  We briefly review the phenomenon of long secondary periods (LSPs) in red
giants, and the LSP variable stars classification introduced in the All-Sky
Automated Survey for Supernovae (ASAS-SN) variable star catalog; they are red
giant Long Period Variables (LPVs) in which their LSP variability is
significantly greater than their pulsational variability. We then describe and
discuss the results of a period and amplitude analysis of a random sample of 35
LSP variables in the ASAS-SN catalog, using ASAS-SN data and the AAVSO VStar
time-series analysis software. The pulsation period and amplitude, and LSP, all
increase with increasing luminosity or size of the star, as expected. The
behavior of the LSP amplitude is more complicated; it appears to be larger in
moderate-luminosity stars, and smaller in low- and high-luminosity stars. In
particular, it is relatively small in a sample of 27 Mira stars, analyzed
separately using AAVSO visual data. These results are discussed in the context
of the current model for the LSP phenomenon, namely that it is caused by
eclipses of the red giant star by a dust-enshrouded companion.
