Due to the conceptual simplicity, the linear filtering framework, notably the autoregressive (AR) process, has a long history in simulating clutter sequences with specified probability density functions (PDFs) and autocorrelation functions (ACFs). However, linear filtering inevitably distorts the input distribution, which may lead to inaccurate PDF reproduction or restrict applicability to very simple ACFs. To address these challenges, this study proposes a series-based analytic continuation strategy that revitalizes AR process clutter simulation by accurately precomputing the input pre-distortion required to compensate for AR filtering. First, the moments and cumulants of the AR input are derived based on the input-output relationship of the AR process, facilitating the moment and cumulant expansions of the Laplace transform (LT) and the logarithmic LT around zero, respectively. Second, both series expansions are analytically continued via the Padé approximation (PA) to recover the LT over the full complex plane. Notably, the PA-based continuation of the moment expansion, a conventional choice, can be highly inaccurate when the LT exhibits strong oscillations. By contrast, given the logarithmic LT generally has a simpler structure, the continuation of the cumulant expansion provides a more stable and accurate alternative. Third, the LT recovered from the cumulant expansion facilitates fast simulation of the AR input non-Gaussian white sequence via a random variable transformation method, thereby enabling an efficient AR process. Finally, simulations demonstrate that the proposed strategy enables accurate and fast simulation of non-Gaussian correlated clutter sequences.