Noncentral Limit Theorems for Bounded Functions of Linear Processes without Finite Mean

We derive noncentral limit theorems for the partial sum processes of K(Xi)‐E{K(Xi)}, where K(x) is a bounded function and {Xi } is a linear process. We assume the innovations of {Xi } are independent and identically distributed and that the distribution of the innovations is an α-stable law (0<α<1) or belongs to the domain of attraction of an α-stable law (0<α<1). Then we establish the finite-dimensional convergence in distribution of the partial sum processes to an αβ-stable Levy motion. The parameter β determines how fast the coefficients of the linear process decay and we assume that 1<αβ<2. We also derive the asymptotic distribution of the kernel density estimator of the marginal density function of {Xi } by exploiting one of the noncentral limit theorems.