  We investigate a family of discrete-time stationary processes defined by
multiple stable integrals and renewal processes with infinite means. The model
may exhibit behaviors of short-range or long-range dependence, respectively,
depending on the parameters. The main contribution is to establish a phase
transition in terms of the tail processes that characterize local clustering of
extremes. Moreover, in the short-range dependence regime, the model provides an
example where the extremal index is different from the candidate extremal
index.
