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In recent years, various early warning signals of critical transition have been presented, such as autocorrelation at lag 1 [AR(1)], variance, the propagator based on detrended fluctuation analysis (DFA-propagator), and so on. Many studies have shown that the climate system has the characteristics of long-term memory (LTM). Will the LTM characteristics of the climate system change as it approaches possible critical transition points? In view of this, the present paper first studies whether the LTM of several folding (folded bifurcation) models changes consistently as they approach their critical points slowly by the rescaled range (R/S) analysis. The results of numerical experiments show that when the control parameters of the folding model are close to its critical threshold, the Hurst exponent H exhibits an almost monotonic increase (significance level α = 0.05). We compare the performance of R/S with the existing indicators, including AR(1), variance, and DFA-propagator, and find that R/S is a perfectly valid alternative. When there is no extra false noise, AR(1) and variance have good early warning effects. After the addition of extra Gaussian white noise of different intensities, the values of AR(1) and variance change significantly. As a result, the DFA-propagator based on AR(1) calibration also changed significantly. Compared with the other three indicators, the early warning effect of H has stronger ability to resist the interference of external false signals. To further verify the validity of increasing H, paleoclimate reconstruction of Cariaco Basin sediment core grayscale record with long trends filtered out is studied by R/S analysis. The other three early warning signals are calculated in the same way. The data contain a well-known abrupt climate change: the transition between the Younger Dryas (YD) and the Holocene. We find that approximately 300 years before this abrupt climate change occurred, before 11.7 kyr BP, the LTM exponents for Cariaco Basin deglacial grayscale data present an obvious increasing trend at a significant level of α = 0.05. Meanwhile, the variation trend of H and DFA-propagator is basically similar. This shows that increasing H by R/S analysis is an effective early warning signal, which indicates that a dynamic system is approaching its possible critical transition points; H is a completely valid alternative signal for AR(1) and DFA-propagator. The main conclusion of this paper is based on numerical experiments. The precise relationship between H and the stability of the underlying state approaching the transition needs to be further studied. [ABSTRACT FROM AUTHOR]