Let be a stochastic sequence. Define the level-crossing time by
Using techniques of large deviations theory the author obtains and under certain regularity conditions and thereby he obtains exponential upper and lower bounds of for large . As examples his regularity conditions are satisfied by sums of i.i.d. sequences, Markov additive processes and partial sums of moving averages. His results extend and complement those of A. Martin-Löf [in: Probability and mathematical statistics, Essays in Hon. of C.-G. Esseen, 129-139 (1983; Zbl 0518.62085)], T. Lehtonen and H. Nyrhinen [Adv. Appl. Probab. 24, No. 4, 858-874 (1992; Zbl 0779.65003) and Scand. Actuarial J. 1992, No. 1, 60-75 (1992; Zbl 0755.62080)] and of many others.