AlwaysOn Statistics getting updated on Readonly node immediately after updating primary node with Fullscan Version:
Microsoft SQL Server 2012 (SP3) (KB3072779) - 11.0.6020.0 (X64) 
    Oct 20 2015 15:36:27 
    Copyright (c) Microsoft Corporation
    Enterprise Edition: Core-based Licensing (64-bit) on Windows NT 6.3  (Build 9600: )

Once I update my statistics with full scan, with 10~20 seconds some of the statistics on the same table are getting update on secondary with a sample. I ran extended event trace to capture auto update stats on secondary node but nothing was logged.
On primary node once I run this:
Update statistics Tablename with fullscan
I see this about my stats status


When I query for the stats on secondary I get this. Look at the rows_sampled and last_updated column.


I also ran an extended event trace to capture the auto update stats but nothing was logged.  Here is the defination of my XE trace.

CREATE EVENT SESSION [AutoUpdateStats] ON SERVER ADD EVENT sqlserver.auto_stats(
ACTION(sqlserver.client_app_name,sqlserver.server_principal_name)
WHERE ([package0].[equal_uint64]([database_id],(25))))WITH (MAX_MEMORY=4096 KB,EVENT_RETENTION_MODE=ALLOW_SINGLE_EVENT_LOSS,MAX_DISPATCH_LATENCY=30 SECONDS,MAX_EVENT_SIZE=0 KB,MEMORY_PARTITION_MODE=NONE,TRACK_CAUSALITY=OFF,STARTUP_STATE=OFF)GO
I did open a case with Microsoft Support. I was notified similar behavior was reported by other customers. For now the solution is to rebuild the index.  Microsoft is investigating further about the root cause.