  The Eigenstate Thermalization Hypothesis (ETH) explains emergence of the
thermodynamic equilibrium by assuming a particular structure of observable's
matrix elements in the energy eigenbasis. Schematically, it postulates that
off-diagonal matrix elements are random numbers and the observables can be
described by Random Matrix Theory (RMT). To what extent physical operators can
be described by RMT, more precisely at which energy scale strict RMT
description applies, is however not fully understood. We study this issue by
introducing a novel numerical approach to probe correlations between matrix
elements for Hilbert-space dimensions beyond those accessible for exact
diagonalization. Our analysis is based on the evaluation of higher moments of
operator submatrices, defined within energy windows of varying width.
Considering nonintegrable quantum spin chains, we observe that genuine RMT
behavior is absent even for narrow energy windows corresponding to time scales
of the order of thermalization time $\tau_\text{th}$ of the respective
observables. We also demonstrate that residual correlations between matrix
elements are reflected in the dynamics of out-of-time-ordered correlation
functions.
