It was recently realized that the persistence exponent appearing in the dynamics of nonequilibrium systems is a special member of a continuously varying family of exponents, describing generalized persistence properties. We propose and solve a simple stochastic spin model, where time intervals between spin flips are independent, and distributed according to a Lévy law. Both the limit distribution of the mean magnetization and the generalized persistence exponents are obtained exactly. We discuss the relevance of this model for phase ordering, spin glasses, and random walks.