Renewal regime switching and stable limit laws

The paper discusses long-memory properties and large sample behavior of partial sums in a general renewal regime switching scheme. The linear model Xt=μt+atXt-1+σtɛt with renewal switching in levels, slope or volatility and general (possibly heavy-tailed) i.i.d. noise ɛt is discussed in detail. Conditions on the tail behavior of interrenewal distribution and the tail index α∈(0,2] of ɛt are obtained, in order that the partial sums process of Xt is asymptotically λ-stable with index λ<α.