  Finite-temperature local spin dynamics within the random spin-1/2
antiferromagnetic Heisenberg chain is studied numerically. The aim is to
explain measured NMR spin-lattice relaxation times in
BaCu_{2}(Si_{0.5}Ge_{0.5})_{2}O_{7}, which is the realization of a random spin
chain. In agreement with experiments we find that the distribution of
relaxation times within the model shows a very large span similar to the
stretched-exponential form. The distribution is strongly reduced with
increasing T but stays finite also in the high-T limit. Our results reveal the
crucial role of the anisotropy (interaction), since the behavior is essentially
contrast with the ones for XX model (equivalent to noninteracting fermions),
where we do not find any significant T dependence of the distribution.
