In the following result, we will use the following Assumption 4 to replace Assumption 2 and give an upper bound of the precise large deviations of the partial sums (S_{n}), (ngeq1), of the WUOD r.v.s.
For the lower bound of the precise large deviations of the partial sums (S_{n}), (ngeq1), of the WOD r.v.s, when (mu_{i}=0), (igeq1), under Assumptions 1 and 3 and some other conditions, Theorem 2 of Wang et al. [20] obtained a lower bound: for every fixed (gamma>0), liminf_{ntoinfty}{inf_{xgeqgamma n}}frac {P(S_{n}>x)}{sum_{i=1}^{n} L_{F_{i}}overline{F_{i}}(x)} geq1.
For the upper bound of the precise large deviations of the partial sums (S_{n}), (ngeq1), of the WUOD r.v.s, when (mu_{i}=0), (igeq1), under Assumptions 1 and 2 and some other conditions, Theorem 1 of Wang et al. [20] gave an upper bound: for every fixed (gamma>0), limsup_{ntoinfty}{sup_{xgeqgamma n}}frac {P(S_{n}>x)}{sum_{i=1}^{n} L_{F_{i}}^{-1}overline{F_{i}}(x)} leq 1.
Under some mild conditions, the lower and upper bounds of the precise large deviations of the partial sums (S_{n}), (ngeq1), are presented.
Comparing Corollary 2.1 and Theorem of Paulauskas and Skučaitė [10], it can be found that they give the lower bound of the precise large deviations of (S_{n}, ngeq1), under different conditions.
When ({X_{i}, igeq1}) are independent and identically distributed r.v.s, some studies of the precise large deviations of the partial sums (S_{n}, ngeq1), can be found in Cline and Hsing [2], Heyde [3, 4], Heyde [5], Mikosch and Nagaev [6], Nagaev [7], Nagaev [8], Ng et al. [9] and so on.
The following result will still consider the WOD r.v.s (X_{i}) with finite means (mu_{i}), (igeq1), and only use Assumption 1 and some other conditions, without using Assumption 3, to obtain a lower bound of the precise large deviations of the partial sums (S_{n}), (ngeq1).
In Paulauskas and Skučaitė [10] and Skučaitė [11], the precise large deviations of a sum of independent but not identically distributed random variables were investigated.
