Chapman-Kolmogorov test for estimating memory length of two coupled processes

Real-world processes often display a prolonged memory, which extends beyond the single-step dependency characteristic of Markov processes. In addition, the current state of an empirical process is often not only influenced by its own past but also by the past states of other dependent processes. This study introduces the generalized version of the Chapman-Kolmogorov equation (CKE) to estimate the memory size in such scenarios. To assess the applicability of our approach, we generate coupled time series with predetermined memory lengths using the autoregressive model. The results show a high degree of accuracy in measuring memory lengths. Subsequently, we employ the generalized CKE to analyze cryptocurrency data as a real-world case study. Our results indicate that the past dynamics of cryptocurrencies significantly impact their current states, thereby highlighting interdependencies among them. The method proposed in this study can be also utilized in forecasting coupled time series.