# THE INVARIANCE PRINCIPLE FOR LINEARLY POSITIVE QUADRANT DEPENDENT SEQUENCES

• Kim, Tae-Sung (Won-Kwang University, Department of Statistics, Iri 570-749) ;
• Han, Kwang-Hee (Chonbuk Sanup University, Department of Computer Science, Kunsan 573-400)
• Published : 1994.10.01
• 28 2

#### Abstract

A sequence ${X_j : j \geq 1}$ of random variables is said to be pairwise positive quadrant dependent (pairwise PQD) if for any real $r-i,r_j$ and $i \neq j$ $$P{X_i > r_i,X_j > r_j} \geq P{X_i > r_i}P{X_j > r_j}$$ (see [8]) and a sequence ${X_j : j \geq 1}$ of random variables is said to be associated if for any finite collection ${X_{i(1)},...,X_{j(n)}}$ and any real coordinatewise nondecreasing functions f,g on $R^n$ $$Cov(f(X_{i(1)},...,X_{j(n)}),g(X_{j(1)},...,X_{j(n)})) \geq 0,$$ whenever the covariance is defined (see [6]). Instead of association Cox and Grimmett's [4] original central limit theorem requires only that positively linear combination of random variables are PQD (cf. Theorem $A^*$).