You can't win the game and your opponents can't lose the game.
11/17/2017 No game effect can cause you to win the game or cause any opponent to lose the game while you control Abyssal Persecutor. It doesn’t matter whether an opponent has 0 or less life, an opponent is forced to draw a card while their library is empty, an opponent has ten or more poison counters, an opponent is dealt combat damage by Phage the Untouchable, you control Felidar Sovereign and have 40 or more life, or so on. You keep playing.
11/17/2017 Other circumstances can still cause an opponent to lose the game, however. An opponent will lose a game if they concede, if that player is penalized with a Game Loss or a Match Loss during a sanctioned tournament due to a DCI rules infraction, or if that player’s Magic Online® game clock runs out of time.
11/17/2017 Effects that say the game is a draw, such as the _Legends_(TM) card Divine Intervention, are not affected by Abyssal Persecutor.
11/17/2017 Abyssal Persecutor won’t preclude an opponent’s life total from reaching 0 or less. It will just preclude that player from losing the game as a result.
11/17/2017 If Abyssal Persecutor leaves the battlefield while an opponent has 0 or less life, that opponent will lose the game as a state-based action. No player can respond between the time Abyssal Persecutor leaves the battlefield and the time that player loses the game.
11/17/2017 Even though your opponents can’t lose the game, a player can’t pay an amount of life that’s greater than their life total. If a player’s life total is 0 or less, that player can’t pay life at all, with one exception: a player may always pay 0 life.