Stamina will never increase without training. So if you have atrocious stamina it will never go up unless you train it.
However, every week stamina decreases a tiny little bit. Once it drops enough over the season to drop a level (i.e. respectable down to average) then it will show a down arrow to signify that the stamina level has dropped that week. This has nothing to do with how many minutes you have played the player. The arrow will disappear at the next training update.
In general, I expect each player to drop about once in stamina per season.