Sure, but what exactly is a "private" war? Wars are supposed to be waged by states or countries, not individuals. So I'll ask you again, what's the difference between a serial killer/mass murderer and someone waging a private war?
Wars are also fought with soliders though. Anyone can do anything and consider themselves a solider for justice. Also terrorist consider themselves as fighting wars even when they don't have the support of a government. A serial killer in theory is just doing it for his own pleasure or to deal with some kind of anti-social behavior he has developed no doubt by having a really terrible life up to that point or some kind of mental illness that someone hasn't been able to help him deal with.
Jason