Extremes 6, 171±188, 2003 # 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory QIHE TANG* Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands E-mail: [email protected] GURAMI TSITSIASHVILI Institute of Applied Mathematics, Far Eastern Scienti®c Center, Russian Academy of Sciences, 690068 Vladivostok, Russia E-mail: [email protected] [Received April 3, 2003; Revised March 3, 2004; Accepted March 4, 2004] Abstract. Let fXk ; 1  k  ng be n independent and real-valued random variables with common subexponential distribution function, and let fyk ; 1  k  ng be other n random variables independent of fXk ; 1  k  ng and satisfying a  yk  b for some 0 < a  b < ? for all 1  k  n. This paper proves that the asymptotic relations ! ! m n n X X X P max yk Xk > x *P yk Xk > x * P…yk Xk > x† 1mn

kˆ1

kˆ1

kˆ1

hold as x ? ?. In doing so, no any assumption is made on the dependence structure of the sequence fyk ; 1  k  ng. An application to ruin theory is proposed. Key words. asymptotics, dominated variation, ruin probability, subexponentiality, uniformity AMS 2000 Subject Classi®cation.

1.

PrimaryÐ62E20 SecondaryÐ91B30, 60G50

Introduction

Let fXk ; 1  k  ng be a sequence of n independent, identically distributed (i.i.d.), and real-valued random variables with common distribution function F, and let each random variable Xk be associated with a positive random variable yk ; 1  k  n, where the sequences fXk ; 1  k  ng and fyk ; 1  k  ng are mutually independent. In this paper *Corresponding author.

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we are interested in the tail behavior of the randomly weighted sums Sym ; 1  m  n, and their maximum Mny , which are de®ned, respectively, by Sym ˆ

m X kˆ1

y k Xk

and

Mny ˆ max Sym : 1mn

…1†

We remark that the limit theory of Sym as m ? ?, sometimes normalized, has been systematically investigated in the literature. In this direction, we refer the reader to Rosalsky and Sreehari (1998) for a complete list of references from 1965 to 1995 and to Hu et al. (2001, Section 1) for a review. However, there are quite few publications which are devoted to the asymptotic behavior of the tail probability of the randomly weighted sums or their maximum. For the special case where the distribution function F is heavy tailed and the weights yk ; 1  k  n, are degenerate at 1, the tail behavior of these quantities has been investigated, for example, by Sgibnev (1996), Embrechts et al. (1997), and Ng et al. (2002), among others. We say a sequence of random variables fzk ; 1  k  ng is bounded 1. of type I if P…a  zk  b† ˆ 1 holds for some 0 < a  b < ? and all 1  k  n; 2. of type II if P…0 < zk  b† ˆ 1 holds for some 0 < b < ? and all 1  k  n; and 3. of type III if P…a  zk < ?† ˆ 1 holds for some 0 < a < ? and all 1  k  n. The standing assumptions of this paper are that the distribution function F is subexponential and that the random weights fyk ; 1  k  ng are bounded of type I. Under these assumptions, this paper shows that P…Mny > x†*P…Syn > x†*

n X kˆ1

P…yk Xk > x†:

…2†

Here and henceforth, all limit relationships are for x ? ? unless stated otherwise, for two positive in®nitesimals a… ? † and b… ? †, we write a…x† < b…x† if lim sup a…x†=b…x†  1 * and write a…x†*b…x† if both a…x† < b…x† and b…x† < a…x†. It is somewhat surprising that in * * establishing the asymptotic relations (2), no any assumption is requested on the dependence structure of the sequence fyk ; 1  k  ng. More explicit formulae can be derived when F has a dominated or rapid variation. The rest of this paper is organized as follows: Section 2 recalls some notions of heavytailed distributions, Section 3 presents the main results, Section 4 proposes an application to the ®nite time ruin probability, and Section 5 proves the theorem after establishing a series of preliminaries. 2.

Heavy-tailed distributions

Throughout this paper, for a distribution function F we write its tail by F ˆ 1 F. Following many recent researchers in the ®elds of applied probability, we restrict our

RANDOMLY WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES

173

interest to the case of heavy-tailed random variables. A distribution function F or its corresponding random variable X is said to be heavy tailed to the right if E expfrXg ˆ ? for any r > 0. A necessary condition for F to be heavy tailed is that F…x† > 0 for any real number x. The most important class of heavy-tailed distribution functions is the subexponential class, denoted by s. By de®nition, a distribution function F concentrated on ‰0; ?† belongs to the class s if Fn …x† ˆn x ? ? F…x† lim

…3†

holds for any (or, equivalently, for some) n  2, where Fn denotes the n-fold convolution of F. That is, for a sequence of i.i.d. random variables fXk ; k  1g with common distribution function F [ s, it holds for any n  2 that  P…Sn > x†*P

 max Xk > x :

1kn

…4†

More generally, a distribution function F concentrated on … ?; ?† is still said to be subexponential to the right if F‡ …x† ˆ F…x†1…0  x < ?† is subexponential, where 1A denotes the indicator function of A. Applying the Proposition of Sgibnev (1988), it is easy to see that relation (3), hence (4), still holds for this general case. Note that if F [ s, then F is long tailed, denoted by F [ l, in the sense that F…x ‡ y† ˆ1 x ? ? F…x† lim

…5†

holds for any (or, equivalently, for some) y > 0; see Chistyakov (1964), or Embrechts et al. (1997, Lemma 1.3.5). Clearly, for any c > 0, we have F… ? † [ s or l if and only if F… ? =c† [ s or l. Another closely related class is the class d of distribution functions with dominated variations. By de®nition, a distribution function F belongs to the class d if lim sup x??

F…xy†
…6†

holds for any (or, equivalently, for some) 0 < y < 1. It is well known that l \ d  s; see Goldie (1978), or Embrechts et al. (1997, Proposition 1.4.4). A famous subclass of the intersection l \ d is r, which is the class of distribution functions with regular

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TANG AND TSITSIASHVILI

variations. By de®nition, a distribution function F belongs to the class r if there exists some 0  a < ? such that lim

x??

F…xy† ˆy F…x†

a

for any y > 0:

…7†

We denote by F [ r a the regularity property in (7). A slightly larger subclass of the intersection l \ d is the so-called extended regular variation (ERV) class. By de®nition, a distribution function F belongs to the class ERV… a; b† for some 0  a  b < ? if y

b

 lim inf x??

F…xy† F…xy†  lim sup y F…x† x ? ? F…x†

a

for any y > 1:

…8†

Note that there are many popular distributions in statistics such as the lognormal, the Weibull, etc., which are subexponential but are excluded by requirement (6). Hence, the class d is not rich enough to model heavy-tailed distributions. We introduce a new distribution class below (see also Tang, 2004): De®nition 2.1: F [ r ? , if lim

x??

A distribution function F is said to have a rapid variation, denoted by

F…xy† ˆ0 F…x†

for any y > 1:

…9†

Studies on the rapid variation can be found in the monographs de Haan (1970, Chapter 1.2) and Bingham et al. (1987, Chapter 2.4). Roughly speaking, the subexponential class s can be divided into two disjoint parts as s & …s \ d† [ …s \ r

? †:

For more details about heavy-tailed distributions and their applications, we refer the reader to Bingham et al. (1987) and Embrechts et al. (1997).

3.

Main results

Recall the randomly weighted sums Sym ; 1  m  n, and their maximum Mny de®ned by (1), where fXk ; 1  k  ng is a sequence of i.i.d. random variables with common distribution function F, and fyk ; 1  k  ng is another sequence of positive random variables independent of fXk ; 1  k  ng. The main result of this paper is the following:

175

RANDOMLY WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES

Theorem 3.1: If F [ s and fyk ; 1  k  ng are bounded of type I (recall this notation in Section 1), then   X n

y y P Mn > x *P Sn > x *P max yk Xk > x * P…yk Xk > x†: …10† 1kn

kˆ1

Comparing (10) with (4), we feel that Theorem 3.1 describes the very nature of the subexponential class. The proof of this theorem is left to Section 5 below. Now we put forward some special cases of Theorem 3.1. We ®rst consider the dominated variation case. Corollary 3.1: If F [ l \ d and fyk ; 1  k  ng are bounded of type II (recall this notation in Section 1), then relations (10) still hold. Proof:

We only prove the relation

n
X P Syn > x * P…yk Xk > x†

…11†

kˆ1

since the result for max1  k  n yk Xk will be addressed in Proposition 5.2 below and the relation for Mny follows from (11) and the two-sided inequality Syn  Mny 

n X kˆ1

yk Xk‡ ;

…12†

where x‡ ˆ maxfx; 0g for any real number x. For any d > 0 such that P…yk  d† > 0 for all 1  k  n, we split the probability P…Syn > x† into two parts as Syn

I1 ‡ I2 ˆ P

n [

> x;

kˆ1

! …0 < yk < d†

‡P

Syn

> x;

n \ kˆ1

! …d  yk  b† :

Clearly, I1  P Since P…

Pn

n X kˆ1

kˆ1

! bXk‡

>x P

n [ kˆ1

! …0 < yk < d† :

bXk‡ > x†*nF…x=b† ˆ O…F…x††, it follows that

lim lim sup

d&0 x??

I1 ˆ 0: F…x†

…13†

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TANG AND TSITSIASHVILI

For I2 , by Theorem 3.1 we have

I2 * ˆ ˆ

n X kˆ1

P yk Xk > x;

n X

n \ kˆ1

P…yk Xk > x†

kˆ1 n X

P…yk Xk > x†

kˆ1

! …d  yk  b†

n X kˆ1

P yk Xk > x;

!

n [ kˆ1

…0 < yk < d†

I3 :

Similarly to the proof of relation (13), lim lim sup

d&0 x??

I3 ˆ 0: F…x†

Simply combining these results gives relation (11). This ends the proof of Corollary 3.1. & Let X be a random variable with a distribution function F [ d. For any y > 0, we set f …y† ˆ lim inf x??

F…xy† ; F…x†

f  …y† ˆ lim sup x??

F…xy† : F…x†

…14†

Let y be another random variable independent of the random variable X and satisfying P…0 < y  b† ˆ 1 for some 0 < b < ?. Applying Theorem 3.3(iv) of Cline and Samorodnitsky (1994), we know that 0 < Ef …y

1

†  lim inf x??

P…yX > x† P…yX > x†  lim sup  Ef  …y P…X > x† x ? ? P…X > x†

1

Hence by Corollary 3.1, we easily obtain the following: Corollary 3.2:

Suppose that fyk ; 1  k  ng are bounded of type II.

1. If F [ l \ d, then F…x†

n X kˆ1

Ef yk

1




n X


< P Mny > x *P Syn > x < F…x† Ef  yk 1 ; *

*

kˆ1

† < ?:

…15†

RANDOMLY WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES

177

2. If F [ ERV… a; b† for some 0  a  b < ?, then n n h n oi h n oi X X

F…x† E min yak ; ybk < P Mny > x *P Syn > x < F…x† E max yak ; ybk ; *

kˆ1

3. If F [ r

a

*

kˆ1

for some 0  a < ?, then

n X

P Mny > x *P Syn > x *F…x† Eyak :

…16†

kˆ1

Next we consider the rapid variation case. Let X and y be two independent random variables distributed by F and G, respectively. Write by y ˆ supfy : P…y  y† < 1g the (upper) endpoint of the random variable y. If F [ r ? ; y is positive with y < ?, and P…y ˆ y† ˆ p > 0, then by the de®nition of the class r ? , P…yX > x† ˆ lim x ? ? F…x=y†

Z lim

F…x=y†

…0;y†[fyg x ? ? F…x=y†

G…dy† ˆ p;

where the interchange of the limit and integral is allowed by the dominated convergence theorem. Hence, we can obtain another consequence of Theorem 3.1 as follows: Corollary 3.3: Suppose that F [ s \ r ? and that fyk ; 1  k  ng are bounded of type I. If P…yk ˆ yk † ˆ pk > 0 for 1  k  n, then with ^y ˆ maxfyk : 1  k  ng,  X   n X x x n y y P…Mn > x†*P…Sn > x†* pk F pk 1…y ˆ ^y† : …17† *F k ^y yk kˆ1 kˆ1 4.

An application to ruin under dependent return rates

We consider a recursive equation S0 ˆ x;

Sn ˆ xn Sn

1

‡ …Zn

Zn †;

n  1;

…18†

which characterizes the surplus process of an insurance company in a discrete time model. Here x  0 is the initial surplus, Zn and Zn are the total incoming premium and total claim amount of the company during the time period n, respectively, and xn is the in¯ation coef®cient from time n 1 to time n related to the stochastic return of an investment. Suppose that the random pairs …Zn ; Zn †, n  1, are i.i.d. and that the sequences fxn ; n  1g and f…Zn ; Zn †; n  1g are mutually independent. Clearly, the recursive equation (18) describes the well-known renewal risk model if xn :1 for all n.

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TANG AND TSITSIASHVILI

Recently, Nyrhinen (1999) and Tang and Tsitsiashvili (2003) investigated the asymptotic behavior of the ®nite and in®nite time ruin probabilities of the model above under the assumption that fxn ; n  1g is a sequence of i.i.d. random variables; Cai (2002) considered a nonstandard situation that fxn ; n  1g follows a dependent autoregressive structure, and established some Lundberg bounds for the in®nite time ruin probability. Now, the results in Section 3 make it possible to derive precise estimates for the ®nite time ruin probability under an arbitrary dependence structure of the sequence fxn ; n  1g. Introduce Xn ˆ Zn Zn ; n  1, which are i.i.d. with common distribution function F, and write Yn ˆ xn 1 ; n  1. Then, iterating (18) yields that S0 ˆ x;

Sn ˆ x

n Y iˆ1

xi

n X kˆ1

Xk

n Y iˆk‡1

xi ;

n  1:

We write the discounted values of the surplus Sn by S~0 ˆ x;

S~n ˆ Sn

n Y iˆ1

Yi ˆ x

n X kˆ1

Xk

k Y iˆ1

Yi ;

n  1:

…19†

Denote the ruin probability within a ®nite horizon n  1 by   c…x; n† ˆ P min Sm < 0 S0 ˆ x : 0mn It follows from (19) that   ~ ~ c…x; n† ˆ P min Sm < 0 S0 ˆ x ˆ P 0mn

max

1mn

m X kˆ1

Xk

k Y iˆ1

! Yi > x :

Clearly, for n  1, if the random variables fxk ; 1  k  ng are bounded of type I/III, then the random variables fYk ; 1  k  ng, hence their products yk ˆ

k Y iˆ1

Yi ;

1  k  n;

are bounded of type I/II. Applying the results in Section 3, we have the following statements:

RANDOMLY WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES

Corollary 4.1:

179

Consider the risk model (18), n  1.

1. If F [ s and fxk ; 1  k  ng are bounded of type I, then c…x; n†*

n X kˆ1

2. If F [ r then

a

P Xk

k Y iˆ1

! Yi > x :

…20†

for some 0  a < ? and fxk ; 1  k  ng are bounded of type III,

c…x; n†*F…x†

n k X Y E Yia :

kˆ1

…21†

iˆ1

3. If F [ s \ r ? , fxk ; 1  k  ng are bounded of type I, and P…yk ˆ yk † ˆ pk > 0 for 1  k  n, then with ^ y ˆ maxfyk : 1  k  ng,  X x n c…x; n†*F pk 1…y ˆ ^y† : k ^ y

…22†

kˆ1

If in (18) the sequences fZn ; n  1g and fZn ; n  1g are mutually independent, then by Lemma 4.2 of Tang (2004), the assumption F [ s=r=s \ r ? in Corollary 4.1 is equivalent to that the common distribution function of fZn ; n  1g belongs to the same class.

5. 5.1.

Proof of Theorem 3.1 On uniform convergence of subexponential tails

In order to prove Theorem 3.1, we need to establish a series of important preliminaries. Let F ˆ F1  F2 , where F1 and F2 are two distribution functions concentrated on ‰0; ?†. It is well known that if F1 [ s, F2 [ l, and F2 …x† ˆ O…F1 …x††, then F [ s and F…x†*F1 …x† ‡ F2 …x†:

…23†

See Cline (1986, Corollary 1); under some additional restrictions it was ®rst obtained by Embrechts and Goldie (1980). The following result, which may be of independent interest on its own right, makes the statement above somewhat stronger:

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TANG AND TSITSIASHVILI

Lemma 5.1: Let X1 and X2 be two independent random variables distributed by F1 and F2 , respectively. If F1 [ s, F2 [ l, and F2 …x† ˆ O…F1 …x††, then for any ®xed 0 < a  1, the relation   x P…X1 ‡ cX2 > x†*F1 …x† ‡ F2 …24† c holds uniformly for c [ ‰a; 1Š, where the uniformity is understood as P…X1 ‡ cX2 > x† lim sup x ? ? c [ ‰a; 1Š F …x† ‡ F …x=c† 1 2 Proof:

1 ˆ 0:

Applying integration by parts and Fatou's lemma, we have Rx

F1 …x t†F1 …dt† F1 …x† x?? A?? R? RA … ? ? ˆ lim sup lim sup

lim sup lim sup

A??

A

x??



 lim sup 2 A??

Z

A

lim inf

? x??

R?

†F1 …x F1 …x†

t†F1 …dt†

x

Z

F1 …x t† F1 …dt† F1 …x†

F1 …0†

0

lim inf

? x??

F1 …x t† F1 …dt† F1 …x†



ˆ 0: Hence, for arbitrarily ®xed e > 0, we can choose some A1 > 0 such that the following inequalities hold simultaneously: Z

x

A1

F1 …0; A1 Š  …1 F1 …x

e†F1 …0†;

…25†

t†F1 …dt†  eF1 …x† for all x  A1 :

…26†

Relying on this A1 > 0, we divide P…X1 ‡ cX2 > x† into four summands as Z J1 ‡ J2 ‡ J3 ‡ J4 ˆ

0 ?

Z ‡

A1 0

Z ‡

x A1

Z ‡

? x

! F2



x

t c

 F1 …dt†:

…27†

First we deal with J1 . By F2 [ l and the dominated convergence theorem we know that Z lim

x??

0

F2 …x t=a† F1 …dt† ˆ F2 …x† ?

Z

0

lim

? x??

F2 …x t=a† F1 …dt† ˆ F1 …0†: F2 …x†

…28†

RANDOMLY WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES

181

Hence, there exists some A2 > 0 such that for all x  A2 and c [ ‰a; 1Š, Z J1 



0 ?

F2

x c

 t F1 …dt†  …1 a

e†F2

  x F1 …0†: c

On the other hand, it is obvious that J1  F2 …x=c†F1 …0†. So for all x  A2 , …1

e†F2

    x x F1 …0†  J1  F2 F1 …0†: c c

…29†

Now we deal with J2 . By (25) we have that   x J 2  F2 F1 …0; A1 Š  …1 c Furthermore, since F2 …x x  A3 and c [ ‰a; 1Š,  J 2  F2

  x e†F2 F1 …0†: c

A1 =a†*F2 …x†, there exists some A3 > 0 such that for all

   A1 x F1 …0†  …1 ‡ e†F2 F1 …0†: c a

x c

This proves that for all x  maxfA1 ; A3 g and c [ ‰a; 1Š, …1

    x x e†F2 F1 …0†  J2  …1 ‡ e†F2 F1 …0†: c c

…30†

As for J3 , we write F2 …x† < ?: x  0 F1 …x†

D ˆ sup

So by (26) we have that for all x  A1 and c [ ‰a; 1Š, Z 0  J3 

x

A1

Z F2 …x

t†F1 …dt†  D

x

A1

F1 …x

t†F1 …dt†  DeF1 …x†:

…31†

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TANG AND TSITSIASHVILI

Finally we turn to J4 . Obviously, J4  F1 …x†. Integration by parts gives Z

0

Z

? 0

J4 ˆ F2 …0†F1 …x† ‡  F2 …0†F1 …x† ‡

?

F1 …x

ct†F2 …dt†

F1 …x

t†F2 …dt†

* F1 …x†; R0 t†F2 …dt†*F2 …0†F1 …x†, which can be veri®ed where we have used a fact that ? F1 …x by the dominated convergence theorem, as done in (28). Hence, there exists some A4 > 0 such that for all x  A4 and c [ ‰a; 1Š, …1

e†F1 …x†  J4  F1 …x†:

…32†

Substituting inequalities (29), (30), (31), and (32) into (27), we eventually obtain that for all x  maxfA1 ; A2 ; A3 ; A4 g and c [ ‰a; 1Š, …1

   x e† F1 …x† ‡ F2  P…X1 ‡ cX2 > x† c  …1 ‡ De†F1 …x† ‡ …1 ‡ eF1 …0††F2

  x : c

The arbitrariness of e > 0 gives result (24). This ends the proof of Lemma 5.1.

&

The following are two straightforward consequences of Lemma 5.1. Corollary 5.1: Let X1 and X2 be two independent random variables distributed by F1 [ s and F2 [ l, respectively. Then for any ®xed 0 < a  b < ?, relation (24) holds uniformly for c [ ‰a; bŠ provided that F2 …x=b† ˆ O…F1 …x††. Proof:

We rewrite the left-hand side of (24) as

P…X1 ‡ cX2 > x† ˆ P…X10 ‡ c0 X2 > x0 †; where X10 ˆ X1 =b; c0 ˆ c=b, and x0 ˆ x=b. By Lemma 5.1, the relations    x0 x P…X10 ‡ c0 X2 > x0 †*P…X10 > x0 † ‡ P X2 > 0 ˆ P…X1 > x† ‡ P X2 > c c hold uniformly for c0 [ ‰a=b; 1Š. Hence, relation (24) holds uniformly for c [ ‰a; bŠ. This ends the proof of Corollary 5.1. &

RANDOMLY WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES

183

Corollary 5.2: Let X1 and X2 be two independent random variables distributed by F1 and F2 , respectively. If there is a distribution function F [ s such that Fi …x†*li F…x† holds for some li > 0 for i ˆ 1; 2, then for any ®xed 0 < a  b < ?, relation (24) holds uniformly for c [ ‰a; bŠ. Proof: By virtue of the closure property of the class s under tail equivalence (see Embrechts et al., 1997, Lemma A3.15), we have that Fi [ s  l for i ˆ 1; 2. By Lemma 5.1 we immediately obtain that (24) holds uniformly for c [ ‰a; 1Š. Hence, Corollary 5.2 has been proved if b  1. Now we assume a  1 < b and derive ! P…X1 ‡ cX2 > x† P…X1 ‡ cX2 > x† sup 1  sup ‡ sup 1 F1 …x† ‡ F2 …x=c† c [ ‰a; bŠ F1 …x† ‡ F2 …x=c† c [ ‰a;1Š c [ …1;bŠ ˆ K1 ‡ K2 : By Lemma 5.1, it holds for any e > 0 and all large x > 0 that K1  e. With c0 ˆ 1=c and x0 ˆ x=c we rewrite K2 as P…c0 X1 ‡ X2 > x0 † : 1 K2 ˆ sup 0 0 0 c0 [ ‰1=b; 1† F1 …x =c † ‡ F2 …x † Applying Lemma 5.1 once again yields that K2  e also holds for all large x0 > 0, or, equivalently, for all large x > 0. The arbitrariness of e > 0 yields that P…X1 ‡ cX2 > x† lim sup x ? ? c [ ‰a; bŠ F …x† ‡ F …x=c† 1 2 This ends the proof of Corollary 5.2.

1 ˆ 0: &

It will be convenient to use the notation cn ˆ …c1 ; c2 ; . . . ; cn † in the following result and its proof. Proposition 5.1: Let fXk ; 1  k  ng be n i.i.d. random variables with common distribution function F [ s. Then for any ®xed 0 < a  b < ?, the relation !   n n X X x P c k Xk > x * F …33† c k kˆ1 kˆ1 holds uniformly for cn [ ‰a; bŠn . Proof: We give the proof of Proposition 5.1 by induction approach. It is trivial that relation (33) holds for n ˆ 1. Now we assume by induction that relation (33) holds for n ˆ m for some positive integer m. We aim to prove that the relation

184

TANG AND TSITSIASHVILI

!

  x P c k Xk > x * F ck kˆ1 kˆ1 m ‡1 X

m ‡1 X

…34†

holds uniformly for cm ‡ 1 [ ‰a; bŠm ‡ 1 . We rewrite (34) as ! m m X X 0 0 P c k Xk ‡ Xm ‡ 1 > x * P…c0k Xk > x0 † ‡ P…Xm ‡ 1 > x0 †; kˆ1

…35†

kˆ1

where c0k ˆ ck =cm ‡ 1 for 1  k  m and x0 ˆ x=cm ‡ 1 . Observe that cm ‡ 1 [ ‰a; bŠ and that 0<

a b  c0k  < ? b a

for all 1  k  m:

Thus, without loss of generality we can assume cm ‡ 1 ˆ 1 in (34). As a result, it suf®ces to prove that the relation !   m m X X x P c k Xk ‡ Xm ‡ 1 > x * F …36† ‡ F…x† ck kˆ1 kˆ1 m

holds uniformly for cm [ ‰a; bŠ . For any e > 0, by the induction assumption, there is some constant B1 > 0 such that the inequalities ! m m m X X X …1 e† P…ck Xk > x†  P ck Xk > x  …1 ‡ e† P…ck Xk > x† …37† kˆ1

kˆ1

kˆ1

hold uniformly for cm [ ‰a; bŠm and x  B1 . We divide the probability on the left-hand side of (36) into two summands as ! Z x B1 Z ? ! m X L1 ‡ L2 ˆ ‡ ck Xk > x t F…dt†: P ?

x

B1

kˆ1

First we deal with L1 . By (37) we have Z x B1 X m L1  …1 ‡ e† P…ck Xk > x ?

ˆ …1 ‡ e†  …1 ‡ e† ˆ …1 ‡ e†

m X kˆ1 m X

t†F…dt†

kˆ1

P…ck Xk‡ ‡ Xm ‡ 1 > x; Xm ‡ 1  x

B1 †

…P…ck Xk‡ ‡ Xm ‡ 1 > x†

P…ck Xk‡ ‡ Xm ‡ 1 > x; Xm ‡ 1 > x††

P…ck Xk‡ ‡ Xm ‡ 1 > x†

…1 ‡ e†mF…x†:

kˆ1 m X kˆ1

…38†

185

RANDOMLY WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES

Symmetrically, Z L1  …1

e†

ˆ …1

e†

x

B1

? m X kˆ1

 …1

e†

m X

m X kˆ1

P…ck Xk > x

P…ck Xk‡ ‡ Xm ‡ 1 > x; Xm ‡ 1  x

e†

B1 †

…P…ck Xk‡ ‡ Xm ‡ 1 > x†

P…Xm ‡ 1 > x

P…ck Xk‡ ‡ Xm ‡ 1 > x†

…1

kˆ1

ˆ …1

t†F…dt†

m X kˆ1

e†mF…x

B1 ††

B1 †:

…39†

By Corollary 5.2 and F [ l, there is some constant B2 > 0 such that the inequalities

…1

        x x ‡ e† F ‡ F…x†  P…ck Xk ‡ Xm ‡ 1 > x†  …1 ‡ e† F ‡ F…x† ck ck

…40†

F…x

…41†

and B1 †  …1 ‡ e†F…x†

hold uniformly for cm [ ‰a; bŠm and x  B2 . Substituting (40) and (41) into (38) and (39) gives that uniformly for cm [ ‰a; bŠm and x  maxfB1 ; B2 g,

…1

  m X x F e† ck kˆ1 2

2…e

  m X x F e †mF…x†  L1  …1 ‡ e† ‡ …e ‡ e2 †mF…x†: c k kˆ1 2

2

…42† Next we turn to L2 . On one hand, by (41) it is trivial that for all x  B2 , L2  F…x

B1 †  …1 ‡ e†F…x†:

186

TANG AND TSITSIASHVILI

On the other hand, we can choose some C > 0 and B3 > 0 such that uniformly for cm [ ‰a; bŠm and x  B3 , Z L2 

?

x

Z 

B1 ?

x‡C

P b  …1

P

m X kˆ1

P b

! ck minfXk ; 0g > x

m X kˆ1

m X kˆ1 2

t F…dt† !

minfXk ; 0g > x

t F…dt†

!

minfXk ; 0g >

C F…x ‡ C†

e† F…x†: m

This proves that uniformly for cm [ ‰a; bŠ and x  maxfB2 ; B3 g, e†2 F…x†  L2  …1 ‡ e†F…x†:

…1

…43†

Combining (42) and (43), we see that the inequalities m X

P

kˆ1

! c k Xk ‡ Xm ‡ 1 > x

 …1

  m X x e† F c k kˆ1 2

…2em

2e2 m

…1

2

e† †F…x†

and m X

P

kˆ1

! c k Xk ‡ Xm ‡ 1 > x

2

 …1 ‡ e†

  m X x F ‡ …em ‡ e2 m ‡ e ‡ 1†F…x† c k kˆ1

m

hold uniformly for cm [ ‰a; bŠ and x  maxfB1 ; B2 ; B3 g. The arbitrariness of e > 0 gives result (36). This ends the proof of Proposition 5.1. & We end this subsection by giving a simple asymptotic property of the tail probability of the maximum of randomly weighted random variables. Proposition 5.2: Let fXk ; 1  k  ng be n independent random variables with P…Xk > x† > 0 for all x > 0 and 1  k  n, and let fyk ; 1  k  ng be other n random variables which are independent of fXk ; 1  k  ng and are bounded of type II. Then,  P

 X n P…yk Xk > x†: max yk Xk > x *

1kn

kˆ1

…44†

RANDOMLY WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES

Proof:

It is trivial that for any x > 0, 

P

187

 max yk Xk > x

n X



1kn

kˆ1

P…yk Xk > x†:

In order to prove the other inequality for (44), we apply an elementary inequality that for n general events E1 ; . . . ; En , n [

P

kˆ1

! Ek



n X kˆ1

X

P…Ek †

1  k 6ˆ l  n

P…Ek El †:

By this inequality we derive that for x > 0,  P

 max yk Xk > x

1kn

n X



kˆ1 n X



kˆ1

*

P…yk Xk > x† P…yk Xk > x†

n X kˆ1

X 1  k 6ˆ l  n

X 1  k 6ˆ l  n

P…yk Xk > x†P…bXl > x†

P…yk Xk > x†:

This ends the proof of Proposition 5.2.

5.2.

P…yk Xk > x; yl Xl > x†

&

Proof of Theorem 3.1

Recalling Proposition 5.2, it remains to prove the asymptotic relations (2). Since fyk ; 1  k  ng are bounded of type I, applying Proposition 5.1 and the dominated convergence theorem, we obtain that P…Syn > x† ˆ E‰P…Syn > x j y1 ; . . . ; yn †Š " # n X P…yk Xk > x j y1 ; . . . ; yn † *E kˆ1

ˆ

n X

kˆ1

P…yk Xk > x†:

Hence, the second asymptotic relation in (2) holds. In view of the two-sided inequality (12), the same relation for Mny is immediate. This ends the proof of Theorem 3.1. &

188

TANG AND TSITSIASHVILI

Acknowledgments Qihe Tang's work was supported by the Dutch Organization for Scienti®c Research (No: NWO 42511013), and Gurami Tsitsiashvili's work was supported by the Russian Fund of Basic Researches Project (No: 03-01-00512). References Bingham, N.H., Goldie, C.M., and Teugels, J.L., Regular variation, Cambridge University Press, Cambridge, 1987. Cai, J., ``Ruin probabilities with dependent rates of interest,'' J. Appl. Probab. 39(2), 312±323, (2002). Chistyakov, V.P., ``A theorem on sums of independent positive random variables and its applications to branching random processes,'' (Russian) Teor. Verojatnost. i Primenen 9, 710±718, (1964); translation in Theor. Probability Appl. 9, 640±648, (1964). Cline, D.B.H., ``Convolution tails, product tails and domains of attraction,'' Probab. Theory Relat. Fields 72(4), 529±557, (1986). Cline, D.B.H. and Samorodnitsky, G., ``Subexponentiality of the product of independent random variables,'' Stochastic Process. Appl. 49(1), 75±98, (1994). Embrechts, P. and Goldie, C.M., ``On closure and factorization properties of subexponential and related distributions,'' J. Austral. Math. Soc. Ser. A 29(2), 243±256, (1980). Embrechts, P., KluÈppelberg, C., and Mikosch, T., ``Modeling extremal events for insurance and ®nance,'' Springer-Verlag, Berlin, 1997. Goldie, C.M., ``Subexponential distributions and dominated-variation tails,'' J. Appl. Probability 15(2), 440±442, (1978). de Haan, L., ``On regular variation and its application to the weak convergence of sample extremes,'' Mathematical Center Tracts, 32 Mathematisch Centrum, Amsterdam, (1970). Hu, T., OrdoÂnÄez Cabrera, M., and Volodin, A.I., ``Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights,'' Statist. Probab. Lett. 51(2), 155±164, (2001). Ng, K.W., Tang, Q., and Yang, H., ``Maxima of sums of heavy-tailed random variables,'' Astin Bull. 32(1), 43± 55, (2002). Nyrhinen, H., ``On the ruin probabilities in a general economic environment,'' Stochastic Process. Appl. 83(2), 319±330, (1999). Rosalsky, A. and Sreehari, M., ``On the limiting behavior of randomly weighted partial sums,'' Statist. Probab. Lett. 40(4), 403±410, (1998). Sgibnev, M.S., ``Banach algebras of measures of class s…g†,'' (Russian) Sibirsk. Mat. Zh. 29(4), 162±171, 225, (1988); translation in Siberian Math. J. 29(4), 647±655, (1988/1989). Sgibnev, M.S., ``On the distribution of the maxima of partial sums,'' Statist. Probab. Lett. 28(3), 235±238, (1996). Tang, Q., ``The ruin probability of a discrete time risk model under constant interest rate with heavy tails,'' Scand. Actuar. J. 4(3), 229±240, (2004). Tang, Q. and Tsitsiashvili, G., ``Precise estimates for the ruin probability in ®nite horizon in a discrete-time model with heavy-tailed insurance and ®nancial risks,'' Stochastic Process. Appl. 108(2), 299±325, (2003).