  We study the projection of an element of fractional Gaussian noise onto its
neighbouring elements. We prove some analytic results for the coefficients of
this projection, in particular, we obtain recurrence relations for them. We
also make several conjectures concerning the behaviour of these coefficients,
provide numerical evidence supporting these conjectures, and study them
theoretically in particular cases. As an auxiliary result of independent
interest, we investigate the covariance function of fractional Gaussian noise,
prove that it is completely monotone for $H>1/2$, and, in particular monotone,
convex, log-convex along with further useful properties.
