Asymptotic expansions for the maximum of random number of random variables (Q1336976)

Let \(Y_ 1,Y_ 2,\dots\) be an i.i.d. sequence of nonnegative r.v.s. Denote \(S_ 0=0\), \(S_ n = \sum_{i \leq n}Y_ i\), \(M_ n = \max_{i \leq n}Y_ i\), and consider \(Z(t) = \max \{t-S_ \nu; \max_{i \leq \nu}Y_ i\}\), where the stopping time \(\nu = \nu (t) = \max \{m \geq 0; S_ m \leq t\}\). The author investigates convergence rates and asymptotic expansions in the limit theorem for \(Z(t)\). This is an extension and considerable improvement of earlier results of the author [Theory Probab. Appl. 36, No. 4, 714-721 (1991); translation from Teor. Veroyatn. Primen. 36, No. 4, 675-681 (1991; Zbl 0748.60029)]. For example, the exponential moment condition is now replaced by the second moment.