  We introduce a definition of long range dependence of random processes and
fields on an (unbounded) index space $T\subseteq \R^d$ in terms of
integrability of the covariance of indicators that a random function exceeds
any given level. This definition is particularly designed to cover the case of
random functions with infinite variance. We show the value of this new
definition and its connection to limit theorems on some examples including
subordinated Gaussian as well as random volatility fields and time series.
