随机阵列最大部分和的矩不等式和概率不等式

MOMENT INEQUALITIES AND PROBABILITY INEQUALITIES FOR THE MAXIMUM PARTIAL SUM OF RANDOM RECTANGULAR ARRAY

设｛Ｘｉｊ，ｉ＝１，…，ｍ；ｊ＝１，…，ｎ｝是任意一个随机变量阵列．令Ｓ（ｉ１，ｊ１；ｉ２，ｊ２）∑ｉ２ｉ＝ｉ１∑ｊ２ｊ＝ｊ１Ｘｉｊ，Ｍ（ｉ１，ｊ１；ｉ２，ｊ２）ｍａｘｉ１≤ｉ≤ｉ２，ｊ１≤ｊ≤ｊ２｛｜Ｓ（ｉ１，ｊ１；ｉ，ｊ）｜｝（１≤ｉ１≤ｉ２≤ｍ，１≤ｊ１≤ｊ２≤ｎ）．本文根据所设Ｅ（ｅｘｐ（ｔ·｜Ｓ（ｉ１，ｊ１；ｉ２，ｊ２）｜）），Ｅ｜Ｓ（ｉ１，ｊ１；ｉ２，ｊ２）｜ｒ和Ｐ（｜Ｓ（ｉ１，ｊ１；ｉ２，ｊ２）｜≥ｔ）的界相应地建立了Ｅ（ｅｘｐｔＭ（１，１；ｍ，ｎ）），ＥＭｒ（１，１；ｍ，ｎ）和Ｐ｛Ｍ（１，１；ｍ。

Let {X ij ,i=1,…,m;j=1,…,n} be arbitrary random rectangular array and put S(i 1,j 1;i 2,j 2)=∑i 2i=i 1∑j 2j=j 1X ij , and M (i 1,j 1;i 2,j 2) = max i 1≤i≤i 2,j 1≤j≤j 2 {|S(i 1,j 1;i,j)|} for 1≤i 1≤i 2≤m,1≤j 1≤j 2≤n. In this paper,bounds for E( exp tM(1,1;M,n),EM r(1,1;m,n) and P{M(1,1;m,n)≥t} are established in terms of assumed bounds for E( exp t|S(i 1,j 1;i 2,j 2)|),E|S(i 1,j 1;i 2,j 2)| r and P(|S(i 1,j 1;i 2,j 2)|≥t) ,respectively.