The present study aims at proving the existence of long memory (or long-range dependence) in the earthquake process through the analysis of time series of induced seismicity. Specifically, we apply alternative statistical techniques borrowed from econometrics to the seismic catalog of The Geysers geothermal field (California), the world’s largest geothermal field. The choice of the study area is essentially guided by the completeness of the seismic catalog at smaller magnitudes (a drawback of conventional catalogs of natural seismicity). Contrary to previous studies, where the long-memory property was examined by using non-parametric approaches (e.g., rescaled range analysis), we assume a fractional integration model for which the degree of memory is defined by a real parameter d, which is related to the best known Hurst exponent. In particular, long-memory behavior is observed for d > 0. We estimate and test the value of d (i.e., the hypothesis of long memory) by applying parametric, semi-parametric, and non-parametric approaches to time series describing the daily number of earthquakes and the logarithm of the (total) seismic moment released per day. Attention is also paid to examining the sensitivity of the results to the uncertainty in the completeness magnitude of the catalog, and to investigating to what extent temporal fluctuations in seismic activity induced by injection operations affect the value of d. Temporal variations in the values of d are analyzed together with those of the b-value of the Gutenberg and Richter law. Our results indicate strong evidence of long memory, with d mostly constrained between 0 and 0.5. We observe that the value of d tends to decrease with increasing the magnitude completeness threshold, and therefore appears to be influenced by the number of information in the chain of intervening related events. Moreover, we find a moderate but significant negative correlation between d and the b-value. A negative, albeit weaker correlation is found between d and the fluid injection, as well as between d and the annual number of earthquakes.

This paper deals with the analysis of the financial stock market in China, investigating its degree of persistence in order to know if shocks affecting them have temporary or permanent effects. For this purpose we examine the closing prices of the Shanghai and Zhenzhen Composite Indices, with data starting at July and April 1991 respectively and ending at March 2020. Looking at the sample before the coronavirus, the results indicate large degrees of persistence with shocks having permanent effects. Meanwhile, during the period of the coronavirus, the results indicate mean reversion with shocks having temporary effects. This may result from the Chinese government's rapid and effective responses during the outbreak of the pandemic.

This paper focuses on the analysis of the time series behaviour of the air quality in the 50 US states by looking at the statistical properties of the particulate matter (PM10 and PM2.5) datasets. We use long daily time series of outdoor air quality indices to examine issues such as the degree of persistence as well as the existence of time trends in data. For this purpose, we use a long memory fractionally integrated framework. The results show significant negative time trend coefficients in a number of states and evidence of long memory in the majority of the cases. In general, we observe heterogeneous results across counties though we notice higher degrees of persistence in the states on the West with respect to those on the East, where there is a general decreasing trend. It is hoped that the findings in the paper will continue to assist in quantitative evidence-based air quality regulation and policies.