(2) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 2 of 18. on (–∞, ∞) belongs to the class D if, for any y ∈ (0, 1), lim sup x→∞. V (xy) V (x). < ∞.. Another slightly smaller class is the class C , which consists of all distributions with consistently varying tails. We say that a distribution V on (–∞, ∞) belongs to the class C if lim lim sup. V (xy). y1 x→∞. V (x). = 1,. or, equivalently,. lim lim inf. y1 x→∞. V (xy) V (x). = 1.. A subclass of the class C is the class of distributions with regularly varying tails. A distribution V on (–∞, ∞) is said to be regularly varying at inﬁnity with index α, denoted by V ∈ R–α , if lim. x→∞. V (xy) V (x). = y–α. holds for some 0 ≤ α < ∞ and all y > 0 (see, e.g., Bingham et al. [1]). For a distribution V , denote the upper Matuszewska index of V by JV+ = – lim. y→∞. log V ∗ (y) , log y. with V ∗ (y) := lim inf x→∞. V (xy) V (x). , y > 1.. Let LV = limy1 V ∗ (y). From Chapter 2.1 of Bingham et al. [1], we know that the following assertions are equivalent: (i) V ∈ D ;. (ii) 0 < LV ≤ 1;. (iii). JV+ < ∞.. From the deﬁnition of the class C , it holds that V ∈ C if and only if LV = 1. When {Xi , i ≥ 1} are independent and identically distributed r.v.s, some studies of the precise large deviations of the partial sums Sn , n ≥ 1, 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. 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. This paper considers the dependent r.v.s with diﬀerent distributions. We investigate the r.v.s with the wide dependence structure, which is introduced in Wang et al. [12]. Deﬁnition 1.1 For the r.v.s {ξn , n ≥ 1}, if there exists a ﬁnite real sequence {gU (n), n ≥ 1} satisfying, for each integer n ≥ 1 and for all xi ∈ (–∞, ∞), 1 ≤ i ≤ n, n n {ξi > xi } ≤ gU (n) P(ξi > xi ), P . i=1. (1.1). i=1. then we say that the r.v.s {ξn , n ≥ 1} are widely upper orthant dependent (WUOD) with dominating coeﬃcients gU (n), n ≥ 1; if there exists a ﬁnite real sequence {gL (n), n ≥ 1}.

(3) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 3 of 18. satisfying, for each integer n ≥ 1 and for all xi ∈ (–∞, ∞), 1 ≤ i ≤ n, n n {ξi ≤ xi } ≤ gL (n) P(ξi ≤ xi ), P . i=1. (1.2). i=1. then we say that the r.v.s {ξn , n ≥ 1} are widely lower orthant dependent (WLOD) with dominating coeﬃcients gL (n), n ≥ 1; if they are both WUOD and WLOD, then we say that the r.v.s {ξn , n ≥ 1} are widely orthant dependent (WOD). Deﬁnition 1.1 shows that the wide dependence structure contains the commonly used notions of the negatively upper/lower orthant dependence (see Ebrahimi and Ghosh [13] and Block et al. [14]) and the extendedly negatively orthant dependence (see Liu [15], Chen et al. [16] and Shen [17]). Here, we present an example of WOD r.v.s, which is the example of Wang et al. [12]. Example 1.1 Assume that the random vectors (ξn , ηn ), n = 1, 2, . . . , are independent and, for each integer n ≥ 1, the r.v.s ξn and ηn are dependent according to the Farlie-GumbelMorgenstern copula with the parameter θn ∈ [–1, 1]: Cθn (u, v) = uv + θn uv(1 – u)(1 – v),. (u, v) ∈ [0, 1]2 ,. which is absolutely continuous with density cθn (u, v) =. ∂ 2 Cθn (u, v) = 1 + θn (1 – 2u)(1 – 2v), ∂u ∂v. (u, v) ∈ [0, 1]2. (see, e.g., Example 3.12 in Nelsen [18]). Suppose that the distributions of ξn and ηn , n = 1, 2, . . . , are absolutely continuous, denoted by Fξn and Fηn , n = 1, 2, . . . , respectively. Hence, by Sklar’s theorem (see, e.g., Chapter 2 of Nelsen [18]), for each integer n ≥ 1 and any xn , yn ∈ (–∞, ∞), P(ξn ≤ xn , ηn ≤ yn ) = Cθn Fξn (xn ), Fηn (yn ) = Fξn (xn )Fηn (yn ) 1 + θn Fξn (xn )Fηn (yn ) and . 1. . 1. P(ξn > xn , ηn > yn ) =. cθn (u, v) du dv Fξn (xn ). Fηn (yn ). = Fξn (xn )Fηn (yn ) 1 + θn Fξn (xn )Fηn (yn ) . Therefore, for each n ≥ 1, we have a(θn ) :=. P(ξn ≤ xn , ηn ≤ yn ) xn ,yn ∈(–∞,∞) P(ξn ≤ xn )P(ηn ≤ yn ) sup. P(ξn > xn , ηn > yn ) xn ,yn ∈(–∞,∞) P(ξn > xn )P(ηn > yn ) ⎧ ⎨1 + θ 0 < θ ≤ 1; n n = ⎩1 –1 ≤ θn ≤ 0,. =. sup.

(4) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. where, by convention,. sup xi ,yi ∈(–∞,∞),i=1,...,n. =. 0 0. Page 4 of 18. = 1. Thus, for each integer n ≥ 1, we have. P(ξ1 ≤ x1 , η1 ≤ y1 , . . . , ξn ≤ xn , ηn ≤ yn ) n i=1 P(ξi ≤ xi )P(ηi ≤ yi ) ≤ xi , ηi ≤ yi ) a(θi ) = n i=1 P(ξi ≤ xi )P(ηi ≤ yi ) i=1 n. n i=1 P(ξi. sup xi ,yi ∈(–∞,∞),i=1,...,n. and sup xi ,yi ∈(–∞,∞),i=1,...,n. =. P(ξ1 ≤ x1 , η1 ≤ y1 , . . . , ξn ≤ xn ) n–1 i=1 P(ξi ≤ xi )P(ηi ≤ yi )P(ξn ≤ xn ). sup xi ,yi ∈(–∞,∞),i=1,...,n. where, by convention,. 0 i=1. n–1 i=1 P(ξi ≤ xi , ηi ≤ yi ) n–1 i=1 P(ξi ≤ xi )P(ηi ≤ yi ). =. n–1 . a(θi ),. i=1. = 1. Similarly, for each integer n ≥ 1, we have. P(ξ1 > x1 , η1 > y1 , . . . , ξn > xn , ηn > yn ) = a(θi ) n xi ,yi ∈(–∞,∞),i=1,...,n i=1 P(ξi > xi )P(ηi > yi ) i=1 n. sup. and P(ξ1 > x1 , η1 > y1 , . . . , ξn > xn ). sup xi ,yi ∈(–∞,∞),i=1,...,n. n–1 i=1 P(ξi. > xi )P(ηi > yi )P(ξn > xn ). =. n–1 . a(θi ).. i=1. Hence, for the r.v.s ξ1 , η1 , . . . , ξn , ηn , . . . , we can take ⎧ ⎨ gL (n) = gU (n) =. ⎩. m i=1 a(θi ) m–1 i=1 a(θi ). n = 2m, n = 2m – 1,. m ≥ 1,. which makes relations (1.1) and (1.2) be satisﬁed. That is to say, the r.v.s ξ1 , η1 , . . . , ξn , ηn , . . . , are WLOD and WUOD. The wide dependent structure has been applied to many ﬁelds such as risk theory (see, e.g., Liu et al. [19], Wang et al. [20], Wang et al. [12], Mao et al. [21]), renewal theory (see, e.g., Wang and Cheng [22], Chen et al. [23]), complete convergence (Wang and Cheng [22], Qiu and Chen [24], Wang et al. [25], Chen et al. [23]), precise large deviations (see, e.g., Wang et al. [26], He et al. [27]) and some statistic ﬁelds (see, e.g., Wang and Hu [28]). Wang et al. [12] gave the following properties of the wide dependent r.v.s. Proposition 1.1 (1) Let {ξn , n ≥ 1} be WLOD (WUOD) with dominating coeﬃcients gL (n)(gU (n)), n ≥ 1. If {fn (·), n ≥ 1} are nondecreasing, then {fn (ξn ), n ≥ 1} are still WLOD (WUOD) with dominating coeﬃcients gL (n)(gU (n)), n ≥ 1. If {fn (·), n ≥ 1} are nonincreasing, then {fn (ξn ), n ≥ 1} are WUOD (WLOD) with dominating coeﬃcients gL (n)(gU (n)), n ≥ 1..

(5) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 5 of 18. (2) If {ξn , n ≥ 1} are nonnegative and WUOD with dominating coeﬃcients gU (n), n ≥ 1, then, for each n ≥ 1,. E. n . ξi ≤ gU (n). i=1. n . Eξi .. i=1. In particular, if {ξn , n ≥ 1} are WUOD with dominating coeﬃcients gU (n), n ≥ 1, then, for each n ≥ 1 and any s > 0, E exp s. n . ξi ≤ gU (n). i=1. n . E exp{sξi }.. i=1. 2 Main results Now many studies of precise large deviations are focused on the dependent r.v.s. One can refer to Wang et al. [29], Liu [30], Tang [31], Liu [15], Yang and Wang [32], Wang et al. [20] and so on. Among them, Yang and Wang [32] consider the precise large deviations for extendedly negatively orthant dependent r.v.s, and Wang et al. [20] investigate the precise large deviations for WUOD and WLOD r.v.s. Their results have used the following assumptions. Assumption 1 For some T > 0, n. i=1 Fi (x). n ≤ lim sup sup. i=1 Fi (x). =: c2 < ∞, nF(x) nF(x) n→∞ x≥T n n Fi (–x) i=1 Fi (–x) 0 < c3 := lim inf inf ≤ lim sup sup i=1 =: c4 < ∞. n→∞ x≥T nF(–x) nF(–x) n→∞ x≥T 0 < c1 := lim inf inf. n→∞ x≥T. Assumption 2 For all i ≥ 1, Fi ∈ D . Furthermore, assume that for any ε > 0, there exist some w1 = w1 (ε) > 1 and x1 = x1 (ε) > 0, irrespective of i, such that for all i ≥ 1, 1 ≤ w ≤ w1 and x ≥ x1 , Fi (wx) Fi (x). ≥ LFi – ε,. or, equivalently, for any ε > 0, there exist some 0 < w2 = w2 (ε) < 1 and x2 = x2 (ε) > 0, irrespective of i, such that for all i ≥ 1, w2 ≤ w ≤ 1 and x ≥ x2 , Fi (wx) Fi (x). ≤ L–1 Fi + ε.. Assumption 3 For all i ≥ 1, Fi ∈ D . Furthermore, assume that for any 0 < δ < 1, there exist some v1 = v1 (δ) > 1 and x1 = x1 (δ) > 0, irrespective of i, such that for all i ≥ 1, 1 ≤ v ≤ v1 and x ≥ x1 , Fi (vx) Fi (x). ≥ δLFi ,.

(6) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 6 of 18. or, equivalently, for any δ > 1, there exist some 0 < v2 = v2 (δ) < 1 and x2 = x2 (δ) > 0, irrespective of i, such that for all i ≥ 1, v2 ≤ v ≤ 1 and x ≥ x2 , Fi (vx) Fi (x). ≤ δL–1 Fi .. For the lower bound of the precise large deviations of the partial sums Sn , n ≥ 1, of the WOD r.v.s, when μi = 0, i ≥ 1, under Assumptions 1 and 3 and some other conditions, Theorem 2 of Wang et al. [20] obtained a lower bound: for every ﬁxed γ > 0, P(Sn > x) lim inf inf n ≥ 1. n→∞ x≥γ n i=1 LFi Fi (x) The following result will still consider the WOD r.v.s Xi with ﬁnite means μi , i ≥ 1, 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 Sn , n ≥ 1. Theorem 2.1 Let {Xi , i ≥ 1} be a sequence of WOD r.v.s with dominating coeﬃcients gU (n) (n ≥ 1) and gL (n) (n ≥ 1) satisfying, for any α ∈ (0, 1), lim gU (n) nF(n). n→∞. α. = 0,. (2.1). and for any β ∈ (0, 1), lim gL (n)n–β = 0.. (2.2). n→∞. The distributions {Fi , i ≥ 1} and F satisfy Assumption 1, F ∈ D and xF(–x) = o F(x) .. (2.3). Suppose that, for some r > 1, sup E (μi – Xi )+. r. < ∞.. i≥1. Then, for every ﬁxed γ > 0, lim inf inf. n→∞ x≥γ n. P(Sn – E(Sn ) > x) nF(x). ≥ c1 LF .. (2.4). For the upper bound of the precise large deviations of the partial sums Sn , n ≥ 1, of the WUOD r.v.s, when μi = 0, i ≥ 1, under Assumptions 1 and 2 and some other conditions, Theorem 1 of Wang et al. [20] gave an upper bound: for every ﬁxed γ > 0, P(Sn > x) lim sup sup n –1 ≤ 1. n→∞ x≥γ n i=1 LFi Fi (x) 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 Sn , n ≥ 1, of the WUOD r.v.s. Assumption 4 is easier to verify than Assumption 2..

(7) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 7 of 18. Assumption 4 The expectations μi , i ≥ 1, satisfy. n. i=1 μi. = O(n).. Note that if supi≥1 μi < ∞ then Assumption 4 is satisﬁed. Particularly, the identically distributed random variables satisfy Assumption 4. Theorem 2.2 Let {Xi , i ≥ 1} be a sequence of WUOD r.v.s with dominating coeﬃcients gU (n) (n ≥ 1) satisfying (2.1). The distributions {Fi , i ≥ 1} and F satisfy Assumptions 1 and 4 and F ∈ D . Then, for every ﬁxed γ > 0, P(Sn – E(Sn ) > x). lim sup sup n→∞ x≥γ n. ≤ c2 LF –1 .. nF(x). (2.5). If we strengthen Assumption 1 to the following assumption, Assumption 4 can be dropped in Theorem 2.2. Assumption 1∗ n. i=1 Fi (x). n ≤ lim sup sup. i=1 Fi (x). =: c6 < ∞, nF(x) nF(x) n→∞ x≥0 n n Fi (–x) i=1 Fi (–x) ≤ lim sup sup i=1 =: c8 < ∞. 0 < c7 := lim inf inf n→∞ x≥0 nF(–x) nF(–x) n→∞ x≥0 0 < c5 := lim inf inf. n→∞ x≥0. Theorem 2.3 Let {Xi , i ≥ 1} be a sequence of WUOD r.v.s with dominating coeﬃcients gU (n) (n ≥ 1) satisfying (2.1). The distributions {Fi , i ≥ 1} and F satisfy Assumptions 1∗ and F ∈ D . Then, for every ﬁxed γ > 0, P(Sn – E(Sn ) > x). lim sup sup n→∞ x≥γ n. nF(x). ≤ c6 LF –1 .. When {Xi , i ≥ 1} are independent but non-identically distributed r.v.s, then gU (n) = gL (n) ≡ 1, n ≥ 1, and (2.1) and (2.2) hold. By Theorems 2.1 and 2.2, the following two corollaries can be obtained. Corollary 2.1 Let {Xi , i ≥ 1} be a sequence of independent but non-identically distributed r.v.s. The distributions {Fi , i ≥ 1} and F satisfy Assumption 1, F ∈ C and (2.3). Suppose that, for some r > 1, supi≥1 E((μi – Xi )+ )r < ∞. Then, for every ﬁxed γ > 0, lim inf inf. P(Sn – E(Sn ) > x). n→∞ x≥γ n. nF(x). ≥ c1 .. (2.6). Corollary 2.2 Let {Xi , i ≥ 1} be a sequence of independent but non-identically distributed r.v.s. The distributions {Fi , i ≥ 1} and F satisfy Assumptions 1 and 4 and F ∈ C . Then, for every ﬁxed γ > 0, lim sup sup n→∞ x≥γ n. P(Sn – E(Sn ) > x) nF(x). ≤ c2 .. (2.7). Remark 2.1 In Theorem of Paulauskas and Skučaitė [10], the case that {Xi , i ≥ 1} is a sequence of independent but non-identically distributed r.v.s was also considered, and the following result was obtained..

(8) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 8 of 18. Let {Xi , i ≥ 1} be a sequence of independent but non-identically distributed r.v.s. Assume that: (1) μi = 0; (2) F ∈ R–α for α > 1 and the distributions {Fi , i ≥ 1} and F satisfy Assumption 1 for ci ≡ 1, i = 1, 2, 3, 4. (3) There exists a sequence of constants an such that an ↑ ∞, sup an n–1 < ∞ and ∞ –p p n=1 an E|Xn | < ∞ for some 1 < p ≤ 2. Then lim. P(Sn > tn ). n→∞. nF(tn ). =1. (2.8). for all sequences tn ∈ (–∞, ∞) satisfying the conditions lim supn→∞ ntn–1 < ∞ and Fi (tn ) = o(nF(tn )), i ≥ 1. From the proof of Theorem of Paulauskas and Skučaitė [10], we note that tn should be positive. If conditions (1) and (2) hold, then the conditions of Corollary 2.2 are satisﬁed. If we let a = limn→∞ sup ntn–1 , then a ∈ (0, ∞) and tn ≥ (a + 1)–1 n for large n. Thus, it follows from (2.7) that lim sup n→∞. P(Sn > tn ). ≤ 1,. nF(tn ). which means that the upper bound of P(Sn > tn ) in (2.8) can be obtained from Corollary 2.2. 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 Sn , n ≥ 1, under diﬀerent conditions. When {Xi , i ≥ 1} and X are identically distributed r.v.s, Assumptions 1 and 1∗ are satisﬁed. The following two corollaries can be obtained directly from Theorems 2.1 and 2.3. Corollary 2.3 Let {Xi , i ≥ 1, X} be identically distributed r.v.s with common distribution F ∈ D . Assume that {Xi , i ≥ 1} are WOD r.v.s with dominating coeﬃcients gU (n) (n ≥ 1) and gL (n) (n ≥ 1) satisfying (2.1) and (2.2). If E(X – )r < ∞ for some r > 1 and relation (2.3) holds, then for every ﬁxed γ > 0, lim inf inf. P(Sn – E(Sn ) > x). n→∞ x≥γ n. nF(x). ≥ LF .. Corollary 2.4 Let {Xi , i ≥ 1, X} be identically distributed r.v.s with common distribution F ∈ D . Assume that {Xi , i ≥ 1} are WUOD r.v.s with dominating coeﬃcients gU (n) (n ≥ 1) satisfying (2.1). Then, for every ﬁxed γ > 0, lim sup sup n→∞ x≥γ n. P(Sn – E(Sn ) > x) nF(x). ≤ L–1 F .. Remark 2.2 (1) We note that, for any ﬁxed d > 0 and r > 1, r E (d – X)+ < ∞. ⇔. E X–. r. < ∞.. (2.9).

(9) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 9 of 18. In fact, on the one hand, r E (d – X)+ = E(d – X)r 1{0≤X≤d} + E(d – X)r 1{X<0} r ≥ E X– . On the other hand, by Cr -inequality, we have r E (d – X)+ ≤ E d + X –. r. ≤ 2r–1 dr + E X –. r. .. Thus (2.9) can be obtained. (2) Corollaries 1 and 2 of Wang et al. [20] consider the identically distributed r.v.s Xi (i ≥ 1) with ﬁnite mean μi = 0 (i ≥ 1). Corollaries 2.3 and 2.4 deal with the case that μi = 0 (i ≥ 1). We note that when μi = 0, i ≥ 1, Corollaries 2.3 and 2.4 cannot be obtained directly from Corollaries 1 and 2 of Wang et al. [20].. 3 Proofs of results 3.1 Some lemmas Before proving the main results, we ﬁrst give some lemmas. The following lemma is a combination of Proposition 2.2.1 of Bingham et al. [1] and Lemma 3.5 of Tang and Tsitsiashvili [33]. Lemma 3.1 If V ∈ D , then (1) for each ρ > JV+ , there exist positive constants A and B such that the inequality ρ x ≤A y V (x) V (y). holds for all x ≥ y ≥ B; (2) it holds for each ρ > JV+ that x–ρ = o V (x) . Lemma 3.2 Let {ξk , k ≥ 1} be WUOD r.v.s with dominating coeﬃcients gU (n) (n ≥ 1), with distributions {Vk , k ≥ 1} and ﬁnite mean 0, satisfying supk≥1 E(ξk+ )r < ∞ for some r > 1. Then, for each ﬁxed γ > 0 and p > 0, there exist positive numbers v and C = C(v, γ ), irrespective of x and n, such that for all n = 1, 2, . . . and x ≥ γ n, P. n k=1. ξk ≥ x ≤. n . Vk (vx) + CgU (n)x–p .. k=1. This lemma extends Lemma 2.3 of Tang [31] to the WUOD r.v.s with diﬀerent distributions. The proof is similar to the proof of Lemma 2.3 of Tang [31]. However, for the completeness of the proof, we give the following proof with some modiﬁcations..

(10) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 10 of 18. Proof If we prove the result is correct for all n > n0 and x ≥ γ n, where n0 is a positive integer, then using the inequality P. n k=1. . n x ξk ≥ x ≤ P ξk ≥ n0 k=1. =. . n . Vk. k=1. x , n0. n = 1, 2, . . . , n0 ,. (3.1). the result can be extended to all n = 1, 2, . . . . For any ﬁxed v > 0, we denote ξk = min{ξk , vx}, k = 1, 2, . . . , by Proposition 1.1(1), they are still WUOD with dominating coeﬃcients gU (n), n ≥ 1. Using a standard truncation argument, we get P. n . . . ξk ≥ x = P. k=1. n . ξk ≥ x, max ξk > vx + P 1≤k≤n. k=1. ≤. n . . . Vk (vx) + P. k=1. n . . n k=1. ξk ≥ x, max ξk ≤ vx 1≤k≤n. ξk ≥ x .. (3.2). k=1. Now, we estimate the second term in (3.2). For a positive number h, which we shall specify later, using Chebyshev’s inequality and Proposition 1.1(2), we have P. n . ξk ≥ x ≤ gU (n)e–hx. n . . Eehξk .. (3.3). i=1. k=1 . For some 1 < q < min{r, 2}, Eehξk is bounded from above by . 0. . . vx. ehu – 1 Vk (du) +. –∞. 0. ehu – 1 – hu q u Vk (du) + ehvx – 1 Vk (vx) + hu+k + 1, q u. (3.4). here u+k = Eξk 1{ξk >0} . For the ﬁrst term in (3.4), since 0≤. ehu – 1 – hu ≤ u ehu – 1 ≤ –u h. for all u ≤ 0,. by the dominated convergence theorem, we have 0. hu –∞ (e. lim. h0. – 1)Vk (du) = lim h0 h. . 0 –∞. ehu – 1 – hu Vk (du) + u–k = u–k , h. where u–k = Eξk 1{ξk ≤0} . Hence, there exists a real function α(·) with α(h) → 0 as h 0 such that . 0. –∞. . ehu – 1 Vk (du) = 1 + α(h) hu–k .. (3.5).

(11) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 11 of 18. By the monotonicity of (ehu – 1 – hu)/uq for u ∈ (0, ∞), the second term in (3.4) is bounded by ehvx – 1 – hvx + q E ξk , (vx)q. (3.6). where ξk+ = max{ξk , 0}. Applying (3.5) and (3.6) to (3.4), from (3.3) we obtain that P. n . ξk ≥ x. k=1 n ehvx – 1 – hvx + 1 + α(h) hu–k + E ξk (vx)q k=1 + ehvx – 1 Vk (vx) + hu+k + 1. ≤ gU (n)e–hx. ≤ gU (n)e–hx. n k=1. = gU (n) exp. ehvx – 1 + exp 1 + α(h) hu–k + E ξk (vx)q. n k=1. ehvx – 1 + α(h)hu–k + E ξk (vx)q n. k=1. q. . q. q. + ehvx – 1 Vk (vx) + hu+k. + ehvx – 1. n . . Vk (vx) – hx , (3.7). k=1. where at the second step we use the inequality s + 1 ≤ es for all s. In (3.7), take h=. 1 vq–1 xq log n + 1 . + q vx k=1 E(ξk ). Since supk≥1 E(ξk )r < ∞, there exists a constant M1 > 0, irrespective of x and n, such that for all n = 1, 2, . . . , n n – uk = u+k ≤ nM1 . k=1. k=1. By some calculation, we know that, for all large n such that for all x ≥ γ n, α(h)M1 ≤ 1 γ . 2 Then, for all large n and x ≥ γ n, n 1 x – α(h)uk ≤ γ n ≤ . 2 2 k=1. Therefore, for all large n and x ≥ γ n, the right-hand side of (3.7) is bounded from above by vq–1 xq 1 1 vq–1 xq nk=1 Vk (vx) gU (n) exp log n + + 1 + n + q + q 2v v k=1 E(ξk ) k=1 E(ξk ) vq–1 xq 1 – log n + 1 + q v k=1 E(ξk ).

(12) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 12 of 18. – 2v1 vq–1 xq 1 vq–1 xq nk=1 Vk (vx) n n + ≤ gU (n) exp + q + q v k=1 E(ξk ) k=1 E(ξk ) – 2v1 q–1 vq–1 x 1 vq–1 xq nk=1 Vk (vx) n + = gU (n) exp x– 2v n + q + q v E(ξ ) E(ξ ) k=1 k=1 k k – 2v1 n q–1 1 vq–1 xq k=1 Vk (vx) vq–1 γ n ≤ gU (n) exp x– 2v . + + q + q v maxk≥1 E(ξk ) k=1 E(ξk ) . (3.8). Since, for each k = 1, 2, . . . and for all x > 0, E ξk+. q. ≥. ∞. uq Vk (du) ≥ xq Vk (x),. x. then the right-hand side of (3.8) is bounded from above by. gU (n)e. 2 v. . vq–1 γ maxk≥1 E(ξk+ )q q–1. 2. – 2v1. x–. q–1 2v. =: CgU (n)x–. q–1 2v. ≤ CgU (n)x–p ,. 1. γ – where C = e v ( max v E(ξ + q ) 2v < ∞. In the last step, we take a proper v > 0 such that k≥1 k) This completes the proof of this lemma.. q–1 2v. > p. . Lemma 3.3 Assumption 1∗ implies Assumptions 1 and 4. Proof It is clear that Assumption 1∗ implies Assumption 1. Now we prove Assumption 1∗ implies Assumption 4. For suﬃciently large n, by Assumption 1∗ , we have n . |μi | ≤. i=1. n . ∞. Fi (y) + Fi (–y) dy. 0. i–1. . ∞. c6 F(y) + c8 F(–y) dy < ∞,. ≤ 2n 0. . which means that Assumption 4 holds. This completes the proof of this lemma.. 3.2 Proof of Theorem 2.1 We use the line of proof of Theorem 3.1 in Ng et al. [9] to prove this result. For any λ > 1, P Sn – E(Sn ) > x n n ≥ P Sn – μk > x, {Xj > λx}. ≥. n j=1. ≥. n . . P Sn – P Sn –. j=1. ≥. n j=1. j=1. k=1 n . μk > x, Xj > λx –. k=1 n . . μk > x, Xj > λx –. P S n – Xj –. P Sn –. 1≤j<l≤n. . n . μk > x, Xj > λx, Xl > λx. k=1. P(Xj > λx, Xl > λx). 1≤j<l≤n. k=1. . . . n k=1. μk > x – λx, Xj > λx – gU (n). . n j=1. 2 Fj (λx).

(13) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. ≥. n . Fj (λx) –. j=1 n . =. . n . P. Sn(j). –. n . j=1. μk ≤ (1 – λ)x – gU (n). k=1. Fj (λx) 1 – gU (n). n . j=1. Page 13 of 18. . Fj (λx) –. j=1. n . P Sn(j) –. j=1. n . 2 Fj (λx). j=1 n . . μk ≤ (1 – λ)x. k=1. =: I1 – I2 ,. (3.9). (j) where Sn = 1≤k =j≤n Xk . In the fourth step, the deﬁnition of WUOD was used, and we use an elementary inequality P(AB) ≥ P(B) – P(Ac ) for all events A and B in the ﬁfth step. We estimate the second term in (3.9). For all large n and x ≥ γ n, we have I2 ≤. n (λ – 1)x . P (μk – Xk ) ≥ 2 j=1 1≤k =j≤n. By Proposition 1.1(1), the r.v.s {μk – Xk , k ≥ 1} are WUOD with dominating coeﬃcients gL (n), n ≥ 1. Then, for arbitrarily ﬁxed γ > 0 and β ∈ (0, 1), by Lemma 3.2, there exist positive constants v0 and C, irrespective of x and n, such that n (λ – 1)x P (μk – Xk ) ≥ 2 j=1 1≤k =j≤n. ≤. n j=1 1≤k =j≤n. ≤n. n . Fk. k=1. (λ – 1)x + P μk – Xk ≥ + CngL (n)x–(β+JF ) 2v0. –(λ – 1)x + + CngL (n)x–(β+JF ) 4v0. holds for all large n and x ≥ γ n. By Lemma 3.1(2), F ∈ D and (2.2), for all large n and x ≥ γ n, β. +. β. +. ngL (n)x–(β+JF ) = ngL (n)x– 2 x–( 2 +JF ) β. β. β. +. ≤ ngL (n)n– 2 γ – 2 x–( 2 +JF ) = o nF(λx) . By Assumption 1, F ∈ D and (2.3), n lim sup sup. k=1 Fk (. n→∞ x≥γ n. ≤ lim sup sup. –(λ–1)x ) 4v0. F(λx) n –(λ–1)x k=1 Fk ( 4v0 ). n→∞ x≥γ n. nF( –(λ–1)x ) 4v0. γ. –1. lim sup x→∞. xF( –(λ–1)x ) 4v0 F( (λ–1)x ) 4v0. lim sup x→∞. F( (λ–1)x ) 4v0 F(λx). = 0. Therefore, it holds that lim sup sup. I2. n→∞ x≥γ n nF(λx). = 0.. (3.10).

(14) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 14 of 18. Now we estimate I1 . By (2.1), F ∈ D and Assumption 1, lim sup sup gU (n). n . n→∞ x≥γ n. Fj (λx). j=1. ≤ lim sup sup gU (n)nF(n). F(λγ n) F(n). n→∞ x≥γ n. ≤ lim sup gU (n)nF(n) lim sup n→∞. n. j=1 Fj (λx). nF(λx). F(λγ n) F(n). n→∞. n. lim sup sup n→∞ x≥γ n. j=1 Fj (λx). nF(λx). = 0. By Assumption 1, we have that. lim inf inf. n. I1. n→∞ x≥γ n nF(λx). ≥ lim inf inf. n→∞ x≥γ n. j=1 Fj (λx). nF(λx). ≥ c1 .. (3.11). Thus, by (3.9)-(3.11), lim inf inf. P(Sn – E(Sn ) > x). ≥ c1 lim lim inf. nF(x). n→∞ x≥γ n. λ1 x→∞. F(λx) F(x). = c1 LF .. The proof of Theorem 2.1 is completed.. 3.3 Proof of Theorem 2.2 m For any ﬁxed positive integer m and for any θ ∈ (0, m+1 ), we deﬁne Xk := min{Xk , θ x}, k ≥ 1, n n Sn := k=1 Xk , n ≥ 1 and xn := x + k=1 μk , n ≥ 1. By a standard truncation argument, we have P Sn – E(Sn ) > x n n μk > x, max Xk > θ x + P Sn – μk > x, max Xk ≤ θ x = P Sn – 1≤k≤n. k=1. ≤. n . k=1. 1≤k≤n. P(Xk > θ x) + P(Sn > xn ).. (3.12). k=1. We estimate the second term in (3.12). Let a = max{–m–1 log(nF(θ x)), 1}, which tends to ∞ uniformly for x ≥ γ n when n tends to ∞. For any ﬁxed h = h(x, n) > 0, when n is suﬃciently large, we have P(Sn > xn ) nF(θ x) ≤ gU (n)ema–hxn. n . . EehXk. k=1. = gU (n)ema–hxn. n k=1. ehy – 1 Fk (dy) + ehθx – 1 Fk (θ x) + 1. θx –∞.

(15) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. . n . ≤ gU (n) exp ma – h xn +. k=1. . θx . Page 15 of 18. n e – 1 Fk (dy) + ehθx – 1 Fk (θ x). –∞. k=1. n hθ x 2 ea xn + ≤ gU (n) exp ma – h n + ehθx – 1 Fk (θ x). . θx a2. hθx. htFk (dy) + e. Fk. –∞. k=1. . hy. θx a2. . . k=1. xn + e ≤ gU (n) exp ma – h. hθ x a2. h. n . hθx. μk + e. k=1. . n k=1. Fk. n hθx θx Fk (θ x) + e –1 a2 k=1. . n n hθ x θx μk + ehθx Fk 2 = gU (n) exp ma – hx + h e a2 – 1 a k=1 k=1 n hθx Fk (θ x) . + e –1. . (3.13). k=1. For ρ > JF+ , take h =. ma–2ρ log a . θx. By Assumption 4, for all large n and x ≥ γ n, it holds that. n n m –ρ log a2 hθ x μk = h e a e a2 – 1 μk h e a2 – 1 k=1. k=1. = o(1)hn = o(hx).. (3.14). For any δ1 > 0, by Assumption 1 and Lemma 3.1(1), for all large n and x ≥ γ n, it holds that ehθx. n . Fk. k=1. θx a2. . ≤ ehθx nF. θx (c2 + δ1 ) a2. ≤ ehθx nAa2ρ F(θ x)(c2 + δ1 ) = A(c2 + δ1 ).. (3.15). For the above δ1 > 0, by Assumption 1, for all large n and x ≥ γ n, it holds that . ehθx – 1. n . Fk (θ x) ≤ ehθx – 1 nF(θ x)(c2 + δ1 ). k=1. = a–2ρ – e–ma (c2 + δ1 ) = o(1).. (3.16). Hence, by (3.13)-(3.16), for all large n and x ≥ γ n, P(Sn > xn ) nF(θ x). ≤ gU (n)e–a exp ma – hx + o(hx) + A(c2 + δ1 ) + o(1) + a . = gU (n) nF(θ x). 1 m. m exp a m + 1 – + o(1) + A(c2 + δ1 ) + o(1) . θ.

(16) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 16 of 18. By (2.1) and F ∈ D , lim sup sup gU (n) nF(θ x). 1 m. = 0.. n→∞ x≥γ n. Since m + 1 –. < 0 and lim infn→∞ infx≥γ n a = ∞, it holds that. m θ. lim sup sup. P(Sn > xn ) nF(θ x). n→∞ x≥γ n. = 0.. Hence, by (3.12),. lim sup sup. P(Sn – E(Sn ) > x) nF(θ x). n→∞ x≥γ n. n ≤ lim sup sup. n→∞ x≥γ n. k=1 Fk (θ x). nF(θ x). ≤ c2 .. (3.17). By the deﬁnition of LF , for any ε > 0, there exist x0 > 0 and 0 < θ0 < 1 such that for all x ≥ x0 and θ0 ≤ θ ≤ 1, F(θ x) F(x). ≤ L–1 F + ε.. (3.18). Take a positive integer m such that θ ≤ lim sup sup. P(Sn – E(Sn ) > x) nF(x). n→∞ x≥γ n. m . Then, for all θ0 m+1. ≤θ ≤. m , by (3.17) and (3.18), m+1. ≤ c2 L–1 F +ε .. By the arbitrariness of ε, it holds that lim sup sup. P(Sn – E(Sn ) > x) nF(x). n→∞ x≥γ n. ≤ c2 L–1 F .. This completes the proof of Theorem 2.2.. 3.4 Proof of Theorem 2.3 By Lemma 3.3, we know that Assumptions 1 and 4 are satisﬁed, and for any ﬁxed T > 0, n lim sup sup n→∞ x≥T. i=1 Fi (x). nF(x). n. i=1 Fi (x). ≤ lim sup sup n→∞. nF(x). x≥0. = c6 .. Thus, by Theorem 2.2, for every ﬁxed γ > 0 and some ﬁxed T > 0,. lim sup sup n→∞ x≥γ n. P(Sn – E(Sn ) > x) nF(x). n ≤ LF. –1. lim sup sup n→∞ x≥T. ≤ c6 LF –1 . This completes the proof of Theorem 2.3.. i=1 Fi (x). nF(x).

(17) Gao et al. Journal of Inequalities and Applications (2018) 2018:21. Page 17 of 18. Acknowledgements The authors wish to thank the referees and Prof. Yuebao Wang for their valuable comments on an earlier version of this paper. This work is supported by the National Natural Science Foundation of China (No. 11401418), the 333 Talent Training Project of Jiangsu Province, the Jiangsu Province Key Discipline in the 13th Five-Year Plan, the Postgraduate Research & Practice Innovation Program of SUST (No. SKCX16_056) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17_2058). Competing interests The authors declare that they have no competing interests. Authors’ contributions MG found the main reference of this paper in the literature study and read it with LC in the corresponding author KW’s workshop. Then KW put forward the main problem and some ideas and methods to deal with the problem. 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