Jan Mielniczuk

Bivariate Densities Having Diagonal Expansion Revisited

853

Abstract

We consider bivariate densities having diagonal expansions and review and generalize some of its known properties. In particular, Mehler's equality and Gebelein's inequality are generalized. Moreover, we consider stationary processes $(X_i)_{i=1}^\infty$ with a covariance function $r(i)$ and with bivariate densities of $(X_1,X_{1+i})$ having diagonal form with coefficients $a_k(i)\,,k=0,1,\ldots$ and state general conditions under which sequences subordinated to $(X_i)_{i=1}^\infty$ are long-range dependent and obey the reduction principle. Furthermore, in the special case $a_k(i)=r(i)^k\,,k=0,1,\ldots$ estimates based on such sequences enjoy some common asymptotic properties under long-range dependence.

Key Words: diagonal expansion of bivariate density, long-range dependence, orthonormal system, mixing coefficients, subordinated sequence, time series.