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Large deviations for randomly weighted sums with dominantly varying tails and widely orthant dependent structure

Abstract

We prove large deviation inequalities for the randomly weighted partial and random sums S n θ = i = 1 n θ i X i , n1; S c θ (t)= i = 1 N ( t ) ( θ i X i +c), cR, where {N(t),t0} is a counting process, { θ i ,i1} is a sequence of positive random variables with two-sided bounds, and { X i ,i1} is a sequence of non-identically distributed real-valued random variables, while the three random sources above are mutually independent. Special attention is paid to the distribution of dominated variation and the widely orthant dependence structure.

MSC:62E20, 62H20, 62P05.

1 Introduction

Let { X k ,k1} be a sequence of real-valued random variables (r.v.s) with X k ’s distribution function (d.f.) F k (x)=1 F ¯ k (x) and μ k =E X k =0 for every k1, and { θ k ,k1} be another sequence of positive random variables, satisfying P r (a θ k b)=1, k1, where 0<ab<. {N(t),t0} denotes a counting process (that is, a non-negative, non-decreasing, and integer-valued stochastic process) with a finite mean function λ(t) for t0 and λ(t) as t. Besides, the three random sources above are mutually independent. Denote S n θ = i = 1 n θ i X i , n1 and S c θ (t)= i = 1 N ( t ) ( θ i X i +c), t0, cR. By convention, the summation over an empty set of indices produces a value of 0. In the present paper, we are interested in the probabilities of large deviations of { S n θ } and { S c θ (t)} in the situation that { X k ,k1} are heavy-tailed and widely orthant dependent.

Since the theory of large deviations with heavy tails is widely used in insurance and finance, in recent decades, there have been a series of articles devoted to related problems. For more details, please refer to Embrechts et al.[1], Klüppelberg and Mikosch [2], Mikosch and Nagaev [3] and references therein. Recently Tang [4] extended the asymptotic behavior of large deviation probabilities of partial sums of heavy-tailed random variables to the case of negatively dependent ones. Under the assumption that random variables are non-identically distributed and extended negatively dependent, Liu [5] obtained a result similar to the one in the above paper, which was promoted to random sums in various situations later by Chen et al.[6]. Specially, Shen and Lin [7] investigated large deviations of randomly weighted partial sums with negatively dependent and consistently varying-tailed random variables, but, unfortunately, there are some flaws in their proofs.

In this paper, motivated by the work of Liu and Hu [8] and Chen et al.[6], on the one hand, we aim to prove that for each fixed γ>0, there exist positive constant M 1 and M 2 such that the inequalities

P r ( S n θ > x ) M 1 ( 1 + o ( 1 ) ) k = 1 n P r ( θ k X k >x)
(1.1)

and

P r ( S n θ > x ) M 2 ( 1 + o ( 1 ) ) k = 1 n P r ( θ k X k >x)
(1.2)

hold uniformly for all xγn as n, respectively; on the other hand, for arbitrarily fixed γ>c (c is an arbitrarily given real number), there are positive constant M ˜ 1 and M ˜ 2 such that

P r ( S c θ ( t ) > x ) M ˜ 1 ( 1 + o ( 1 ) ) E [ k = 1 N ( t ) P r ( θ k X k > x c λ ( t ) ) ]
(1.3)

and

P r ( S c θ ( t ) > x ) M ˜ 2 ( 1 + o ( 1 ) ) E [ k = 1 N ( t ) P r ( θ k X k > x c λ ( t ) ) ]
(1.4)

hold uniformly for all xγλ(t) as t, respectively.

The paper is organized as follows. Section 2 presents our main results after recalling some preliminaries. Sections 3 and 4 prove Theorems 2.1 and 2.2, respectively.

2 Main results

We say that a random variable X or its distribution function is heavy-tailed if E e t X = for all t>0. An important class of heavy-tailed distributions is , which consists of all distributions with dominated variation in the sense that the relation lim sup x F ¯ (xy)/ F ¯ (x)< holds for some (hence for all) 0<y<1. Recall the upper/lower Matuszewska index of distribution F, defined as J F + = lim y (log F ¯ (y)/logy) and J F = lim y (log F ¯ (y)/logy), where F ¯ (y)= lim inf x F ¯ (xy)/ F ¯ (x) and F ¯ (y)= lim sup x F ¯ (xy)/ F ¯ (x) for any y>0. From Lemma 3.5 of Tang and Tsitsiashvili [9], we know that if FD, then 0 J F J F + <, and for arbitrary p 1 < J F and p 2 > J F + , there exist positive constant C ˜ i and D ˜ i , i=1,2, such that

F ¯ (y)/ F ¯ (x) C ˜ 1 ( x / y ) p 1
(2.1)

holds for all xy D ˜ 1 , and

F ¯ (y)/ F ¯ (x) C ˜ 2 ( x / y ) p 2
(2.2)

holds for all xy D ˜ 2 . Hence, for any p 1 < J F , we have

F ¯ (x)=o ( x p 1 ) ;
(2.3)

and for any p 2 > J F + ,

x p 2 =o ( F ¯ ( x ) ) .
(2.4)

Furthermore, if the distribution F + (x)=F(x) 1 ( x 0 ) has a finite mean, then J F + 1.

Now, we present some new dependence structures introduced in Wang et al.[10].

Definition 2.1 We say that the r.v.s { η n ,n1} are widely upper orthant dependent (WUOD) if there exists a finite real sequence { g U (n),n1} satisfying for each n1 and for all x i (,+), 1in,

P r ( i = 1 n { η i > x i } ) g U (n) i = 1 n P r ( η i > x i );
(2.5)

we say that the r.v.s { η n ,n1} are widely lower orthant dependent (WLOD) if there exists a finite real sequence { g L (n),n1} satisfying for each n1 and for all x i (,+), 1in,

P r ( i = 1 n { η i x i } ) g L (n) i = 1 n P r ( η i x i );
(2.6)

if they are both WUOD and WLOD, then we say that the r.v.s { η n ,n1} are widely orthant dependent (WOD). WUOD, WLOD, and WOD r.v.s are called, by a joint name, wide dependence (WD) r.v.s, and g U (n), g L (n), n1, are called dominating coefficients.

Wang et al.[10] also gave some examples of WD r.v.s with various dominating coefficients which show that WD r.v.s contain some common negatively dependent r.v.s, some positively dependent r.v.s and some others.

From the definitions of WD, the following proposition can be obtained directly (see, e.g., Wang et al.[10]).

Proposition 2.1 (1) Let{ η n ,n1}be WUOD (WLOD) with dominating coefficients g U (n), n1 ( g L (n), n1). If{ f n (),n1}are non-decreasing, then{ f n ( η n ),n1}are still WUOD (WLOD) with dominating coefficients g U (n), n1 ( g L (n), n1); if{ f n (),n1}are non-increasing, then{ f n ( η n ),n1}are WLOD (WUOD) with dominating coefficients g U (n), n1 ( g L (n), n1).

(2) If{ η n ,n1}are non-negative and WUOD with dominating coefficients g U (n), n1, then for eachn1,

E i = 1 n η i g U (n) i = 1 n E η i .

In particular, if{ η n ,n1}are WUOD with dominating coefficients g U (n), n1, then for eachn1and anys>0,

Eexp { s i = 1 n η i } g U (n) i = 1 n Eexp{s η i }.

For convenience, we introduce some notation. For two positive functions a() and b(), we write a(t)b(t) if 0< lim inf t a(t)/b(t) lim sup t a(t)/b(t)<. For two positive bivariate functions a(,) and b(,), we say that the asymptotic relation a(n,x)b(n,x) or a(n,x)b(n,x) holds uniformly over all x in a nonempty set Δ, if lim sup n sup x Δ a(n,x)/b(n,x)1 or lim inf n inf x Δ a(n,x)/b(n,x)1. For a real number x, we write x + =max{x,0} and x =min{x,0}.

Before we state our main results, we will introduce some basic assumptions, to be used in this paper.

(A1) There exist a real-valued random variable Y with its d.f. F Y (x)D, and some positive integer n 0 , positive constants C 1 and T such that for all n> n 0 ,

1 n i = 1 n F ¯ i (x) C 1 F ¯ Y (x),
(2.7)

holds uniformly for all xT, and E ( Y + ) s < for some s>1.

(A2) There exist a real-valued random variable Z with its d.f. F Z (x)D, and positive constants C 2 , C 3 and T such that for every n1,

1 n i = 1 n F ¯ i (x) C 2 F ¯ Z (x)
(2.8)

holds uniformly for all xT, and for n large enough,

1 n i = 1 n F i (x) C 3 F Z (x)
(2.9)

holds uniformly for all xT; and EZ<, J F Z >1.

(A3)

lim sup x F ¯ Z (x)/ F ¯ Y (x)<.
(2.10)

(A4)

F Z (x)=o ( F ¯ Y ( x ) ) .
(2.11)

Remark 2.1 According to (2.7), (2.8), and (2.10), we can see that the r.v. Z’s and Y’s right tails are weak equivalent, i.e., F ¯ Z (x) F ¯ Y (x). The assumption (A4), which is equivalent to F Z (x)=o( F ¯ Z (x)), shows the r.v. Z’s left tails are lighter than the r.v. Y’s right tails. It is clear that all assumptions (A1)-(A4) are easily satisfied.

The main results of this paper are given below.

Theorem 2.1 Let the random variables{ X n ,n1}introduced in Section  1 be WOD and, for somer>1, E ( X n ) r <, n1. If the assumptions (A1)-(A4) hold and there exists a positive numberβ< J F Z 1such that

g L (n)=o ( n β ) and g U (n)=o ( n β ) ,
(2.12)

then (1.1) and (1.2) hold, respectively.

Theorem 2.2 In addition to the conditions of Theorem  2.1, if one of the following two conditions is satisfied:

  1. (I)

    whenc0, we have for arbitrarily fixedω>0and somer> J F Y +

    E [ N ( t ) r 1 ( N ( t ) > ( 1 + ω ) λ ( t ) ) ] =O ( λ ( t ) ) ;
    (2.13)
  2. (II)

    whenc<0, we have for all0<ω<1

    lim t P r ( N ( t ) ( 1 ω ) λ ( t ) ) F ¯ Y ( λ ( t ) ) =0,
    (2.14)

then (1.3) and (1.4) hold, respectively.

Remark 2.2 According to (3.13), (3.15), and (3.16), we can take

M 1 =( C 1 / C 2 ) lim inf x ( F ¯ Y ( u x / a ) / F ¯ Z ( x / b ) )

and

M 2 =( C 2 / C 1 ) lim sup x ( F ¯ Z ( ν x / b ) / F ¯ Y ( x / a ) )

in Theorem 2.1, respectively, where a, b, u, ν, C 1 , and C 2 are some fixed positive constants. For the given distribution functions F ¯ Z (x) and F ¯ Y (x), we can obtain the sharp lower and upper bound M 1 , M 2 . Hence, though the above expressive forms are not nice-looking, causes no trouble for real applications. For M ˜ 1 and M ˜ 2 , by the proof of Theorem 2.2, we can make a similar remark.

3 Proof of Theorem 2.1

We start with a series of lemmas based on which Theorem 2.1 will be proved. The proofs of the following Lemmas 3.1 and 3.2 are straightforward and are therefore omitted.

Lemma 3.1 IfE ( X + ) s <for somes>0, then the relation

lim n sup x γ n n s F ¯ (x)=0

holds for arbitrarily fixedγ>0.

Lemma 3.2 IfE ( X i ± ) q <, i1, andE ( Z ± ) q <for someq1, and (2.8), (2.9) hold, then there exists positive constant μ ˆ q ± <such that

i = 1 n E ( X i ± ) q n μ ˆ q ±

holds for anyn=1,2, .

Lemma 3.3 Let{ X n ,n1}be a sequence of real-valued and WUOD r.v.s with X n ’s d.f. F n (x)=1 F ¯ n (x)and μ n =E X n =0for everyn1, and let{ c 1 , c 2 ,}be a sequence of real numbers satisfying0<a c i b<, i1. If the assumptions (A1)-(A3) hold and there exists a positive numberβ>0such that

g U (n)=O ( n β ) ,
(3.1)

then there exists a constant M 2 >0 such that the relation

lim sup n sup x γ n sup c ̲ n [ a , b ] n P r ( i = 1 n c i X i > x ) i = 1 n P r ( c i X i > x ) M 2
(3.2)

holds for arbitrarily fixed constantγ>0, where c ̲ n =( c 1 , c 2 ,, c n ).

Proof For arbitrarily fixed 0<ν<1, let X ˜ i =min{ X i ,νx/ c i }, i1 and S ˜ n = i = 1 n c i X ˜ i . Using the standard truncation technique, we have

P r ( i = 1 n c i X i > x ) i = 1 n P r ( c i X i >νx)+ P r ( S ˜ n >x).
(3.3)

Now, we deal with the second term on the right-hand side of (3.3). Let c=c(n,x, c ̲ n )=max{log k = 1 n P r ( c k X k >νx),1}, then we can obtain lim sup n sup x γ n sup c ̲ n [ a , b ] n c= according to (2.8) and Lemma 3.1. Write W= P r ( S ˜ n >x)/ i = 1 n P r ( c i X i >νx). For a positive number h=h(n,x, c ̲ n )>0, which we shall specify later, by Chebyshev’s inequality and Proposition 2.1, we have

W g U ( n ) exp { h x + c + i = 1 n log E e h c i X ˜ i } g U ( n ) exp { h x + c + i = 1 n log [ ν x / c i ( e h c i y 1 ) d F i ( y ) + ( e h ν x 1 ) P r ( c i X i > ν x ) + 1 ] } g U ( n ) exp { h x + c + i = 1 n [ ν x / c i ( e h c i y 1 ) d F i ( y ) + ( e h ν x 1 ) P r ( c i X i > ν x ) ] } g U ( n ) exp { h x + c + i = 1 n 0 ( e h c i y 1 ) d F i ( y ) + i = 1 n ( 0 ν x c i c κ + ν x c i c κ ν x c i ) ( e h c i y 1 ) d F i ( y ) + ( e h ν x 1 ) i = 1 n P r ( c i X i > ν x ) } = g U ( n ) exp { h x + c + I 1 + I 2 + I 3 + ( e h ν x 1 ) i = 1 n P r ( c i X i > ν x ) } ,
(3.4)

where κ>1 is an arbitrarily fixed constant. Take h=(cκ p 2 logc)/νx, where p 2 > J F Y + . By (2.4) and (2.7),

lim sup n sup x γ n sup c ̲ n [ a , b ] n h lim sup n sup x γ n log C 1 n F ¯ Y ( ν x / a ) ν x lim sup n sup x γ n log C 1 n ( ν x a ) p 2 ν x = 0 .

For I 1 , we have

I 1 h n = 0 ( 1 n i = 1 n F i ( y ) c i e h c i y ) d y b ( T + T 0 ) 1 n i = 1 n F i ( y ) ( 1 e h c i y ) d y 1 n i = 1 n c i E X i .
(3.5)

Denote g(n,h,y, c ̲ n )=( i = 1 n F i (y)(1 e h c i y ))/n and g n (y)= sup x γ n sup c ̲ n [ a , b ] n g(n,h,y, c ̲ n ). By (2.9), we see that there exists N>0 such that for nN and all yT, | g n (y)| C 3 F Z (y). And for any [s,t](,T], | g n (y)| sup x γ n sup c ̲ n [ a , b ] n (1 e b h s )0, n for all y[s,t]. Hence,

lim sup n sup x γ n sup c ̲ n [ a , b ] n T g ( n , h , y , c ̲ n ) d y lim sup n T g n ( y ) d y = T lim n g n ( y ) d y = 0 .
(3.6)

From the definition of g n (y), we know that | g n (y)|1 for every n and all y[T,0), and g n (y)0, n for all y[T,0). So,

lim sup n sup x γ n sup c ̲ n [ a , b ] n T 0 g ( n , h , y , c ̲ n ) d y lim sup n T 0 g n ( y ) d y = T 0 lim n g n ( y ) d y = 0 .
(3.7)

Combining (3.6), (3.7) with (3.5), we obtain

I 1 =φnhh i = 1 n c i E X i ,
(3.8)

where lim sup n sup x γ n sup c ̲ n [ a , b ] n φ=0. For I 2 and I 3 , we have

I 2 + I 3 i = 1 n ( 0 ν x c i c κ h c i y e h c i y d F i ( y ) + ν x c i c κ ν x c i e h c i y d F i ( y ) ) i = 1 n ( h c i e h ν x c κ E X i + + e h ν x F ¯ i ( ν x c i c κ ) ) ,
(3.9)

where lim sup n sup x γ n sup c ̲ n [ a , b ] n x/ c κ according to (2.4) and (2.7). Plugging (3.8) and (3.9) into (3.4) yields

W g U ( n ) exp { h x + c + φ h n + h b i = 1 n E X i + ( e h ν x c κ 1 ) + C 2 n F ¯ Z ( ν x b c κ ) e h ν x + C 2 n F ¯ Z ( ν x b ) ( e h ν x 1 ) } g U ( n ) exp { h x + c + [ φ + b μ ˆ 1 + ( e h u ν / c κ 1 ) ] h n + C 2 C ˜ 2 c κ p 2 n F ¯ Z ( ν x b ) e h ν x + C 2 n F ¯ Z ( ν x b ) e h ν x } g U ( n ) exp { h x + c + [ φ + b μ ˆ 1 + ( e h u ν / c κ 1 ) ] h n + B ˆ ( c κ p 2 + 1 ) n F ¯ Z ( ν x b ) e h ν x } ,
(3.10)

where in the first step we apply (2.8), in the second step we use Lemma 3.2 and (2.2), and B ˆ is an appropriately chosen positive number. By the value of h, we have

lim sup n sup x γ n sup c ̲ n [ a , b ] n B ˆ ( c κ p 2 + 1 ) n F ¯ Z ( ν x b ) e h ν x <

and

lim sup n sup x γ n sup c ̲ n [ a , b ] n [ φ + b μ ˆ 1 + ( e h u ν / c κ 1 ) ] h n + ( κ p 2 log c ) / ν c =0.

Hence,

W g U (n) n β exp { β log n + ( 1 1 ν ) c + o ( c ) + O ( 1 ) } .
(3.11)

Using (2.3) and (2.8), we obtain

lim sup n sup x γ n sup c ̲ n [ a , b ] n log n c 1 p 1 1 ,

where 1< p 1 < J F Z . Taking 0<ν<( p 1 1)/(β+ p 1 1)<1, from (3.11), we have

lim sup n sup x γ n sup c ̲ n [ a , b ] n W lim sup n sup x γ n sup c ̲ n [ a , b ] n g U ( n ) n β exp { ( β p 1 1 + 1 1 ν ) c + o ( c ) + O ( 1 ) } = 0 .
(3.12)

Applying (2.7), (2.8), and (2.10), we know that there exists M 2 >0 such that

lim sup n sup x γ n sup c ̲ n [ a , b ] n i = 1 n P r ( c i X i > ν x ) i = 1 n P r ( c i X i > x ) M 2 .
(3.13)

Combining (3.12), (3.13) with (3.3), we can obtain (3.2). □

Lemma 3.4 Let{ X n ,n1}be a sequence of real-valued and WOD r.v.s with X n ’s d.f. F n (x)=1 F ¯ n (x)and μ n =E X n =0for everyn1, and let{ c 1 , c 2 ,}be a sequence of real numbers satisfying0<a c i b<, i1. If the assumptions (A1)-(A4) hold and there exists a positive numberβ< J F Z 1such that

g U (n)=o ( n β ) and g L (n)=o ( n β ) ,
(3.14)

then there exists a constant M 1 >0 such that the relation

lim inf n inf x γ n inf c ̲ n [ a , b ] n P r ( i = 1 n c i X i > x ) i = 1 n P r ( c i X i > x ) M 1
(3.15)

holds for arbitrarily fixed constantγ>0, where c ̲ n =( c 1 , c 2 ,, c n ).

Proof It is sufficient to prove that

lim inf n inf x γ n inf c ̲ n [ a , b ] n P r ( i = 1 n c i X i > x ) i = 1 n P r ( c i X i > u x ) 1
(3.16)

holds for arbitrarily fixed u>1. We write A k ={ c k X k >ux} and B k = 1 i k n A i c . Observing that A k B k , k=1,2,,n, are mutually disjoint, we have

P r ( i = 1 n c i X i > x ) k = 1 n [ P r ( A k ) P r ( A k B k c ) P r ( i = 1 n c i X i x , A k B k ) ] k = 1 n [ P r ( A k ) 1 i k n P r ( A k A i ) P r ( i : 1 i n , i k c i X i ( 1 u ) x , A k B k ) ] k = 1 n P r ( A k ) [ 1 g U ( n ) k = 1 n P r ( A k ) ] k = 1 n P r ( i : 1 i n , i k c i X i ( 1 u ) x , A k B k ) = J 1 J 2 ,
(3.17)

where at the third step we used Proposition 2.1.

For J 1 , by (2.3), (2.8), and (3.14), we have for arbitrarily fixed γ>0

lim inf n inf x γ n inf c ̲ n [ a , b ] n J 1 k = 1 n P r ( c k X k > u x ) 1 lim sup n sup x γ n g U ( n ) n C 2 F ¯ Z ( u x b ) 1 lim sup n C 2 g U ( n ) n 1 p 1 ( b u γ ) p 1 = 1 ,
(3.18)

where p 1 =β+1.

Now we deal with J 2 . For fixed 0<w<1, Let W k = X k and W ˜ k =min{ W k ,(wx)/ c k }, then

J 2 k = 1 n P r ( A k B k , i = 1 n { c i W i > w x } ) + k = 1 n P r ( i : 1 i n , i k c i X i ( 1 u ) x , j = 1 n { c j W j w x } ) i = 1 n P r ( c i W i > w x ) + k = 1 n P r ( c k W ˜ k ( u 1 ) x ) = K 1 + K 2 .
(3.19)

For K 1 , by (2.7), (2.9)-(2.11), we have

lim sup n sup x γ n sup c ̲ n [ a , b ] n K 1 i = 1 n P r ( c i X i > u x ) lim sup x C 3 F Z ( w b x ) C 1 F ¯ Y ( u a x ) =0.
(3.20)

For K 2 , using Chebyshev’s inequality and Proposition 2.1 again, we have

K 2 k = 1 n g L ( n 1 ) exp { h ( u 1 ) x + i : 1 i n , i k log E e h c i W ˜ i } k = 1 n g L ( n 1 ) exp { h ( u 1 ) x + i : 1 i n , i k [ w x c i ( e h c i y 1 ) d F W i ( y ) + ( e h w x 1 ) P r ( c i W i > w x ) ] } k = 1 n g L ( n 1 ) exp { h ( u 1 ) x + i : 1 i n , i k [ 0 ( e h c i y 1 ) d F W i ( y ) + 0 w x c i e h c i y 1 h c i y ( c i y ) s ( c i y ) s d F W i ( y ) + h c i E X i + ( e h w x 1 ) P r ( c i W i > w x ) ] } ,
(3.21)

where h=(log( w s 1 x s /n μ ˆ s +1))/wx and s>1. Using similar techniques as in (3.8), we can obtain

| i : 1 i n , i k [ 0 ( e h c i y 1 ) d F W i ( y ) + h c i E X i ] | |α|nh,
(3.22)

where α0 holds uniformly for xγn, as n. By (2.9) and (2.11),

lim sup n sup x γ n sup c ̲ n [ a , b ] n i : 1 i n , i k ( e h w x 1 ) P r ( c i W i > w x ) lim sup x w s 1 μ ˆ s C 3 x s F ¯ Y ( b w x ) 0 .
(3.23)

Take sufficiently large n such that |α|(u1)γ/2. Combining (3.21)-(3.23) and observing the monotonicity of 0( e h c i y 1h c i y)/ ( c i y ) s for all y>0, we have

K 2 k = 1 n g L ( n 1 ) n β exp { β log n h ( u 1 ) x + | α | n h + e h w x 1 h w x ( w x ) s i : 1 i n , i k c i s E ( X i ) s + o ( 1 ) } k = 1 n g L ( n 1 ) n β exp { log ( x γ ) β + log ( w s 1 x s n μ ˆ s + 1 ) u 1 2 w + e h w x 1 ( w x ) s b s n μ ˆ s + o ( 1 ) } C ˜ n x β ( s 1 ) ( u 1 ) 2 w ,
(3.24)

where C ˜ is some positive constant. For fixed u>1, we take 0<w<1 such that ( s 1 ) ( u 1 ) 2 w >β+ J F Y + . By (2.4) and (2.7), we have

lim sup n sup x γ n sup c ̲ n [ a , b ] n K 2 k = 1 n P r ( c k X k > u x ) lim sup x C ˜ x β ( s 1 ) ( u 1 ) 2 w C 1 F ¯ Y ( u a x ) =0.
(3.25)

Combing (3.17)-(3.20) with (3.25), we can obtain (3.16). This ends the proof of Lemma 3.4. □

Proof of Theorem 2.1 By Lemma 3.3, for arbitrarily fixed γ>0, we have uniformly for xγn

P r ( k = 1 n θ k X k > x ) = E [ P r ( k = 1 n θ k X k > x | θ 1 , θ 2 , , θ n ) ] M 2 E [ k = 1 n P r ( θ k X k > x | θ 1 , θ 2 , , θ n ) ] = M 2 k = 1 n P r ( θ k X k > x ) .
(3.26)

According to Lemma 3.4 and using a similar method of proof as in (3.26), we can obtain the remainder of Theorem 2.1. □

4 Proof of Theorem 2.2

For proving Theorems 2.2, we first give two lemmas.

Lemma 4.1 Let{ X n ,n1}be a sequence of real-valued and WUOD r.v.s with X n ’s d.f. F n (x)=1 F ¯ n (x)and0<E X n + <0for everyn1, and Let{ θ i ,i1}be a sequence of non-negative r.v.s satisfying P r (a θ i b)=1, i1, 0<ab<and independent of{ X n ,n1}. If (2.8) andE Z + <hold and g U (n)=O( n β )for some positive number β, then for every fixedu>0, there is D ˆ = D ˆ (u)>0such that

P r ( i = 1 n θ i X i > x ) i = 1 n P r ( θ i X i >ux)+ D ˆ ( n x ) 1 u n β
(4.1)

holds for large n and allx>0.

Proof Using the techniques similar to Lemma 3.3 with some obvious modifications, we can prove the lemma. □

Combining Lemma 2.1 of Chen et al.[6] with Lemma 3.1 of Ng et al.[11], we can obtain the following lemma.

Lemma 4.2 If a non-negative random process{ζ(t),t0}satisfiesEζ(t)1, t, then (i)-(iv) are mutually equivalent:

  1. (i)

    ζ(t) P r 1, ast;

  2. (ii)

    for every fixedθ>0, Eζ(t) 1 { ζ ( t ) 1 > θ } =o(1);

  3. (iii)

    for every fixedθ>0, Eζ(t) 1 { | ζ ( t ) 1 | > θ } =o(1);

  4. (iv)

    for every fixed0<θ<1, P r (1ζ(t)θ)=o(1).

By Lemma 4.2 and (2.13), we know that

N ( t ) λ ( t ) P r 1.
(4.2)

Proof of Theorem 2.2 Now, we prove Theorem 2.2 under condition (I). Using Theorem 2.1 and (2.8), we obtain for any fixed σ>0 the result that there exists a positive integral number N such that when nN, for sufficiently large x,

P r ( i = 1 n θ i X i > x ) ( M 2 +σ) i = 1 n P r ( θ i X i >x)( M 2 +σ) C 2 n F ¯ Z ( x b ) .
(4.3)

It is clear that for every n=1,2,,N and all large x,

P r ( i = 1 n θ i X i > x ) i = 1 n P r ( θ i X i >x/N) C 2 n F ¯ Z ( x N b ) .
(4.4)

Hence, by (4.3) and (4.4), there exists some positive number D, for every n=1,2, , and all sufficiently large x,

P r ( i = 1 n θ i X i > x ) Dn F ¯ Z (x).
(4.5)

Take 0<ω<1 such that c(1+ω)<γ. Throughout this proof, we suppose that x[γλ(t),). Consider the following decomposition:

P r ( S c θ ( t ) > x ) = ( n < ( 1 ω ) λ ( t ) + ( 1 ω ) λ ( t ) n ( 1 + ω ) λ ( t ) + n > ( 1 + ω ) λ ( t ) ) P r ( S n θ > x n c ) P r ( N ( t ) = n ) = L 1 + L 2 + L 3 .
(4.6)

Firstly, we deal with L 1 . For sufficiently large t, by (4.5), we have

L 1 n < ( 1 ω ) λ ( t ) P r ( S n θ > x ( 1 ω ) c λ ( t ) ) P r ( N ( t ) = n ) D F ¯ Z ( x ( 1 ω ) c λ ( t ) ) n < ( 1 ω ) λ ( t ) n P r ( N ( t ) = n ) .

For convenience, write H=E[ i = 1 N ( t ) P r ( θ i X i >xcλ(t))]. According to x(1ω)cλ(t)xcλ(t), (2.7), (2.10), and (iv) of Lemma 4.2, we have

lim sup t sup x γ λ ( t ) L 1 H lim sup t sup x γ λ ( t ) D F ¯ Z ( x ( 1 ω ) c λ ( t ) ) n < ( 1 ω ) λ ( t ) n P r ( N ( t ) = n ) C 1 F ¯ Y ( x c λ ( t ) a ) E [ N ( t ) 1 { N ( t ) > n 0 } ] D ˜ lim sup t n < ( 1 ω ) λ ( t ) n P r ( N ( t ) = n ) E N ( t ) n 0 D ˜ lim sup t E [ N ( t ) λ ( t ) 1 { N ( t ) < ( 1 ω ) λ ( t ) } ] 1 n 0 λ ( t ) D ˜ ( 1 ω ) lim t P r ( N ( t ) λ ( t ) < 1 ω ) = 0 ,
(4.7)

where D ˜ is some positive constant.

Secondly, we deal with L 2 . On the one hand, by Theorem 2.1, for arbitrary ε 1 >0 and sufficiently large t,

L 2 ( M 1 ε 1 ) ( 1 ω ) λ ( t ) n ( 1 + ω ) λ ( t ) i = 1 n P r ( θ i X i > x n c ) P r ( N ( t ) = n ) ( M 1 ε 1 ) C 1 F ¯ Y ( x ( 1 ω ) c λ ( t ) a ) ( 1 ω ) λ ( t ) n ( 1 + ω ) λ ( t ) n P r ( N ( t ) = n ) .
(4.8)

By (2.8), we have H C 2 F ¯ Z ((xcλ(t))/b)λ(t). Using (2.10) and (iii) of Lemma 4.2, we know that there is a positive constant M ˜ 1 such that

lim inf t inf x γ λ ( t ) L 2 H M ˜ 1 lim t E [ N ( t ) λ ( t ) 1 { | N ( t ) λ ( t ) 1 | ω } ] = M ˜ 1 .
(4.9)

On the other hand, using similar techniques as in (4.8), for any ε 2 >0 and sufficiently large t, we have

L 2 ( M 2 + ε 2 ) C 2 F ¯ Z ( x ( 1 + ω ) c λ ( t ) b ) ( 1 ω ) λ ( t ) n ( 1 + ω ) λ ( t ) n P r ( N ( t ) = n ) .
(4.10)

By (2.7), we have

H C 1 F ¯ Y ( x c λ ( t ) a ) E [ N ( t ) 1 { N ( t ) > n 0 } ] .
(4.11)

Then, by (4.10) and (4.11), there exists a positive number M ˜ 2 such that

lim sup t sup x γ λ ( t ) L 2 H M ˜ 2 lim t E [ N ( t ) λ ( t ) 1 { | N ( t ) λ ( t ) 1 | ω } ] 1 n 0 λ ( t ) = M ˜ 2 .
(4.12)

Finally, we deal with L 3 . Taking 0<υ<1 and splitting L 3 into two parts, we obtain

L 3 = ( ( 1 + ω ) λ ( t ) < n ( 1 υ ) x / c + n > max { ( 1 + ω ) λ ( t ) , ( 1 υ ) x / c } ) P r ( i = 1 n θ i X i > x n c ) P r ( N ( t ) = n ) = R 1 + R 2 ,
(4.13)

where R 1 is understood as 0 in case (1+ω)λ(t)>(1υ)x/c. For R 1 , taking u=1/p in (4.1) and letting p> J F Y + , we have

R 1 ( 1 + ω ) λ ( t ) < n ( 1 υ ) x / c P r ( i = 1 n θ i X i > υ x ) P r ( N ( t ) = n ) ( 1 + ω ) λ ( t ) < n ( 1 υ ) x / c ( i = 1 n P r ( θ i X i > υ x / p ) + D ˆ ( υ x ) p n p + β ) P r ( N ( t ) = n ) C 2 F ¯ Z ( υ x b p ) E [ N ( t ) 1 { N ( t ) > ( 1 + ω ) λ ( t ) } ] + D ˆ ( υ x ) p E [ N ( t ) p + β 1 { N ( t ) > ( 1 + ω ) λ ( t ) } ] .
(4.14)

Hence, according to (ii) of Lemma 4.2, (2.13), and (2.4), we have

lim sup t sup x γ λ ( t ) R 1 H lim sup t sup x γ λ ( t ) C 2 F ¯ Z ( υ x b p ) E [ N ( t ) λ ( t ) 1 { N ( t ) > ( 1 + ω ) λ ( t ) } ] C 1 F ¯ Y ( x c λ ( t ) a ) ( 1 n 0 λ ( t ) ) + lim sup t sup x γ λ ( t ) D ˆ ( υ x ) p C 1 F ¯ Y ( x c λ ( t ) a ) × lim sup t sup x γ λ ( t ) E [ N ( t ) p + β 1 { N ( t ) > ( 1 + ω ) λ ( t ) } ] / λ ( t ) ( 1 n 0 λ ( t ) ) = 0 .
(4.15)

For R 2 , we have

lim sup t sup x γ λ ( t ) R 2 H lim sup t sup x γ λ ( t ) n > max { ( 1 + ω ) λ ( t ) , ( 1 υ ) x / c } n p ( ( 1 υ ) x / c ) p P r ( N ( t ) = n ) C 1 F ¯ Y ( x c λ ( t ) a ) E [ N ( t ) 1 { N ( t ) > n 0 } ] lim sup t sup x γ λ ( t ) ( ( 1 υ ) x / c ) p E [ N ( t ) p 1 { N ( t ) > ( 1 + ω ) λ ( t ) } ] / λ ( t ) C 1 F ¯ Y ( x c λ ( t ) a ) ( 1 n 0 λ ( t ) ) = 0 .
(4.16)

Combing (4.6), (4.7), (4.9), (4.12), (4.13), and (4.15) with (4.16), we finish the proof under condition (I).

Finally, we prove Theorem 2.2 under condition (II). Without loss of generality, we assume c<γ<0. We still take 0<ω<1 such that c(1+ω)<γ<0 and use the decomposition (4.6). For L 1 , we take γ 1 >0 and divide the interval [γλ(t),) into two parts, which are [γλ(t), γ 1 λ(t)) and [ γ 1 λ(t),). When x[ γ 1 λ(t),), we have

F ¯ Z ( x c λ ( t ) ) F ¯ Z ( ( 1 c / γ 1 ) x ) F ¯ Z (x).
(4.17)

When x[γλ(t), γ 1 λ(t)), we have

F ¯ Y ( x c λ ( t ) ) F ¯ Y ( ( γ 1 c ) λ ( t ) ) F ¯ Y ( λ ( t ) ) .
(4.18)

Applying (2.14), (4.5), (4.17), and (4.18), we obtain

lim sup t sup x γ λ ( t ) L 1 H lim sup t sup x γ 1 λ ( t ) D ( 1 ω ) F ¯ Z ( x ) P r ( N ( t ) ( 1 ω ) λ ( t ) ) C 1 F ¯ Y ( x c λ ( t ) a ) ( 1 n 0 λ ( t ) ) + lim sup t sup γ λ ( t ) x < γ 1 λ ( t ) D ( 1 ω ) P r ( N ( t ) ( 1 ω ) λ ( t ) ) C 1 F ¯ Y ( x c λ ( t ) a ) ( 1 n 0 λ ( t ) ) = 0 .
(4.19)

For L 3 , since xcnγλ(t)cn(γ ( 1 + ω ) 1 c)n, then according to Theorem 2.1, (2.10), and Lemma 4.2, we have

lim sup t sup x γ λ ( t ) L 3 H lim sup t sup x γ λ ( t ) M 2 n > ( 1 + ω ) λ ( t ) i = 1 n P r ( θ i X i > x c n ) P r ( N ( t ) = n ) C 1 F ¯ Y ( x c λ ( t ) a ) E [ N ( t ) 1 { N ( t ) > n 0 } ] lim sup t sup x γ λ ( t ) M 2 C 2 F ¯ Z ( x c λ ( t ) b ) E [ N ( t ) λ ( t ) 1 { N ( t ) > ( 1 + ω ) λ ( t ) } ] C 1 F ¯ Y ( x c λ ( t ) a ) ( 1 n 0 λ ( t ) ) = 0 .
(4.20)

For L 2 , observing λ(t) ( γ c ) 1 (xcλ(t)) and c<0, by Theorem 2.1, (2.14), and (iii) of Lemma 4.2, we know that, on the one hand, there exists a positive number M ˜ 1 such that

lim inf t inf x γ λ ( t ) L 2 H lim inf t inf x γ λ ( t ) M 1 ( 1 ω ) λ ( t ) n ( 1 + ω ) λ ( t ) i = 1 n P r ( θ i X i > x c n ) P r ( N ( t ) = n ) C 2 F ¯ Z ( x c λ ( t ) b ) λ ( t ) lim inf t inf x γ λ ( t ) M 1 C 1 F ¯ Y ( x c ( 1 + ω ) λ ( t ) a ) E [ N ( t ) λ ( t ) 1 { | N ( t ) λ ( t ) 1 | ω } ] C 2 F ¯ Z ( x c λ ( t ) b ) lim inf t inf x γ λ ( t ) M 1 C 1 F ¯ Y ( a 1 ( 1 c ω ( γ c ) 1 ) ( x c λ ( t ) ) ) E [ N ( t ) λ ( t ) 1 { | N ( t ) λ ( t ) 1 | ω } ] C 2 F ¯ Z ( x c λ ( t ) b ) M ˜ 1 ;
(4.21)

on the other hand, there exists a positive number M ˜ 2 such that

lim sup t sup x γ λ ( t ) L 2 H lim sup t sup x γ λ ( t ) M 2 C 2 F ¯ Z ( b 1 ( 1 + c ω ( γ c ) 1 ) ( x c λ ( t ) ) ) E [ N ( t ) λ ( t ) 1 { | N ( t ) λ ( t ) 1 | ω } ] C 1 F ¯ Y ( x c λ ( t ) a ) ( 1 n 0 λ ( t ) ) M ˜ 2 .
(4.22)

Combing (4.19)-(4.22), we finish the proof under condition (II). □

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Acknowledgements

The authors would like to thank the two referees for careful reading of our manuscript and for helpful and valuable comments and suggestions, which helped us improve the earlier version of the paper. The research of the authors was supported by the Natural Science Foundation of the Inner Mongolia Autonomous Region (No. 2013MS0101), the National Natural Sciences Foundation of China (No. 11201317), and the Beijing Municipal Education Commission Foundation (No. KM201210028005).

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Bai, X., Song, L. & Hu, T. Large deviations for randomly weighted sums with dominantly varying tails and widely orthant dependent structure. J Inequal Appl 2014, 140 (2014). https://doi.org/10.1186/1029-242X-2014-140

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Keywords

  • large deviations
  • randomly weighted sums
  • widely orthant dependence
  • dominated variation