I experienced the same behavior after a year and a half of usage of my s3 mini.
After some research and testing I found 2 reasons for this.
1. The main reason was that the battery was gradually holding less charge, so I decided to replace it with a cheap 1900mAh non-Samsung replacement, which turned out to be a bad idea....It worked ok for about 2 months but then my s3 would start randomly rebooting when the charge was below 75%. After some checking with a battery monitor app I found that the voltage was dropping below 3.0 volts (normally it should be running at between 3.5-4.0 volts). So I bought an original Samsung replacement (1500mAh), and this really made a big difference.
2. The second reason for faster discharging compared to when it was new is that a whole lot of apps are running in the background. I set my wifi advanced option 'keep wifi on during sleep' to only when plugged in (to a charger), which does help, and I only enable mobile data (3G) when I really need it.
This has definitely restored my battery life back to normal