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 τ 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 λ_{0} of the source is greater than τ. 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 λ_{0}>τ, 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.