Limit theorems in a boundary crossing problem for random walks (Q1975810)

Let \(\xi_1, \xi_2, \dots\) be independent identically distributed random variables satisfying the Cramér condition, \(S_n=\xi_1+\cdots+\xi_n\). For \(a\geq 0\) and \(b>0\), put \(N=N_{a,b}=\min\{n\geq 1:S_n\notin[-a,b)\}\). Asymptotic representations for the joint distribution of the pair \((N,S_N)\), as \(a+b\to\infty\), are suggested. The results are formulated in terms of the factorization components. The special case in which \({\mathbb E} \xi_1=0\) and \(a,b\to\infty\) was previously treated by the author [Theory Probab. Appl. 24, 480-491 (1980), translation from Teor. Veroyatn. Primen. 24, 475-485 (1979; Zbl 0408.60050); ibid. 24, 869-876 (1980) resp. ibid. 24, 873-879 (1979; Zbl 0416.60059) and Tr. Inst. Mat. 1, 18-25 (1982; Zbl 0513.60069)].