Abstract: In this paper, we study the persistence change problem of heavy-tailed observations with infinite variance by constructing Ratio two-sided test statistic based on M-estimates. It is shown that the asymptotic distribution of the statistic under the null hypothesis is functional for Brownian motion, which is independent of the tail index, and its consistency is given under the alternative hypothesis. The Bootstrap sampling method is used to approximate the asymptotic distribution to obtain the accurate critical values. The numerical simulation results show the Ratio test based on M-estimates has a satisfactory empirical size without significant distortion, and significantly improves empirical power compared with the test based on least square estimate, especially when the tail features of time series are thicker. Finally, a set of gold ETF volatility index data verifies the validity and feasibility of our proposed method.