  The observed radiation from hot gas accreting onto a black hole depends on
both the details of the flow and the spacetime geometry. The lensing behavior
of a black hole produces a distinctive pattern of autocorrelations within its
photon ring that encodes its mass, spin, and inclination. In particular, the
time autocorrelation of the light curve is expected to display a series of
peaks produced by light echoes of the source, with each peak delayed by the
characteristic time lapse $\tau$ between light echoes. However, such peaks are
absent from the light curves of observed black holes. Here, we develop an
analytical model for such light curves that demonstrates how, even though light
echoes always exist in the signal, they do not produce autocorrelation peaks if
the characteristic correlation timescale $\lambda_0$ of the source is greater
than $\tau$. We validate our model against simulated light curves of a
stochastic accretion model ray traced with a general-relativistic code, and
then fit the model to an observed light curve for Sgr A*. We infer that
$\lambda_0>\tau$, providing an explanation for the absence of light echoes in
the time autocorrelations of Sgr A* light curves. Our results highlight the
importance for black hole parameter inference of spatially resolving the photon
ring via future space-based interferometry.
