SQL Server keeps track of how many inserts and deletes since last stats update – when the number of inserts/deletes exceeds the stats update threshold the next time a query requests those stats it’ll qualify for an update. Trace flag 2371 alters the threshold function before SQL Server 2016. With 2016 compatibility mode, the T2371 function becomes default behavior. Auto-stats update and auto-stats update async settings of the database determine what happens once the stats qualify for an update. But whether an auto-stats update or a manual stats update, the density, histogram, etc are all updated.
Trace flags 2389, 2390, 4139, and the ENABLE_HIST_AMENDMENT_FOR_ASC_KEYS hint operate outside the full stats framework, bringing in the quickstats update. They have slightly different scope in terms of which stats qualify for quickstats updates – but in each case its *only* stats for indexes, not stats for non-indexed columns that can qualify. After 3 consecutive stats updates on an index, SQL Server “brands” the stats type as ascending or static, until then it is branded ‘unknown’. The brand of a stat can be seen by setting trace flag 2388 at the session level and using dbcc show_statistics.
Right now there are just a few details and several links, but it does look like he’s going to expand it out.