Grishchuk has shown that the stochastic background of gravitational waves p
roduced by an inflationary phase in the early Universe has an unusual prope
rty: it is not a stationary Gaussian random process. Because of squeezing,
the phases of the different waves are correlated in a deterministic way, ar
ising from the process of parametric amplification that coated them. The re
sulting random process is Gaussian but non-stationary. This provides a uniq
ue signature that could in principle distinguish a background created by in
flation from stationary stochastic backgrounds created by other types of pr
ocesses. We address the question: could this signature be observed with a g
ravitational wave detector? Sadly, the answer appears to be no: an experime
nt which could distinguish the non-stationary behavior would have to last a
pproximately the age of the Universe at the time of measurement. This rules
out direct detection by ground and space based gravitational wave detector
s, but not indirect detections via the electromagnetic cosmic microwave bac
kground radiation.