  Over the past decades much effort has been devoted towards understanding and
forecasting natural hazards. However, earthquake forecasting skill is still
very limited and remains a great scientific challenge. The limited earthquake
predictability is partly due to the erratic nature of earthquakes and partly to
the lack of understanding the underlying mechanisms of earthquakes. To improve
our understanding and potential forecasting, here we study the spatial and
temporal long-term memory of interevent earthquakes above a certain magnitude
using lagged conditional probabilities. We find, in real data, that the lagged
conditional probabilities show long-term memory for both the interevent times
and interevent distances and that the memory functions obey scaling and decay
slowly with time, while, at a characteristic time, the decay crossesover to a
fast decay. We also show that the ETAS model, which is often used to forecast
earthquake events, yields scaling functions of the temporal and spatial
interevent intervals which are not consistent with those of real data.
