the tail behaviour of a random sum of subexponential ...

THE TAIL BEHAVIOUR OF A RANDOM SUM OF SUBEXPONENTIAL RANDOM VARIABLES AND VECTORS D. J. DALEY Centre for Mathematics and its Applications, The Australian National University, ACT, 0200, Australia [email protected] EDWARD OMEY Economische Hogeschool Sint-Aloysius, Brussels, Belgium [email protected] REIN VESILO Department of Electronics, Macquarie University, NSW, 2019, Australia [email protected] Abstract. Let {X, Xi , i = 1, 2, ...} denote independent positive random variables having common distribution function (d.f.) F (x) and, independent of X, let ν denote an integer valued random variable.
ν
∞ Using X0 = 0, the random sum Z = i=0 Xi has d.f. G(x) = n=0 Pr{ν = n}F n∗ (x) where F n∗ (x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each ε > 0 we can find a constant K such that the inequality 1 − F n∗ (x) ≤ K(1 + ε)n (1 − F (x)) ,

n ≥ 1, x ≥ 0 ,

holds. When F is subexponential and E(1 + ε)ν < ∞, it is a standard result in risk theory that G(x) satisfies 1 − G(x) ∼ E(ν)(1 − F (x)) ,

x → ∞.

(∗)

In this paper, we show that (*) holds under weaker assumptions on ν and under stronger conditions on F . Stam (1973) considered the case where F (x) = 1 − F (x) is regularly varying with index −α. He proved that if α > 1 and E(ν α+ε ) < ∞, then relation (*) holds. For 0 < α < 1, it is sufficient that E(ν) < ∞. In this paper we consider the case where F (x) is an O-regularly varying subexponential function. If the lower Matuszewska index β(F ) < −1, then the condition E(ν |β(F )|+1+ε ) < ∞ is sufficient for (*). If β(F ) > −1, then again E(ν) < ∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio F n∗ (x)/F (x). In the paper, we also consider (*) in the special case where X is a positive stable random variable or has a compound Poisson distribution derived from such a random variable and, in this case, we show that for n ≥ 2, the ratio F n∗ (x)/F (x) ↑ n as x ↑ ∞. In Section 3 of the paper, we briefly discuss an extension of Kesten’s inequality. In the final section of the paper, we discuss a multivariate analogue of (*).

2 Key words. Heavy tails, subexponential distribution, regularly varying distribution, O-regularly varying distribution, stable distribution, bounds, monotonicity AMS 2000 Subject Classification.

Primary – 60G70 Secondary – 60F10, 60G50, 60E07

3 1. Introduction Let F be a distribution function with support a subset, possibly all, of [0, ∞), F = 1 − F , F n∗ the n-fold convolution of F and F n∗ = 1 − F n∗ . The mean of F (if it exists) will be denoted by µ. The class S of subexponential distributions consists of those heavy-tailed distribution functions satisfying the condition that F 2∗ (x) = 2. x→∞ F (x)

(1.1)

lim

We will use the notation F ∈ S. It can be shown (see e.g. Embrechts et al. (1997) Lemma 1.3.4) that for n ≥ 2, (1.1) implies F n∗ (x) = n. x→∞ F (x)

(1.2)

lim

General introductions to subexponential distributions are contained in Embrechts et al. (1997), Rolski et al. (1999) and Asmussen (2000). Let G(x) be the weighted sum of convolutions of the form G(x) =

∞ 

pn F n∗ (x) ,

n=0

for some probability distribution {pn }. Unless stated otherwise, we assume lems involving the tail of G(x), G(x) =

∞ 

pn F n∗ (x) ,


∞ 0

npn < ∞. Prob(1.3)

n=1

occur in a number of areas such as the analysis of GI/GI/1 queueing systems with heavy-tailed service times and risk processes with heavy-tailed claim sizes (see e.g. Asmussen (2000) and Embrechts et al. (1997)). The usual method of deriving the limit of (1.3) is to obtain a bound on F n∗ (x)/F (x) and apply the dominated convergence theorem to give ∞ ∞  G(x) F n∗ (x)  = lim = pn npn . lim x→∞ F (x) x→∞ F (x) n=1 n=1

(1.4)

A commonly used bound on F n∗ (x)/F (x) is Kesten’s geometric bound (Athreya and Ney, 1972, p. 149), given below; it can be found with proof in Embrechts et al. (1997, Lemma 1.3.5(c)) and Rolski et al. (1999, Lemma 2.5.3). Kesten’s Lemma. If F ∈ S, then for each  > 0 there exists a constant K ≡ K() < ∞ such that for all n ≥ 2

F n∗ (x) ≤ K(1 + )n , F (x)

x ≥ 0.

By associating the distribution {pn } with an integer valued random variable, ν say, for which Pr{ν = n} = pn , (1.4) can be written as lim

x→∞

G(x) = E(ν) . F (x)

(1.5)

4 This result can be more succinctly stated as G(x) ∼ E(ν)F (x) .

(∗)

Appeal to Kesten’s Lemma requires E(z ν ) < ∞ for some z > 1, so the random variable ν must have finite moments of all orders. In the queueing examples given by Asmussen (2000), this is often the case: e.g. in Asmussen (2000) Theorem 3.1 of Chapter IX (p. 261), pn has a geometric distribution with parameter the defect of a ladder height distribution. This paper is concerned with conditions weaker than requiring that all moments of ν should be finite, considering only some sub-classes of subexponential distributions. This expands the range of applications where interchange of limit operations is required for limits of weighted sums of the form (1.3). Recall that F is said to be regularly varying with index −α if F (x) = L(x)x−α , where L(x) is slowly varying (i.e. limx→∞ L(ax)/L(x) = 1 for every finite a > 0). We then write F ∈ R−α , so L ∈ R0 . Stam (1973) was amongst the first to consider (∗) in the case of regular variation. Let V be a regularly varying function in Rρ with V (x) → ∞ as x → ∞, and consider the following relations as x → ∞ for some constants a, b, c : (a) V (x)F (x) → a ≥ 0; (b) V (x)G(x) → b; and (c) V (x) Pr{ν > x} → c ≥ 0. Stam (1973) obtained the results consolidated in Theorem A (his theorems are identified in parentheses). Theorem A. (i) (Theorem 1.4) If ρ > 1 and (c) holds, then (a) holds if and only if (b) holds with b = cµρ +aE(ν). (ii) (Theorem 4.1). If 0 ≤ ρ < 1 and E(ν) < ∞, then (a) holds if and only if (b) holds with b = aE(ν). Remark. (1) Stam (1973, Theorems 4.4 and 4.5) also treated the case where ρ = 1 and provided conditions under which (c) holds. See also Embrechts et al. (1985, Theorems 2 and 3) or Mallor and Omey (2006, Section 2.3). (2) Note that in case (i), where ρ > 1, the assumptions imply that µ < ∞. In the case where 0 < ρ < 1, the underlying mean, µ, is not finite. In Case (i), if V (n) Pr{ν > n} → 0 then (∗) follows. If F ∈ R−α for some α > 1, we can choose V (x) = 1/F (x) ∈ Rα and a = 1. If now E(ν α+ ) < ∞, then nα+ Pr{ν > n} → 0 (n → ∞). This implies V (n) Pr{ν > n} → 0 (n → ∞), again giving (∗). We show that similar conclusions hold under the weaker condition that the tail F is a subexponential function of O-regular variation, where (see p. 65 of Bingham, Goldie and Teugels (1989),

5 hereafter [BGT]) a positive measurable function f is of O-regular variation, denoted by f ∈ OR, if for every finite λ > 1, f (λx) f (λx) ≤ lim sup < ∞. f (x) f (x) x→∞

0 < lim inf x→∞

The lower Matuszewska index β(f ) of a function f ∈ OR is defined as the supremum of those β for which, for some constant D = D(β) > 0 and all Λ > 1, f (λx) ≥ D(1 + o(1))λβ , f (x)

x → ∞, uniformly in λ ∈ [1, Λ].

(See [BGT] p. 68). If f ∈ OR then β(f ) is finite ([BGT] Theorem 2.1.7). It is well known that if F ∈ S, then F ∈ L, i.e. for all real y we have that F (x + y)/F (x) → 1, as x → ∞. The converse, in general, is false. On the other hand, it is well known that F ∈ OR ∩ L implies that F ∈ S. We show that for F ∈ OR and β(F ) < −1, then F n∗ (x)/F (x) ≤ An|β(F )|+1+ for some constant A, and  > 0. For F ∈ OR and −1 < β(F ) < 0, we have F n∗ (x)/F (x) ≤ An for some constant A. We also show that if F is either a stable distribution with support [0, ∞) or a compound Poisson distribution derived from such a stable distribution then F n∗ (x)/F (x) is monotonic increasing in x for fixed n and the monotone convergence theorem can be used to justify (1.5), with the only restriction being that E(ν) should be finite. Since limx→∞ F n∗ (x)/F (x) = n the bound F n∗ (x)/F (x) ≤ n follows. In summary, this paper establishes Theorem 1, as we now state. Theorem 1. Let G(x) =


∞

n=1 pn F

n∗

(x) for F (x) some subexponential distribution and a prob-

ability distribution {pn : n = 0, 1, 2, . . .}.
∞ (a) If n=1 npn = ∞ then lim inf x→∞ G(x)/F (x) = ∞.
∞ (b) If E(ν) = n=1 npn < ∞ then ∞

 G(x) → npn < ∞ , F (x) n=1

x → ∞,

(1.6)

when also
∞ n (i) n=1 pn (1 + ) < ∞ for some  > 0; or (ii) F (x) = L(x)x−α is regularly varying with index −α where α > 1 and E(ν α+ ) =
∞ α+

pn < ∞ for arbitrarily small  > 0; or n=1 n (iii) F ∈ R−α , where 0 < α < 1; or (iv) F ∈ OR, with β(F ) < −1, F ∈ L ∩ OR ⊂ S and E(ν −β(F )+1+ ) < ∞ for arbitrarily small  > 0; or (v) F ∈ OR, with β(F ) > −1 and F ∈ L ∩ OR ⊂ S; or

6 (vi) F is the distribution function of a non-negative stable random variable or F is a compound distribution function of the form (4.2) with H the distribution function of a non-negative strictly stable random variable. Proof. Part (a) follows from (2.5.12) of Rolski et al. or else from Fatou’s Lemma and using the limit in (1.2): ∞=

∞  n=1

∞ 

npn =

pn lim inf

n=1

x→∞

∞  F n∗ (x) pn F n∗ (x) ≤ lim inf . x→∞ F (x) F (x) n=1

Case (i) of part (b) is standard in e.g. Embrechts et al. (1997), as already noted. Case (ii) follows from Theorem A (i) and Case (iii) follows from Theorem A (ii). Case (iv) follows from Theorem 2 (Section 3) where we show that F n∗ (x)/F (x) ≤ An|β(F )|+1+

(x > 0) for some constant A > 0 and, since F ∈ L ∩ OR, application of the dominated convergence theorem proves the result. Case (v) follows similarly but with Theorem 3 giving the bound F n∗ (x)/F (x) ≤ An (x > 0) for some constant A > 0 in lieu of Theorem 2. Case (vi) follows from Theorem 7 and Theorem 8 (Section 5), respectively, where in both cases F n∗ (x) ≤n F (x)

and

F n∗ (x) F (x)

is monotonic increasing

and so the results follow from the monotonic convergence theorem.




Remark. (1) We note that Case (v) of Theorem 1 includes Case (iii) since R−α ∈ OR. (2) Theorem A shows the precise interplay between the different distribution functions and in the case that c > 0 the constant E(ν) should be replaced by another constant. We leave it to the reader to complete the details of this case. We also present an extension of Kesten’s Lemma for subexponential distribution functions; it aims to identify more clearly the steps in developing the bound. In our original work, we used this extension to develop bounds for extended regularly varying distribution functions (denoted ER) before finding the stronger bounds given below for the larger class of O-regularly varying distribution functions (R ⊂ ER ⊂ OR). In the rest of the paper, Section 2 presents bounds on F n∗ (x)/F (x) for F ∈ OR. Section 3 presents a general bound for subexponential distributions. Section 4 proves the results for the stable distribution cases. Section 5 briefly discusses some multivariate analogues of these results. In related work by Kalma (1972), Embrechts et al. (1985), Rolski et al. (1999) and Teugels and Omey (2002), the authors relate G(x) to a renewal counting process and prove several results conunas cerning the asymptotic behaviour of G(x). Rates of convergence in (1.5) are provided by Baltr¯ and Omey (1998, 2002). A survey of results in the univariate and multivariate cases can be found in Mallor and Omey (2006). In the case of regularly varying distribution functions, Mikosch and Nagaev (1998, 2001) show that these rates of convergence can be arbitrarily slow. In this paper, we do not consider rates of convergence.

7 Fa¨ y et al. (2006) have proved a result similar to (1.6) under the condition that Pr{ν > x} =   o F (x) and F regularly varying; this condition is certainly met if E(ν β ) < ∞ for β > α + 2. They also look at the asymptotic tail behaviour of G under the condition that the tail of the distribution {pn } is heavier than that of F . Indeed, from their work it follows that the tail behaviour of G is dominated by the heavier of the tails of {pn } and F . 2. Bounds for Regularly Varying and OR Distribution Functions This section obtains bounds for F ∈ R and F ∈ OR of the form F n∗ (x)/F (x) ≤ g(n). If F ∈ R−α , α > 0, then g(n) = Anα+1+ , where  > 0 and A does not depend on x and n. If F ∈ OR similar results are obtained if α is replaced by the lower Matuszewska index |β(F )|. For F ∈ R−α , 0 < α < 1, or F ∈ OR and β(F ) > −1, we obtain the stronger result that g(n) = An by considering the Laplace–Stieltjes Transform (L-ST) of the integrated tail of F . Theorem 2. If F ∈ OR with lower Matuszewska index β = β(F ) and  > 0 then we can find a finite positive constant A such that F n∗ (x)/F (x) ≤ An|β|+1+ , all x ≥ 0. Proof. If F ∈ OR, we can use [BGT] Theorem 2.2.1 to find a positive constant D and a constant X = X(D, ) such that D

 y −|β|−

x

≤

F (y) , F (x)

y≥x≥X.

(2.1)

Now, we have F n∗ (x) ≤ nF (x/n) (e.g. see Chistyakov (1964), Lemma 1) and it follows that: F n∗ (x) ≤

n1+|β|+ F (x) , D

x ≥ nX .

If x < nX, then we have F n∗ (x) ≤ 1 and F (nX) ≤ F (x) and it follows that 1 F n∗ (x) ≤ , F (x) F (nX)

x ≤ nX

and, consequently (using (2.1)), also that 1 1 F n∗ (x) ≤ n|β|+ ≤ n|β|+1+ , F (x) DF (X) DF (X)

x ≤ nX .

Hence, if we let A = 1/(DF (X)) we have that F n∗ (x) ≤ An|β|+ +1 , F (x)

x > 0.


8 Remark. Since we only use the lower bound, the result also applies to distribution functions which have bounded decrease defined as those functions whose lower Matuszewska index is finite (see BGT p. 71). Theorem 2 improves Theorem 1 of Shneer (2004) who assumes that F (x) ∈ L ∩ OR and does not mention Matuszewska indices. We prove the inequality of Theorem 2 only by assuming the existence of β(F ) (which appears explicitly in the upper bound). Theorem 3. (i) If F ∈ R−α , for 0 ≤ α < 1, then we can find a finite positive constant A such that F n∗ (x)/F (x) ≤ An. (ii) If F ∈ OR and β(F ) > −1 then we can find a finite positive constant A such that F n∗ (x)F (x) ≤ An, all x ≥ 0. Proof. The proof follows immediately from Lemma 4(i) and Lemma 5(ii), below.




To formulate the results for Lemmas 4 and 5, associate to each d.f. the integrated tail as follows: 

x

mF (x) =

F (t)dt . 0

It is clear that mF (x) ≥ xF (x) and also it is clear that the L-ST is given by m  F (s) =

1 − F(s) . s

Now we establish the following result: Lemma 4. (i) For each n ≥ 1, we have F n∗ (x) ≤ nmF (x)/x. (ii) We have xG(x) ≤ mG (x) ≤ E(ν)mF (x). (iii) If µ = E(X) = ∞, then mG (x) ∼ E(ν)mF (x). Note that we always have xG(x) ≤ mG (x) and that lim inf G(x)/F (x) ≥ E(ν). If µ < ∞, we have mG (∞) = E(ν)µ. Proof. (i) Using L-STs we have m  F n∗ (s) =

1 − Fn (s) m  F (s) . 1 − F (s)

Using (1 − z n )/(1 − z) = 1 + z + z 2 + ... + z n−1 we obtain that mF n∗ (x) =

n−1  k=0

F k∗ ∗ mF (x) .

9 (Where ∗ denotes convolution, i.e. u ∗ v(x) =

x 0

u(x − y)dv(y).) Since F k∗ (x) ≤ 1, it follows that

mF n∗ (x) ≤ nmF (x) ,

n ≥ 1, x ≥ 0 .

(2.2)

Since we also have xF n∗ (x) ≤ mF n∗ (x), the first result follows. (ii) and (iii) Using (2.2) it easily follows that mG (x) ≤ E(ν)mF (x). To obtain (iii), recall that lim inf G(x)/F (x) ≥ E(ν). Since µ = ∞, we find that lim inf mG (x)/mF (x) ≥ E(ν).




Lemma 4 shows that xG(x) ≤ E(ν)mF (x). To find a relation with F (x), we have to make an assumption about F (x). Lemma 5. (i) For 0 ≤ α < 1, we have F (x) ∈ R−α if and only if mF (x) ∈ R1−α ; and, both statements imply that mF (x) ∼ xF (x)/(1 − α). (ii) If F (x) ∈ OR with β(F ) > −1, then mF (x) ≤ AxF (x), for some constant A. Proof. (i) This is a standard result in regular variation theory, (e.g. [BGT], Theorem 1.6.4). (ii) This is a standard result in the OR-theory, (e.g. [BGT], Corollary 2.6.2).




Lemmas 4 and 5 can now be used to prove Theorem 3. 3. General Bounds This section obtains general results on bounds on F n∗ (x)/F (x) for F ∈ S. Define F n∗ (x) . F (x)

αn = sup x≥0

(3.1)

The force of Theorem 6 below is that it gives bounds on αn in terms of the tail behaviour of just F and F 2∗ . Let {Tn : n = 1, 2, . . .} be some sequence of values, to be determined later, and choose n to satisfy sup x≥Tn

F 2∗ (x) − F (x) F 2∗ (x) = sup − 1 ≤ 1 + n , F (x) x≥Tn F (x)

(3.2)

where, because F ∈ S, we can make n arbitrarily small by choosing Tn large enough. Define cn = F (Tn )/F (Tn ), so that 1 + cn = 1 + F (Tn )/F (Tn ) = 1/F (Tn ). Theorem 6. For n = 1, 2, . . . , let Tn , cn and n be as given above. Then αn ≤

n−1 

n−1

j=1

k=j+1

(1 + cj )

(1 + k ) +

n−1

(1 + k ) ,

k=1

10 where the empty product

n−1

k=1 (1

+ k ) = 1.

Proof. Our proof modifies the approach used in deriving the geometric bound in Kesten’s Lemma. Using F (n+1)∗ (x) = F (x) + F ∗ F n∗ (x),

 x n∗  x n∗ F (x − y) F (x − y) F (x − y) dF (y), sup · dF (y) sup αn+1 = 1 + max F (x) F (x − y) F (x) 0≤x 0 and that F n∗ (x)/F (x) ≤ n. An alternative proof of the upper bound is to prove Theorem 7 for just n = 2. Then, since all stable distributions are absolutely continuous (e.g. Lukacs (1960) Theorem 5.7.4) it follows that F (0) = 0 and the upper bound comes from Corollary 6.2. ∞
∞ As n ≥ 0 for every n, it follows that k=1 (1 + k ) is finite if and only if k=1 k < ∞. This gives the following: Corollary 6.3. If A ≡

∞

k=1 (1

+ k ) < ∞ then

αn ≤ A + A

n−1  (1 + cj ) = A 1 +

n−1  j=1

j=1

1 F (Tj )

.

11 4. Monotone Convergence of Stable Distributions A random variable X is stable (see e.g. Samorodnitsky and Taqqu (1994)) if for any positive numbers d

A and B there exist a positive number C and real number D such that AX1 + BX2 = CX + D, where X1 and X2 are independent copies of X. A strictly stable random variable is one for which D = 0. The index α of a stable random variable is the value of α such that C α = Aα + B α ; then α ∈ (0, 2]. For α = 1, the sum of n ≥ 2 i.i.d. strictly stable random variables X1 , . . . , Xn , where X1 , X2 , . . . are independent copies of X, satisfies d

X1 + · · · + Xn = n1/α X.

(4.1)

A stable random variable can be characterised by four parameters: α ∈ (0, 2], σ > 0, β ∈ [−1, 1] and µ ∈ (−∞, ∞), where α is the index, σ the scale parameter, β the skewness parameter and µ the location parameter. X can denoted by d

X = Sα (σ, β, µ). For strictly stable random variables, µ = 0. Further, if X ≥ 0 a.s., then 0 < α < 1 and β = 1, so d

that for a positive strictly stable random variable X, we have X = Sα (σ, 1, 0). The characteristic function of a stable distribution is of the form φ(t) = exp [iµt − σ α |t|α {1 + β sgn(t)ω(|t|, α)}] 

where ω(|t|, α) =

tan( 12 πα) (2 log t)/n

(α = 1), (α = 1).

Using the normalisation (see e.g. Lukacs (1960, pp. 102–103)) K 2 = cos2 ( 12 πα) + β 2 sin2 ( 12 πα), −β sin( 12 πα) , K 1/α  cos( 12 πα) , σ= K

sin( 12 πγ) =

cos( 12 πγ) =

β cos( 12 πα) , K

the characteristic function can be written in the form φ(t) = exp[−|t|α (cos γ − i sgn(t) sin γ)] , for α = 1. Denote the probability density of such a random variable by p(x; α, γ) (see e.g. Lukacs, 1960). For positive strictly stable random variables, γ = −α.

12 From [BGT] equation (8.3.12), stable distribution functions with support [0, ∞) and index α are regularly varying and belong to R−α , and, hence, are subexponential. A more elaborate case involving stable random variables can be constructed using compound Poisson distributions. The compound Poisson distribution F governed by a Poisson distribution with parameter λ and distribution H is defined as −λ

F (x) = e

∞  λk k=0

The characteristic function of F is  φF (t) =

∞

−∞

k!

H k∗ (x) .

(4.2)

  eitx dF (x) = exp − λ[1 − φH (t)] ,

where φH (t) is the characteristic function of H. It follows that the characteristic function of F n∗ (t)   is exp − nλ[1 − φH (t)] and so F n∗ (x) = e−nλ

∞  (nλ)k k=0

H k∗ (x) = e−nλ + e−nλ

k!

∞  (nλ)k k=1

k!

H k∗ (x) .

We now give two monotonicity results for stable distributions. Theorem 7. If F is a stable non-negative random variable then, for all n ≥ 2, 1≤

F n∗ (x) ↑ n, F (x)

0 ≤ x → ∞.

Theorem 8. If F is a compound Poisson distribution of the form (4.2) with H the distribution function of a non-negative strictly stable random variable then, for all n ≥ 2, 1≤

F n∗ (x) ↑ n, F (x)

0 ≤ x → ∞.

In the proofs of these results, we use properties of totally positive functions: k(x, y) is said to be totally positive of order 2 (TP2 ) if the determinant |k(xi , yj )| ≥ 0, where x1 < x2 and y1 < y2 (see Karlin (1968)). Let fn (x) be the density function of F n∗ (x) and define ζn (u, v) = fn (v)f (u) − f (v)fn (u) = −ζn (v, u) . Lemma 9. When F has a density f , the condition that F n∗ (x)/F (x) is monotonic increasing is equivalent to the condition

 I3 ≡



y

du x

y

∞

ζn (u, v) dv ≥ 0,

13 which is certainly satisfied when ζn (u, v) ≥ 0 for 0 < u < v. Proof of Lemma 9. The condition that F n∗ (x)/F (x) is monotonic increasing can be written as I ≡ F n∗ (y)F (x) − F n∗ (x)F (y) ≥ 0 ,

0 < x < y < ∞.

This is equivalent to  ∞  ∞  ∞  ∞ fn (v) dv f (u) du − f (v) dv fn (u) du 0≤I= y x y x  [fn (v)f (u) − f (v)fn (u)] du dv = u>x,v>y  u  ∞  ∞  y   ∞ du ζn (u, v) dv + du ζn (u, v) dv + du = y

y

y

= I1 + I2 + I3 , Now,

 I2 =



∞

dv y

x

ζn (u, v) dv

y

say. 

∞

u

∞



∞

ζn (v, u) du =

u

du

v

y

ζn (v, u) dv = −I1 ,

y

since ζn (u, v) = −ζn (v, u), giving I = I3 .




Proof of Theorem 5. Since α is fixed, write f (x) ≡ p(x; α, −α). Using the property (4.1) of positive strictly stable random variables, fn (x) = and ζn (u, v) =

1



1 n1/α

f (x/ n1/α )

 f (v/ n1/α )f (u) − f (v)f (u/ n1/α ) .

n1/α Lemma 1(iv) of Gawronski (1984) states that for 0 < α < 1, the kernel K(x, y) ≡ p(ex−y ; α, −α) ,

−∞ < x, y < ∞ ,

is strictly totally positive for all orders. To prove I3 ≥ 0 in Lemma 9, applying the strictly totally positive result of Gawronski (1984) to K(x, y) = ζn (u, v) gives f (ex1 −y1 )f (ex2 −y2 ) − f (ex2 −y1 )f (ex1 −y2 ) ≥ 0. This result can be applied if an association can be made between u, v and x1 , x2 , y1 , y2 such that the conditions for Gawronski’s Lemma are true. Using the substitutions v , n1/α u = 1/α , n

ex1 −y1 = u,

ex2 −y2 =

ex2 −y1 = v,

ex1 −y2

14 we then have ey1 −y2 =

ex2 −y2 ex1 −y2 1 = = 1/α x −y x −y 2 1 1 1 e e n

and

ex1 −x2 =

ex1 −y1 ex1 −y2 u = = . x −y x −y 2 1 2 2 e e v

Here, in the first relation, n ≥ 2 and α > 0 so y1 < y2 , while the other relation gives x1 < x2 precisely when u < v. Total positivity therefore implies that in the region u < v, ζn (u, v) ≥ 0, giving I3 ≥ 0. Proof of Theorem 8. Since H is strictly stable, with density h(·) say, we have ∞  (nλ)k

F n∗ (x) = e−nλ + e−nλ

k=1

k!

H(x/ k1/α ),

fn (x) = e−nλ

∞  (nλ)k h(x/ k1/α ). 1/α k! k k=1

Following the proof of Theorem 7,   ∞  ∞  (nλ)k  λm −nλ 1/α −λ 1/α h(v/ k ) e h(u/ m ) fn (v)f (u) = e k1/α k! m1/α m! m=1 k=1 = e−(n+1)λ

∞  ∞ 

nk λk+m h(v/ k1/α ) h(u/ m1/α ). 1/α m1/α k! m! k k=1 m=1

A similar derivation for f (v)fn (u) gives −(n+1)λ

fn (u)f (v) = e

∞ ∞  

nk λk+m h(u/ k1/α ) h(v/ m1/α ). 1/α m1/α k! m! k k=1 m=1

˜ v), where Thus, ζn (u, v) = fn (v)f (u) − f (v)fn (u) = e−(n+1)λ ζ(u, ˜ v) = ζ(u, =

  nk λk+m 1/α 1/α 1/α 1/α h(v/ k )h(u/ m ) − h(u/ k )h(v/ m ) k1/α m1/α k! m! k=1 m=1 ∞ ∞  

  nk λk+m 1/α 1/α 1/α 1/α h(v/ k )h(u/ m ) − h(u/ k )h(v/ m ) +0 1/α m1/α k! m! k k=1 m=k+1 ∞ ∞  

+ =

  nk λk+m 1/α 1/α 1/α 1/α h(v/ k )h(u/ m ) − h(u/ k )h(v/ m ) k1/α m1/α k! m! k=1 m=2 ∞ k−1  

  nk λk+m 1/α 1/α 1/α 1/α h(v/ k )h(u/ m ) − h(u/ k )h(v/ m ) k1/α m1/α k! m! k=1 m=k+1 ∞ ∞  

+

∞ ∞   k=1 m=k+1

=

  nm λk+m 1/α 1/α 1/α 1/α h(v/ m )h(u/ k ) − h(u/ m )h(v/ k ) k1/α m1/α k! m!

  m  λk+m k 1/α 1/α 1/α 1/α n − n )h(u/ k ) − h(u/ m )h(v/ k ) . h(v/ m k1/α m1/α k! m! k=1 m=k+1 ∞ ∞  

As in Theorem 7, Gawronski’s result can be applied to show that ξk,m (u, v) ≡ h(v/ m1/α )h(u/ k1/α ) − h(v/ k1/α )h(u/ m1/α ) ≥ 0.

15 Using the substitutions

v , m1/α v = 1/α , k

ex1 −y1 = ex2 −y1

u

ex2 −y2 = ex1 −y2 =

k1/α

,

u , m1/α

we then have ey1 −y2 =

ex2 −y2 ex1 −y2 u = = , x −y x −y 2 1 1 1 e e v

ex1 −x2 =

ex1 −y1 ex1 −y2 k1/α = = . ex2 −y1 ex2 −y2 m1/α

In the first relation, y1 < y2 is equivalent to u < v, and in the second, x1 < x2 is equivalent to k < m. Thus, in the region u < v and under the condition k < m, in the last summation of the derivation of ζn (u, v) above, ξk,m(u, v) ≥ 0. Since nm − nk > 0 for m > k, it follows that I3 ≥ 0.
5. Multivariate Case

In this section we briefly discuss some multivariate analogues of our results. For convenience and without loss of generality, we only discuss the two-dimensional case. Let F (x, y) = Pr{X ≤ x, Y ≤ y} denote a bivariate d.f. with marginals F1 (x) and F2 (x) and (1)

(2)

suppose that X ≥ 0, Y ≥ 0. Partial sums will be denoted by (Sn , Sn ) and we use the notation (1)

(2)

F ∗n (x, y) = 1 − F n∗ (x, y), where F n∗ (x, y) is the d.f. of (Sn , Sn ). Proposition 10. (i) If the marginals F1 and F2 are subexponential (in ) then, for each ε > 0, we can find a constant K such that F ∗n (x, y) ≤ K(1 + ε)n F (x, y) ,

x ≥ 0, y ≥ 0 .

(ii) If F1 and F2 are in the class OR with lower Matuszewska indices β(1) and β(2) then, for each ε > 0, we can find a constant K such that F ∗n (x, y) ≤ Knmax(|β(i)|,|β(2)|)+ε+1 F (x, y) ,

x ≥ 0, y ≥ 0 .

(iii) If for i = 1, 2 we have Fi ∈ R−α(i) , where 0 ≤ α(i) < 1, or if Fi ∈ OR with β(i) > −1, then we can find a constant K such that F ∗n (x, y) ≤ KnF (x, y) ,

x ≥ 0, y ≥ 0 .

Proof. (i) We have F ∗n (x, y) ≤ F1∗n (x) + F2n∗ (y). Using Kesten’s inequality we obtain that F ∗n (x, y) ≤ K(1 + ε)n (F1 (x) + F2 (y)),

x ≥ 0, y ≥ 0 .

16 Since F1 (x) + F2 (y) ≤ 2F (x, y), the result follows. (ii) Similar to (i) but now using Theorem 2.


(iii) Similar to (i) but now using Theorem 3.
∞

Now consider the random vector of random sums

n=0

pn F

n∗

(1) (2) (Sν , Sν ).

Its d.f. is given by G(x, y) =

(x, y) and the tail is given by G(x, y) =

∞ 

pn F n∗ (x, y) .

n=1

As in the univariate case it is straightforward to prove that as min(x, y) → ∞, we have lim inf

min(x,y)→∞

G(x, y)/F (x, y) ≥ E(ν) .

To obtain an upper bound, we use the following result of Baltrunas et al. (2006, Proposition 11). Theorem 11. If the marginals Fi are subexponential, then as min(x, y) → ∞ we have F n∗ (x, y) → n. F (x, y) We now combine this result with the inequalities found earlier. Corollary 11.1. (i) Suppose that the marginals Fi are subexponential and assume that E(1 + ε)ν < ∞. Then, as min(x, y) → ∞, we have G(x, y) → E(ν) . F (x, y)

(5.1)

(ii) If the marginals F1 and F2 are in the class OR ∩L with lower Matuszewska indices β(1) and β(2), and if E(ν max(|β(1)|,|β(2)|)+ε+1 ) < ∞, then (5.1) holds. (iii) Suppose that for i = 1, 2 we have Fi ∈ R−α(i) where 0 ≤ α(i) < 1, or suppose that Fi ∈ OR ∩ L with β(i) > −1. If E(ν) < ∞, then we have (5.1). 6. Concluding remarks

1) In most cases we can assume that X is a real random variable by defining X + = max(0, X). Since, for x > 0, we have Pr{X > x} = Pr{X + > x}, the results can be reformulated in an obvious
n way. (Taking sums we have Sn ≤ 1 Xi+ so that F n∗ (x) ≤ F+n∗ (x).) 2) Suppose that F (x, y) is a multivariate regularly varying function such that for some h(x) ∈ R−α , α > 0, we have F (tx, ty)/h(t) → λ(x, y),

∀x, y

with min(x, y) < ∞ .

17 In this case, Corollary 11 (i) and (ii) yield results of the type G(tx, ty) → E(ν)λ(x, y) . h(t) Such type of results were established in Omey (1990). 3) Theorem 11 shows a multivariate form of subexponential behaviour which follows from only univariate assumptions about the marginals of F . There are other papers in which the authors discuss multivariate subexponential distributions. We mention here the paper of Cline and Resnick (1992) and more recently the papers of Omey (2006) and Omey et al. (2006). In these papers, the authors define subexponential distributions by using relations of the form (see Omey et al. (2006) and Mallor and Omey (2006)) F n∗ (tx + a, ty + b) = ϑ(x, y, a, b) t→∞ F (tx, ty) lim

or Cline and Resnick (1992) lim tF n∗ (a(t)x + b(t), c(t)y + d(t)) = ϑ(x, y) .

t→∞

18 Acknowledgement. We thank Charles Goldie for noting an error in an earlier version of the manuscript.

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