The origin of power-law distributions in self-organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process. An avalanche is then regarded as a first-return random-walk process in a one-dimensional lattice. Power-law distributions of the lifetime and spatial size are found when the random walk is unbiased with equal probability to move in opposite directions. This shows that power-law distributions in selforganized criticality may be caused by the balance of competitive interactions. At the mean time, the mean spatial size for avalanches with the same lifetime is found to increase in a power law with the lifetime.