Sums of dependent lognormal random variables: asymptotics and simulation Søren Asmussen∗ Leonardo Rojas-Nandayapa† Department of Theoretical Statistics, Department of Mathematical Sciences Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark
December 20, 2005
Abstract Let (Y1 , . . . , Yn ) have a joint n-dimensional Gaussian distribution with a general mean vector and a general covariance matrix, and let Xi = eYi , Sn = X1 + . . . + Xn . The asymptotics of P(Sn > x) as n → ∞ is shown to be the same as for the independent case with the same lognormal marginals. Further, a number of simulation algorithms based on conditional Monte Carlo ideas are suggested and their efficiency properties studied. Key words: Bounded relative error, complexity, conditional Monte Carlo, Gaussian copula, logarithmic efficiency, rare event, simulation, subexponential distribution, tail asymptotics, value at risk
∗ †
[email protected]; http://home.imf.au.dk/asmus
[email protected]
1
1
Introduction
Tail probabilities P(Sn > x) of a sum Sn = X1 + · · · + Xn of heavy-tailed risks X1 , . . . , Xn is of major importance in applied probability and its applications in risk management, such as the determination of risk measures like the value-at-risk (VaR) for given portfolios of risks, evaluation of credit risk, aggregate claims distributions in insurance, operational risk ([14], [19] etc. Under the assumption of independence among the risks, the situation is well understood. In particular, from the very definition of subexponential distributions, given identical marginal distributions, the maximum among the involved risks determines the distribution of the sum and, on the other hand, for non-identical marginals the distribution of the sum is determined by the component with the heaviest tail (see e.g. Asmussen [4, Ch.IX]). Over the last few years, several results in this direction have been developed. A survey and some new results are given in Albrecher & Asmussen [2]. For regularly varying marginals and n = 2, [2] gives bounds in terms of the tail dependence coefficient ¯ ¡ ¢ λ = lim P F2 (X2 ) > u ¯ F1 (X1 ) > u u→1
and it is noted that the asymptotics of P(Sn > x) is the same as in the independent case when λ = 0. For general discussion of bounds, see further Denuit et al. [15], Cossette et al. [13] and Embrechts & Puccetti [17]. Finally W¨ uthrich [25] and Alink et al. [3] gave sharp asymptotics of the tail of Sn in the case of an Archimedean copula. The overall picture is that, except for some special cases, the situation seems best understood with regularly varying marginals. This paper deals with the basic case of lognormal marginals with a multivariate Gaussian copula, which appears particularly important for applications to insurance and finance. That is, we can write Xk = eYk where the random vector (Y1 , . . . , Yn ) has a multivariate Gaussian distribution with E Yk = µk , Var Yk = σk2 and Cov(Yk , Y` ) = σk` (here σkk = σk2 ). The marginal density of Xk is n (log x − µ )2 o 1 k exp − 2 2σk x 2πσk √
Our investigations go in two directions, asymptotics of P(Sn > x) as x → ∞, and how to develop simulation algorithms which are efficient for large x. We next state and discuss our results in these two directions. Numerical illustrations are then given in Section 4, and the proofs of the theoretical results can be found in Sections 5–6.
2
Asymptotic Results
The asymptotics of P(Sn > x) is a problem with a well-known solution when the Xk are independent. More precisely, let © ª 2 2 σ 2 = max σk2 , µ = max µ (1) k , mn = # k : σk = σ , µk = µ . 2 k=1,...,n
k: σk =σ 2
2
Since the tail behaviour is well-known to be P(Xk > x) ∼ √
n (log x − µ )2 o σk k exp − 2 2σ 2π log(x − µk ) k
it follows that the lognormal distribution F µ,σ (x) with parameters µ, σ given by (1) is the heaviest among the marginal distributions of the Xk . Therefore by standard subexponential theory P(Sn > x) ∼ mF µ,σ (x) ∼ √
n (log x − µ)2 o σmn exp − 2σ 2 2π(log x − µ)
where F µ,σ (x) is the heaviest tail and mn the number of summands with this tail. Our main asymptotic result on P(Sn > x) states that this remains true in the dependent setting under consideration: Theorem 2.1. Let Xk = eYk where the Yk ’s have multivariate normal distribution with µk = E[Yk ] and σk2 = Var[Yk ]. Let Sn = X1 + . . . + Xn and let σ 2 , µ, mn be defined by (1). Then P(Sn > x) ∼ mn F µ,σ (x) (2) Remark 2.1. As will be seen, our proof that mn F µ,σ (x) is an asymptotic lower bound for P(Sn > x) essentially just uses the well-known tail independence of Gaussian copulas. It seems therefore reasonable to ask whether Theorem 2.1 remains valid when considering the same marginals but a different copula with tail independence. An example in Albrecher & Asmussen [2] shows that this is not the case. 2 Remark 2.2. Our numerical results show that the approximation in Theorem 2.1 is rather accurate but tends to underestimate. This could be explained by some r.v. Xi with tails being only slightly lighter than the heaviest one. This suggests considering the adjusted approximation P(Sn > x) ∼
n X
P(Xi > x)
(3)
i=1
which obviously has the same asymptotics.
3
2
Simulation Algorithms
We next consider simulation of P(Sn > x). By an estimator, we understand a r.v. Z(x) that can be generated by simulation and is unbiased, i.e. E Z(x) = P(Sn > x). The standard efficiency concepts in rare event simulation are bounded relative error, meaning Var Z(x) x) 3
and the slightly weaker concept of logarithmic efficiency, where one only requires lim sup x→∞
Var Z(x) x)2−²
for all ² > 0; for background, see Asmussen & Rubinstein [11], Heidelberger [22] and Asmussen & Glynn [8]. For the i.i.d. case, the first logarithmically efficient algorithm with heavy tails was given by Asmussen & Binswanger [6] in the regularly varying case. It simulates X1 , . . . , Xn , forms the order statistics X(1) < · · · < X(n) , discards the largest and returns the conditional Monte Carlo estimator ¯ ¡ ¢ F ((x − S(n−1) ) ∨ X(n−1) ) Z1 (x) = P Sn > n ¯ X(1) , . . . , X(n−1) = F (X(n−1) ) where S(n−1) = X(1) +. . .+X(n−1) . For the i.i.d. lognormal case, logarithmic efficiency was established in Binswanger [12]. The algorithm generalizes immediately to the dependent setting, with the modification that the conditional expectation comes out in a different way. For each k, let Fkc denote the conditional distribution of Xk given the X` with ` 6= k. Well-known formulas for conditioning in the normal distribution c show that Fkc is a lognormal distribution with parameters µck , σk2 where ¡ ¢ ¡ ¢t µc1 = µ1 + σ12 . . . σ1n Σ−1 log X2 − µ2 . . . log Xn − µn 1 σ12 σ22 · · · σ2n ¡ ¢ c .. .. .. .. σ12 = σ12 − σ12 . . . σ1n Σ−1 . . 1 . where Σ1 = . σ1n
σn2 · · ·
σnn
and similar formulas hold for k > 1. Let K be the k with Xk = X(n) . Theorem 3.1. Let X1 , .., Xn be lognormal random variables with Gaussian copula, X(1) , . . . , X(n) the corresponding order statistics, S(n−1) = X(1) + . . . + X(n−1) . Then c
Z2 (x) =
F K ((x − S(n−1) ) ∨ X(n−1) ) c
F K (X(n−1) )
(4)
is an unbiased and logarithmically efficient estimator of P(Sn > x). A somewhat different conditional Monte Carlo idea was suggested by Asmussen & Kroese [9], who used a (trivial) symmetry argument to note that the following estimator is unbiased in the i.i.d. case. It simulates X1 , . . . , Xn−1 , takes the maximum Mn−1 and returns the estimator nF (Mn−1 ∨ (x − Sn−1 )). The next modification provides a first approach to the exchangeable case: c
Z3 (x) = nF (Mn−1 ∨ (x − Sn−1 ))
(5)
where F c = F1c = . . . = Fnc . The algorithm is shown to have bounded relative error in the i.i.d. case with regularly varying marginals in [9]. That the same conclusion is true with lognormal marginals is a special case of the following result and was shown also independently by Kortschak [24] (but note that only independence is considered there). In fact: 4
Theorem 3.2. The estimator Z3 (x) has bounded relative error in the exchangeable lognormal case. We next consider the extension for the general non-exchangeable dependent case. For each i, simulate independently the set {Xj,i : j 6=Pi} with the same distribution as {Xj : j 6= i}, let Mn,i = max{Xj,i : j 6= i}, Sn,i = Xj,i and return j6=i
Z4 (x) =
n X
c
F i (Mn,i ∨ (x − Sn,i ))
(6)
i=1
A faster and simpler version of this algorithm is obtained when simulating P just the set {X1 , . . . , Xn } and considering Mn,i = max{Xj : j 6= i} and Sn,i = j6=i Xj . In practice, the first version showed smaller variances. Theorem 3.3. Both versions of the estimator Z4 (x) are unbiased and have bounded relative error. To improve the efficiency of this algorithm we can combine these estimators with other variance-reduction methods. Among these, importance sampling is wellknown. Under some technical conditions bounded relative error and logarithmic efficiency remain: Proposition 3.1. Let X be a random variable with density f (possibly a random vector). And Z(x, X) = Z(x) be an unbiased estimator of z(x) with bounded relative error (logarithmically efficient). If g is a density such that: f (t) x) with the associated 95% confidence interval corresponding to R = 5000 replications using the first version of estimator Z4 (x). The second graph shows how fast the relative difference (difference among the two approximations divided by the largest one) goes to 0.
Point estimate Z (x) 4
Confidence interval Approximation Theorem 1.1 Approximation Remark 1.1
−3
10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5 4
x 10 0.24 Relative Difference 0.22 0.2 0.18 0.16 0.14 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5 4
x 10
Figure 1: Tail probability P(S10 > x). Comparison of approximations and simulations.
6
Example 2 We use the same n = 8 and the same parameters as in Example 1. Figure 2 shows the comparison of the estimator Z2 (x) and the two versions of the estimator Z4 (x) using R = 5000 replications (no importance sampling technique used). A general algorithm by Glasserman, Heidelberger and Shahabuddin [21] was used for verification purposes using R = 50000 replications. Variances and times elapsed are similar, however, the implementation of the second version requires less effort and it has the advantage that in most of the cases it is easier to combine it with other variance reduction techniques.
Point Estimates
Variances 0
10 Z2(x) Z4(x) Version 1 Z4(x) Version 2
−2
10
GHS
Z2(x) Z4(x) Version 1 Z4(x) Version 2 GHS
−2
10
−3
10
−4
10
−4
10
2000
4000
6000
8000
10000
2000
Times elapsed
4000
6000
8000
10000
Times*variances
5
10 Z2(x) Z4(x) Version 1 Z4(x) Version 2
400 300
Z2(x) Z4(x) Version 1 Z4(x) Version 2
0
10
GHS
GHS
200 −5
10 100 0
−10
0
2000
4000
6000
8000
10
10000
0
2000
4000
6000
Figure 2: Tail probability P(S10 > x). Comparison of estimators Z2 (x), Z4 (x) and GHS.
7
8000
10000
Example 3 In the third example we consider 100 lognormal r.v. with multivariate gaussian copula. Let Fi ∼ N (i − 9, i + 1) with i = 0, . . . , 9 and X10i+1 , . . . , X10(i+1) with common distribution Fi . All correlations equal to 0.4. Figure 3 provides a comparison of the estimators Z2 (x) and the two versions of Z4 (x) using R = 10, 000 replications. Although the estimator Z2 (x) has a bigger variance, it provides a considerable faster algorithm than any version of Z4 (x), and for this example it is the most efficient in terms of time-variance comparisons. Point Estimates
Variances −2
10
Z (x) 2 Z4(x) Version 1 Z (x) Version 2
−2
10
Z (x) 2 Z4(x) Version 1 Z (x) Version 2
4
4
−4
10 −3
10
0.5
1
1.5
2
2.5
0.5
1
1.5
2
5
5
x 10
x 10
Times elapsed
Times*variances
Z2(x) Z (x) Version 1 4 Z (x) Version 2
80
Z2(x) Z (x) Version 1 4 Z (x) Version 2
4
60
2.5
4
40 20 0
0
0.5
1
1.5
2
2.5
0.5
1
5
1.5
2
2.5 5
x 10
x 10
Figure 3: Tail probability P(S100 > x). Comparison of estimators Z2 (x) and Z4 (x).
8
Example 4 Consider two lognormal r.v. with multivariate gaussian copula and parameters µ1 = µ2 = 0, σ12 = σ22 = 1 and σ12 = 0.7. We approximate the tail probability P(S2 > x) using the estimator Z2 (x) plus importance sampling. The new measure employed for sampling was a lognormal r.v. with parameters µ = σ12 x and σ 2 = 1. Results with R = 5000 replications are shown in figure 4. For this particular example, comparison is favorable: lower variance and shorter simulation times are observed in the new algorithm. Point estimates
Variances
GHS Z2(x) with IS Z2(x) without IS
GHS Z2(x) with IS Z2(x) without IS
−4
10
−3
10
−6
10
25
30
35
40
45
50
25
Times elapsed
30
35
40
45
50
Variance−Time Comparison (var*time)
0.5 GHS Z2(x) with IS Z2(x) without IS
0.4
GHS Z2(x) with IS Z2(x) without IS −6
10
0.3 0.2
−8
10
0.1 0 20
30
40
50
25
30
Figure 4: Tail probability P(S2 > x)
9
35
40
45
50
5
Proof of Asymptotic Results
Proof of Theorem 2.1. W.l.o.g. consider that X1 , . . . , Xn are ordered such that X1 ∼ Fµ,σ and µ = 0 (otherwise replace Xi and x by Xi e−µ and xe−µ ). We use induction. The case n = 1 is straightforward. Assume the theorem is true for n − 1. We need the following lemmas: Lemma 5.1. Let 0 < β < 1. Then P(Sn−1 > x − xβ ) ∼ P(Sn−1 > x) as x → ∞. Proof. By the hypothesis of the induction, we have P(Sn−1
σ1 mn−1 log2 (x − xβ ) >x−x )∼ √ exp{− } 2σ12 2π log(x − xβ ) β
Now, just note that log(x − xβ ) = log x + log(1 − 1/x1−β ) ∼ log x log2 (x − xβ ) = log2 x + log2 (1 − 1/x1−β ) + 2 log x log(1 − 1/x1−β ) 2 log x = log2 x + o(1) − 1−β = log2 x + o(1) . x Lemma 5.2. There exists 0 < β < 1 such that P(Sn−1 > xβ , Xn > xβ ) is asymptotically dominated by P(X1 > x) as x → ∞. Proof. If σ1 > σn choose σσn1 < β < 1. Then P(Sn > xβ , Xn > xβ ) ≤ P(Xn > xβ ), and this has lighter tail than P(X1 > x) since β > σσn1 . If σ1 = σn choose β such that β 2 > max{1/2, γ} and (β − βγ )2 + β 2 > 1 where γ = max{ σσkn 2 } (observe that this is possible since γ ∈ (−1, 1)). Let α = 1/β and n consider P(Sn−1 > xβ , Xn > xβ ) = P(Sn−1 > xβ , xα > Xn > xβ ) + P(Sn−1 > xβ , Xn > xα ) ≤ P(Sn−1 > xβ , xα > Xn > xβ ) + P(Xn > xα ) It follows that P(Xn > xα ) is asymptotic dominated by P(X1 > x) since α > 1 and c σ1 = σn . Denote Y (y) = (Y1 , . . . , Yn−1 |Yn = y) and the corresponding definitions c (y) and consider for X c (y) and Sn−1 αZlog x
P(Sn−1 > xβ , xα > Xn > xβ ) =
c P(Sn−1 (y) > xβ )fYn (y)dy β log x
10
Using the standard fact that Y c (y) ∼ N ({µi + σσin2 (y − µn )}i , {σij − bij }ij ), where n σ σ bij = inσ2 jn , we can bound the last by n
αZlog x c P(Sn−1 (0)eγy > xβ )fYn (y)dy
(7)
β log x
Let K be the index of the Ykc (y) with the heaviest tail. If γ < 0, then last expression c is bounded by P(Sn−1 (0) > xβ )P(Xn > xβ ) which is asymptotically bounded by c c β m P(XK (0) > x )P(Xn > xβ ) (hypothesis of induction), and this is asymptotically c and Xn do not have heavier tails than bounded by mc P2 (X1 > xβ ) (because XK X1 ). It follows that mc P2 (X1 > xβ ) ∼
σ1 mc 2β 2 log2 x } exp{− 2σ12 log2 xβ
is asymptotically dominated by P(X1 > x) since 2β 2 > 1 Now, consider the case γ > 0, then the expression (7) is bounded by: α
c c P(Sn−1 (0)eγ log x > xβ )P(Xn > xβ ) = P(Sn−1 (0) > xβ−γα )P(Xn > xβ )
By the hypothesis of induction last expression is asymptotically equivalent to c mc P(XK (0) > xβ−γα )P(Xn > xβ )
which is asymptotically bounded above by mc P(X1 > xβ−γα )P(X1 > xβ ) Observe that β − γα = β −
γ β
> 0 since we choose β 2 > γ. It follows that
mc P(Xn > xβ−γα )P(Xn > xβ ) ∼
σ12 mc [(β − γα)2 + β 2 ] log2 x exp{− } 2π log xβ−γα log xβ 2σ12
is asymptotically dominated by P(X1 > x) since (β − γ/β)2 + β 2 > 1. For an asymptotic lower bound of P(Sn > x), consider X P(Sn > x) ≥ P(Xi > x, Xj < x; j 6= i) i
=
X
P(Xi > x) − P(
i
≥
X X
P(Xi > x) −
=
i
X
P(Xi > x, Xj > x)
j6=i
P(Xi > x) −
i
X
{Xi > x, Xj > x})
j6=i
i
=
[
X
P(Xj > x|Xi > x)P(Xi > x)
j6=i
P(Xi > x) −
X j6=i
11
o(1)P(Xi > x) ∼ mn P(X1 > x)
where we use the fact that in the multivariate case Xi and Xj are tail independent. For an asymptotic upper bound, choose β as in the hypotesis of Lemma 5.2 and consider P(Sn > x) ≤ P(Sn > x, Sn−1 < xβ ) + P(Sn > x, Sn−1 > xβ , Xn < xβ ) + P(Sn > x, Sn−1 > xβ , Xn > xβ ) ≤ P(Xn > x − xβ ) + P(Sn−1 > x − xβ ) + P(Sn−1 > xβ , Xn > xβ ) We use succesively Lemma 5.1 twice, the hypotesis of induction and Lemma 5.2 to get the asymptotic upper bound P(Xn > x) + P(Sn−1 > x) + P(Sn−1 > xβ , Xn > xβ ) ∼ P(Xn > x) + mn−1 P(X1 > x) + P(Sn−1 > xβ , Xn > xβ ) ∼ P(Xn > x) + mn−1 P(X1 > x) ∼ mn P(X1 > x) completing the proof.
6
Proofs of Efficiency of the Simulation Algorithms
Proof of Theorem 3.1. We start by proving that estimator Z2 (x) is unbiased. Let F(n−1) = σ(K, Xi : i 6= K), hence P(Sn > x) = E[P(Sn > x|F(n−1) )] = E[P(X(n) + S(n−1) > x|F(n−1) )] = E[P(XK > x − S(n) |F(n−1) )] · c ¸ F K ((x − S(n−1) ) ∨ X(n−1) ) =E c F K (X(n−1) ) To prove that Z2 (x) is logarithmically efficient we need the following lemmas: Lemma 6.1. Let F1 and F2 lognormal distributions such that F2 has a heavier tail than F1 . Then, there exists c ∈ R such that F 1 (x) F 2 (x) ≤c F 1 (y) F 2 (y) for all y ≤ x. Proof. We will prove the case where σ1 < σ2 . Let λ1 (x), λ2 (x) the corresponding failure rate functions and consider the next function [λ1 (t) − λ2 (t)]+ = λ1 (t) − λ2 (t) + [λ2 (t) − λ1 (t)] I{t:λ1 (t) log x x
x
Lemma 6.2. Let X1 , X2 lognormal random variables with Gaussian copula. Then h(x) =
P(X1 > x, X2 > x) P(X1 > x)P(X2 > x)
is bounded.
13
Proof. h(x) is a continuous function. The result follows by observing that h(0) = 1 and lim h(x) = 1: x→∞
P(X1 > x, X2 > x) x→∞ P(X1 > x)P(X2 > x) P(X1 > X2 > x) + P(X2 > X1 > x) = lim x→∞ P(X1 > x)P(X2 > x) ∞ R∞ R P(X1 > t|X2 = t)f2 (t)dt + P(X2 > t|X1 = t)f1 (t)dt x = lim x x→∞ P(X1 > x)P(X2 > x) lim
Applying L’Hopital’s rule we obtain: P(X1 > x|X2 = x)f2 (x) + P(X2 > x|X1 = x)f1 (x) x→∞ P(X1 > x)f2 (x) + P(X2 > x)f1 (x)
= lim
Recall that F µ,σ is the distribution of the random variable with the heaviest tail. Lemma 6.3. The estimator Z2 (x) satisfies ¤ £ 2 Var(Z2 (x)) ≤ F µ,σ (x/n) c2 | log F µ,σ (x/n)| + c3 with c2 , c3 positive constants. Proof. Consider: · E
c 2
F K ((x − Sn−1 ) ∨ X(n−1) )
¸
· =E
c 2
F K (X(n−1) )
c 2
F K ((x − S(n−1) ) ∨ X(n−1) ) c 2
· +E
F K (X(n−1) )
; X(n−1)
x < n
c 2
F K ((x − S(n−1) ) ∨ X(n−1) ) c 2
F K (X(n−1) )
c
c
; X(n−1)
¸
x > n
¸
If X(n−1) < x/n then F K (x − S(n−1) ) < F K (x/n), so the last expression can be bounded by · c2 ¸ · ¸ F K (x/n) x x E c2 ; X(n−1) < + E 1; X(n−1) > n n F K (X(n−1) ) By Lemma 6.1 this can be bounded by · c1 E
=
2
F µ,σ (x/n) 2
F µ,σ (X(n−1) )
2 c1 F µ,σ (x/n)
Zx/n
; X(n−1)
f(n−1) (y) 2
0
F µ,σ (y)
¸ · ¸ x x < + E 1; X(n−1) > n n
dy + F (n−1) (x/n)
14
Using partial integration we can write this as · 2 c1 F µ,σ (x/n)
¯ ¸ Zx/n F (n−1) (y) ¯¯x/n F (n−1) (y)f (y) − dy + F (n−1) (x/n) 2 3 ¯ +2 F µ,σ (y) 0 F µ,σ (y) 0
· ¸ Zx/n F (n−1) (y)f (y) 2 ≤ c1 F µ,σ (x/n) 1 + 2 dy + F (n−1) (x/n) 3 F µ,σ (y)
(8)
0
Next, we use Lemma 6.2 to get X X 2 F (n−1) (x) ≤ P(Xi > x, Xj > x) ≤ c P(Xi > x)P(Xj > x) ≤ cn(n − 1)F µ,σ (x) i>j
i>j
hence, equation (8) is bounded by ·
¸ fµ,σ (y) 2 1 + 2cn(n − 1) dy + cn(n − 1)F µ,σ (x/n) F µ,σ (y) 0 ¤ £ 2 = F µ,σ (x/n) c2 | log F µ,σ (x/n)| + c3 Zx/n
2 c1 F µ,σ (x/n)
By Lemma 6.3: £ ¤ 2 | log(F µ,σ (x/n) c2 | log F µ,σ (x/n)| + c3 ) | log Var(Z2 (x))| lim inf ≥ lim inf 2 x→∞ | log P2 (Sn > x)| x→∞ log(F (x)) µ,σ
= lim inf x→∞
since lim inf
log(F µ,σ (x/n)) log(F µ,σ (x))
(9)
£ ¤ log( c2 | log F µ,σ (x/n)| + c3 ) 2 log(F µ,σ (x))
x→∞
=0
Using L’Hopital rule on expression (9) we obtain: log(F µ,σ (x/n)) lim inf = lim inf x→∞ x→∞ log(F µ,σ (x))
fµ,σ (x/n) F µ,σ (x/n) fµ,σ (x) F µ,σ (x)
= lim inf x→∞
λ(x/n) λ(x)
where λ(x) is the failure rate of a lognormal random variable and is well known to x be asymptotically equivalent to log . So the last limit is equal to σ2 x lim inf x→∞
log(x/n) σ 2 x/n log x σ2 x
= n lim inf x→∞
15
log(x/n) =n log x
Proof of Theorem 3.2. lim sup x→∞
Var(Z3 (x)) E(Z32 (x)) ≤ lim sup 2 P2 (Sn > x) x→∞ P (X1 > x) 2
≤ lim sup
(n + 1)2 E(F (Mn ∨ (x − Sn ))) 2
F (x)
x→∞
(10)
If Mn < x/2n, then Mn < x/2 < x − nMn < x − Sn , so 2
E[F (Mn ∨ (x − Sn ))] 2
F (x) 2
2
E[F (Mn ∨ (x − Sn )); Mn < x/2n] + E[F (Mn ∨ (x − Sn )); Mn > x/2n]
=
2
F (x) h F 2 (x − nM ) i h F 2 (M ) i n n ≤E ; Mn < x/2n + E ; Mn > x/2n 2 2 F (x) F (x)
(11)
We consider these two terms separately. First Integral: The first term in (11) is equivalent to Zx/2
2
F (x − y) 2
0
F (x)
fMn (y/n)dy
x/2 Zx/2 2 nF (x − y)F Mn (y/n) F (x − y)f (x − y) + 2n =− F Mn (y/n)dy 2 2 F (x) F (x) 0 0
2
σ we obtain that c t is an asymptotic upper bound for λ(t). Zx n Zx o n F (x − y) 1 o = exp λ(t)dt < exp c log x dt t F (x) x−y x−y n o = exp c log x(log x − log(x − y)) Using a first order Taylor expansion of log around (x − y) and the fact that it is y , so the last expression is a concave function we have that log x < log(x − y) + x−y y bounded by exp{c log x x−y }. y We claim that log(2y) > log x x−y when y ∈ (y0 , x/2). (This is true since both y functions are increasing and equal when y = x/2, but log(2y) is concave and log x y−x is convex). So, we have proved that
F (x − y) < c1 exp{c log y} F (x) Using this result and considering x > 2y0 we get Zx/2
2
y0
Z∞
2
F (x − y) F (x)
f (y/n)dy < y0
c log2 y √ 3 exp{c4 log y − }dy < ∞ 2 2π
For y ∈ (0, y0 ) we simply use the bound: Zy0
2
F (x − y) 2
0
F (x)
2
f (y/n)dy
(Mn,i ∨ (x − Sn,i ))] i
X
=
i c E−i [F i (Mn,i
∨ (x − Sn,i ))]
(12)
i
Next, we will prove that it has bounded relative error.
lim sup x→∞
Var(Z4 (x)) P2 (Sn > x) Var(
n P
i=1
= lim sup
P2 (Sn > x)
x→∞ n P
< lim sup i=1
n P
c
F i (Mn,i ∨ (x − Sn,i )))
= lim sup i=1
P2 (Sn > x)
x→∞
c2
E(F i (Mn,i ∨ (x − Sn,i )))
c < lim sup
P2 (Sn > x)
x→∞
c
Var(F i (Mn,i ∨ (x − Sn,i )))
n P
2
i=1
E(F µ,σ (Mn,i ∨ (x − Sn,i ))) 2
F µ,σ (x)
x→∞
where we used that the sets {Xji : j 6= i} where simulated independently. Last expression has the same form than expression (10). Proof is completed in the same way but checking that F Mn,i (x) < nF µ,σ (x). Proof of Theorem 3.3 (Version 2). Consider E−i the expectation taken over the set of r. v. {Xj : j 6= i}. X X P(Sn > x) = P(Sn > x, Xi = Mn ) = P[Xi > (Mn,i ∨ (x − Sn,i ))] i
=
X
i c E−i [F i (Mn,i
i
=E
hX
c F i (Mn,i
∨ (x − Sn,i ))] =
i ∨ (x − Sn,i ))
i
18
X i
c
E[F i (Mn,i ∨ (x − Sn,i ))]
proving that it is an unbiased estimator of P(Sn > x). Now, for the bounded relative error: Var(Z4 (x)) 2 x→∞ P (Sn > x) n P c Var( F i (Mn,i ∨ (x − Sn,i ))) i=1 = lim sup P2 (Sn > x) x→∞ h³ P ´2 i n c E F i (Mn,i ∨ (x − Sn,i )) i=1 ≤ lim sup P2 (Sn > x) x→∞ ·X n 1 c2 ≤ lim sup 2 F i (Mn,i ∨ (x − Sn,i )) E x→∞ P (Sn > x) i=1 ¸ X c c + F j (Mn,j ∨ (x − Sn,j ))F k (Mn,k ∨ (x − Sn,k ))
lim sup
j,k
we use Cauchy-Schwarz inequality: ·X n 1 c2 ≤ lim sup EF i (Mn,i ∨ (x − Sn,i )) x→∞ P(Sn > x) i=1 ¸ X £ c2 ¤ 12 c2 + EF j (Mn,j ∨ (x − Sn,j ))EF k (Mn,k ∨ (x − Sn,k )) j,k
c
n P i=1
≤ lim sup
c2
E(F i (Mn,i ∨ (x − Sn,i ))) P(Sn > x)
x→∞
c ≤ lim sup x→∞
n P i=1
2
E(F µ,σ (Mn,i ∨ (x − Sn,i ))) 2
F µ,σ (x)
The last expression has the same form than the expression in (10). Proof follows in the same way but checking that F Mn,i (x) < nF µ,σ (x). Proof of Proposition 3.1. Take n ∈ N such that
f (t) g(t)
< m. Then:
h³ h f (X) ´2 i f (X) i = Eg Z(x) − z 2 (x) Varg Z(x) g(X) g(X) h f (Z) i 2 ≤ mEg Z (x) − z 2 (x) g(Z) = mEg [Z 2 (x)] − z 2 (x) = mVarg [Z(x)] + (m − 1)z 2 (x) Dividing by z 2 (x), taking limits and using the fact that Z(x) has bounded relative error we obtain the desired result. 19
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