An interesting topic! I think the age of the child when they're being introduced to the game is very significant. A child's (such as your 9 year old's) brain simply isn't developed enough to draw clear lines between fantasy and reality, which means they'll take lessons from video games into the real world more readily than an adult would. We can play GTA and relax, knowing that when we turn off the game, that mentality of sandbox violence ends, and we can return to the real world.
The others made very valid points about other factors in the child's life (including things as simple as personality), but in the end I do think that children should be kept to age-appropriate games.PG for children, T for Teen, etc. I've seen young children (9-12) become extremely obsessed with M-rated video games, and worry about their development as a result.