Samsung Galaxy s3 mini continuously rebooting
First method-


Reset the device (press and hold POWER button for 8-10 seconds).  


Second method-


Press and hold VOLUME UP and Home(center) button, then press Power until the phone vibrates. Wait until you see the Android logo then release all the buttons.
Select Wipe data/Factory reset with VOLUME DOWN key, press Power (right hand-side). 
Select YES -- this deletes all user data.
After format, press Power again to reboot phone (select reboot system now).
If your device is frozen and unresponsive, press and hold Samsung Galaxy S3 hard reset for 8-10 seconds. The device will reboot automatically.  


Third method-


Factory format entering sgs3 mini format code. Enter *2767*3855#.  


Note - Warning! All your data including contacts, messages etc. will be lost! Copy all your necessary data/Contacts/Messages etc. to SIM or make backup to your PC before full reset! Take out SIM card before full reset. Source: Hard-reset.com.