Covariance and comparison inequalities under quadrant dependence
- PDF / 183,487 Bytes
- 10 Pages / 439.37 x 666.142 pts Page_size
- 7 Downloads / 155 Views
Covariance and comparison inequalities under quadrant dependence Przemysław Matuła · Maciej Ziemba
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract We study the difference between a distribution of the random vectors [X, Y ] and [X , Y ], where X and Y are independent and X has the same law as X and Y as Y . Particular interest is focused on positively quadrant dependent random variables X and Y , in this case the bounds for the difference in question are expressed in terms of the covariance of X and Y . Keywords Covariance · Positive and negative dependence · Probabilistic inequalities · Comparison theorems Mathematics Subject Classification
60E15 · 62H20
1 Introduction and the notation Positive and negative dependence concepts play very important role not only in mathematical statistics but also in applications of probability theory, in particular in mathematical physics. The definitions of positively and negatively quadrant dependent random variables (r.v.’s) were introduced by Lehmann in 1966 (cf. [5]) and soon after extended to the multivariate case by Esary at al. and Joag-Dev and Proschan, who introduced the notion of positive and negative association (cf. [3,4]). Nowadays, a comprehensive study of this topic is contained in the monographs of Bulinski and Shashkin (cf. [2]), Oliveira (cf. [10]) and Prakasa Rao (cf. [11]).
P. Matuła (B) Institute of Mathematics, Marie Curie-Skłodowska University, pl. M.C.-Skłodowskiej 1, 20-031 Lublin, Poland e-mail: [email protected] M. Ziemba Department of Mathematics, Lublin University of Technology, ul. Nadbystrzycka 38d, 20-618 Lublin, Poland e-mail: [email protected]
123
P. Matuła, M. Ziemba
Let us recall that the random variables X, Y are positively quadrant dependent (PQD) if H X,Y (t, s) := P(X ≤ t, Y ≤ s) − P(X ≤ t)P(Y ≤ s) ≥ 0 for all t, s ∈ R and X, Y are negatively quadrant dependent (NQD) if H X,Y (t, s) ≤ 0. It is well known that X, Y are NQD iff X, −Y are PQD. In view of this duality, we shall focus only on the PQD case later on. Denote by ∞ ∞ CovH (X, Y ) = H X,Y (t, s)dtds −∞ −∞
the so-called Hoeffding covariance (it is always well defined for PQD or NQD r.v.’s, even though it may be infinite and if the usual product moment covariance exists, then it is equal to the Hoeffding covariance). From this fact it follows that uncorrelated PQD (or NQD) r.v.’s are independent. Therefore, in the study of limit theorems, covariance is usually used to ”control” the dependence of r.v.’s. It is also well known that monotonic functions of positively (negatively) dependent r.v.’s inherit such properties. In particular, the indicators I(−∞,t (X ), I(−∞,s (Y ) are PQD (NQD), provided X, Y are PQD (NQD). Here and in the sequel I A (x) denotes the indicator function of a set A. It is easy to see that Cov I(−∞,t (X ), I(−∞,s (Y ) = H X,Y (t, s), thus, in the study of limit theorems for empirical processes based on positively or negatively dependent observations, it is important to cont
Data Loading...
