I was battling with frequent reboots of my Sony XPERIA V. Particularly under high load when playing a 3D game or making heavy use of the camera my phone would get really hot and simply switch off.
My first (and it turns out guilty) suspect was the battery, but the annoying thing was that this behaviour just happened to co-incide with a recent upgrade to Android 4.3. I wanted to be sure It was the battery before I forked out the cash for a replacement so I diligently downgraded back to Android 4.1.2 and experienced the same stability problems.
I am now running Android 4.3 on my Sony XPERIA V with a new Sony BA800 battery and just completed 5 levels of Real Racing 3 without any trouble - I couldn't even complete one level with the old battery. So Problem solved!
P.S. Walk-in retail stores in South Africa don't carry a lot of Sony XPERIA accessories but I was able to find the battery online at xperiastore.co.za for about as much as I was expecting to pay.