This paper proposes a stochastic predator-prey model with infinite delay. It demonstrates that the dynamics of the model are governed by two thresholds, \( \mathscr{R}_{01} \) and \( \mathscr{R}_{02} \) . We draw conclusions: (1) if \( \mathscr{R}_{01} < 0 \) , both predator and prey populations become extinct; (2) if \( \mathscr{R}_{01} > 0 \) and \( \mathscr{R}_{02} < 0 \) , the predator population undergoes exponential extinction, while the prey population persists for a long time; (3) if \( \mathscr{R}_{02} > 0 \) , both predator and prey populations coexist for a long time. The established coexistence approach improves the existing literature Jiang et al., (2024). The approximate expression of the probability density function is derived and its unknown parameters are estimated by employing the maximum likelihood estimation.
