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 X t = μ t + a t X t - 1 + σ t ɛ t with renewal switching in levels, slope or volatility and general (possibly heavy-tailed) i.i.d. noise ɛ t is discussed in detail. Conditions on the tail behavior of interrenewal distribution and the tail index α ∈ ( 0 , 2 ] of ɛ t are obtained, in order that the partial sums process of X t is asymptotically λ -stable with index λ < α .