Impact of previous one-step variation in positively long-range correlated processes

In a positively long-range correlated process, variations among consecutive steps are interdependent, especially the influence of previous one-step variation on next steps. How to quantify this kind of impact is of great importance to predict the future variations. In this paper, we demonstrate that this kind of impact depends on the memory strength of underlying processes from two aspects based on the theoretical and observational calculations. More precisely, the conditional calculations and the marginal distribution of the next step variation with given distribution of the previous one-step variation. Both the theoretical and observational calculations demonstrate that the previous one-step variation affect greatly the variation for the next one-step, and the expectation of next step variation will shift to larger value as the increase of memory strength but with a much smaller uncertainty. This is beneficial for our one-step ahead prediction, and will be especially beneficial for multi-step ahead prediction.