Yes, certainly, that was how the Unabomber - who didn't even live with his family and strung his killings out over some time - was finally caught. And it's hard to imagine, that in the case of this sniper, no family member would notice something strange all of a sudden in his routines and habits.
Ping to other folks interested in this idea.
Read this:
"There has been no contact between him and his family since he left for the United States," he added.
Soldier Is Sniper Suspect
A French soldier who went missing on a holiday trip to America is reported to have matched an image of the serial sniper terrorising Washington.
The soldier is reported to have been identified as the man in a US police sketch of the gunman who has killed nine people in the Washington area since October 2...Bureau did not give the man's name or town of origin but said he was French.
"There has been no contact between him and his family since he left for the United States," he added.
The man was reported as a deserter to police in October and his details were sent to the international police force, Interpol.
http://www.sky.com/skynews/article/0,,0-12148625,00.html