We show that a non-trivial continuous-time strictly α-stable, α (0,2), stationary process cannot be represented in distribution as a discrete linear process n=-~~f t (n)ε n ,t R,where {f t } t R is a collection of deterministic functions and {ε n } n Z are independent strictly α-stable random variables. Analogous results hold for self-similar strictly α-stable processes and for strictly α-stable processes with stationary increments. As a consequence, the usual wavelet decomposition of Gaussian self-similar processes cannot be extended to the α-stable, α<2 case.