An estimate for probabilities of large deviations (Q1103262)

Let \(X_{nk}\), \(1\leq k\leq k_ n\), \(n\geq 1\), be a sequence of series of i.i.d., for each n, random variables. Denote \[ S_ n=\sum^{k_ n}_{k=1}X_{nk},\quad \bar S_ n=\max_{1\leq \ell \leq k_ n}\sum^{\ell}_{k=1}X_{nk},\quad \theta_ n(u)=\sum^{K_ n}_{k=1}P(X_{nk}<u), \] \[ Q_ n(u)=\int_{| y| >u}d\theta_ n(y),\quad Q^+_ n(u)= \int^{\infty}_{u} dQ_ n(y). \] Assuming that \(\bar S_ n/\kappa_ n\to^{P}0\) for some \(\kappa_ n\), new propositions of the type \(P(S_ n>x)\sim Q^+_ n(x)\) or \(P(\bar S_ n>x)\sim Q^+_ n(x)\) are proved. Results of \textit{N. N. Amosova} [ibid. 35, No.1, 125-131 (1984; Zbl 0549.60025), English translation in Math. Notes 35, 68-71 (1984)], \textit{A. V. Nagaev} [Izvestija Akad. Nauk UzSSR, Ser. fiz.-mat. Nauk 13, No.6, 17- 22 (1970; Zbl 0226.60043)] and \textit{S. V. Nagaev} [Teor. Veroyatn. Primen. 26, 369-372 (1981; Zbl 0457.60020)] are generalized and refined.