We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation X
t
= a
t
X
t−1 + ε
t
with random (renewal-reward) coefficient, a
t
, taking independent, identically distributed values A
j
∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, ε
t
, belonging to the domain of attraction of an α-stable law (0 < α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of A
j
near the unit root a = 1, we show that the partial sums process of X
t
converges to a λ-stable Lévy process with index λ < α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance X
t
to that of infinite-variance X
t
.
On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise
