  Let $\mathbf{X}=(\mathbf{X}_t)_{t \geq 0}$ be a stochastic process issued
from $x \in \mathbb R$ that admits a marginal stationary measure $\nu$, i.e.
$\nu \mathbf{P}_t f = \nu f$ for all $t \geq 0$, where $\mathbf{P}_t f(x)=
\mathbb{E}_x[f(\mathbf{X}_t)]$. In this paper, we introduce the (resp.
biorthogonal) spectral projections correlation functions which are expressed in
terms of projections into the eigenspaces of $\mathbf{P}_t$ (resp. and of its
adjoint in the weighted Hilbert space $L^2(\nu)$). We obtain closed-form
expressions involving eigenvalues, the condition number and/or the angle
between the projections in the following different situations: when
$\mathbf{X}=X$ with $X=(X_t)_{t \geq 0}$ is a Markov process, $\mathbf{X}$ is
the subordination of $X$ in the sense of Bochner, and $\mathbf X$ is a
non-Markovian process which is obtained by time-changing $X$ with an inverse of
a subordinator. It turns out that these spectral projections correlation
functions have different expressions with respect to these classes of processes
which enables to identify substantial and deep properties about their dynamics.
This interesting fact can be used to design original statistical tests to make
inferences, for example, about the path properties of the process (presence of
jumps), distance from symmetry (self-adjoint or non-self-adjoint) and
short-to-long-range dependence. To reveal the usefulness of our results, we
apply them to a class of non-self-adjoint Markov semigroups studied in Patie
and Savov [28], and then time-change by subordinators and their inverses.
