  We report results of a muon spin relaxation study of slow magnetic
fluctuations in the pseudogap phase of underdoped single-crystalline
YBa$_{2}$Cu$_{3}$O$_{y}$, $y = 6.77$ and 6.83. The dependence of the dynamic
muon spin relaxation rate on applied magnetic field yields the rms
magnitude~$B\mathrm{_{loc}^{rms}}$ and correlation time~$\tau_c$ of fluctuating
local fields at muon sites. The observed relaxation rates do not decrease with
decreasing temperature~$T$ below the pseudogap onset at $T^\ast$, as would be
expected for a conventional magnetic transition; both $B\mathrm{_{loc}^{rms}}$
and $\tau_c$ are roughly constant in the pseudogap phase down to the
superconducting transition. Corresponding NMR relaxation rates are estimated to
be too small to be observable. Our results put strong constraints on theories
of the anomalous pseudogap magnetism in YBa$_{2}$Cu$_{3}$O$_{y}$.
