ≥１，证明（Ｓ_（ｋ，ｎ），ｎ≥１）和（Ｓ_ｎ，ｎ≥１）两个序列的Ｌ~Ｐ（Ｐ＞１）和几乎必然收敛定理．

Let (X_n,n ≥ 1) is a class of strongly positive dependent (SPD) stochastic sequence which was introduced by author[1]. Its dependence condition is weaker than the association introduced by Esary et al[2]; Suppose EX_n = 0 (n ≥ 1), let S_n = X_1 +… + X_n, and S_(1,n)≤… S_（k,n）≤… S_（n,n） denote (S_1,… S_n) corresponding order statistics. The main results of the paper are following: (i) Some inequalities of moment of S_(k,n) (1≤ k ≤ n, n ≥ 1), particularly Doob's inequality, are derived; (ii) fixed k ≥ 1, L^p (p > 1) and almost sure covergence theorem for (S_（k,n）, n ≥ l) and (S_n,n ≥ 1) two sequences are proved.