Abstract. We present bounds of the error of the exponential approximation of the first occurrence time of a rare event in a stationary stochastic process with finite alphabet with $\alpha$-mixing property with a summable function $\alpha$ or with general $\phi$-mixing property. We prove a lower bound for this error in terms of the measure of the cylinder and an upper bound as a function of the measure of the cylinder plus the decay of correlation of the process.