  We present measurements of magnetic field and frequency dependences of the
low temperature (T = 1.8 K) AC-susceptibility, and temperature and field
dependences of the longitudinal field positive muon spin relaxation ({\mu}SR)
for LiY$_{1-x}$Ho$_x$F$_4$ with x = 0.0017, 0.0085, 0.0408, and 0.0855. The
fits of numerical simulations to the susceptibility data for the x = 0.0017,
0.0085 and 0.0408 show that Ho-Ho cross-relaxation processes become more
important at higher concentrations, signaling the crossover from single-ion to
correlated behavior. We simulate the muon spin depolarization using the
parameters extracted from the susceptibility, and the simulations agree well
with our data for samples with x = 0.0017 and 0.0085. The {\mu}SR data for
samples with x = 0.0408 and 0.0855 at low temperatures (T < 10 K) cannot be
described within a single-ion picture of magnetic field fluctuations and give
evidence for additional mechanisms of depolarization due to Ho$^{3+}$
correlations. We also observe an unusual peak in the magnetic field dependence
of the muon relaxation rate in the temperature interval 10 - 20 K that we
ascribe to a modification of the Ho$^{3+}$ fluctuation rate due to a field
induced shift of the energy gap between the ground and the first excited
doublet crystal field states relative to a peak in the phonon density of states
centered near 63 cm$^{-1}$.
