  We compute the probability distribution for the spin of a black hole
following a series of minor mergers with isotropically distributed,
non-spinning, inspiraling compact objects. By solving the Fokker-Planck
equation governing this stochastic process, we obtain accurate analytical fits
for the evolution of the mean and standard deviation of the spin distribution
in several parameter regimes. We complement these analytical fits with
numerical Monte-Carlo simulations in situations when the Fokker-Planck analysis
is not applicable. We find that a ~150 solar-mass intermediate-mass black hole
that gained half of its mass through minor mergers with neutron stars will have
dimensionless spin parameter chi=a/M~0.2 \pm 0.08. We estimate the effect of
the spin of the central black hole on the detection range for
intermediate-mass-ratio inspiral (IMRI) detections by Advanced LIGO and
extreme-mass-ratio inspiral(EMRI) detections by LISA. We find that for
realistic black hole spins, the inclination-averaged Advanced-LIGO IMRI
detection range may be increased by up to 10% relative to the range for IMRIs
into non-spinning intermediate-mass black holes. For LISA, we find that the
detection range for EMRIs into 10^5 solar-mass massive black holes (MBHs) is
not significantly affected by MBH spin, the range for EMRIs into 10^6
solar-mass MBHs is affected at the ~ 10% level, and EMRIs into maximally
spinning 10^7 solar-mass MBHs are detectable to a distance ~25 times greater
than EMRIs into non-spinning black holes. The resulting bias in favor of
detecting EMRIs into rapidly spinning MBHs will play a role when extracting the
MBH spin distribution from EMRI statistics.
