On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation X-t = a(t)X(1-1) + epsilon(t) with random (renewal-reward) coefficient, a(t), taking independent. identically distributed values A(j) is an element of [0.1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution. and independent. identically distributed innovations, epsilon(t), belonging to the domain of attraction of an alpha-stable law (0 < alpha <= 2. alpha not equal 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 lambda-stable Levy process with index lambda < alpha. 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.