Sums of dependent lognormal random variables: asymptotics and simulation

Transcription

1 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, 8 Aarhus C, Denmark December 2, 25 Abstract Let (Y 1,..., Y n ) have a joint n-dimensional Gaussian distribution with a general mean vector and a general covariance matrix, and let X i = e Y i, S n = X X n. The asymptotics of P(S n > 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 asmus@imf.au.dk; leorojas@imf.au.dk 1

2 1 Introduction Tail probabilities P(S n > x) of a sum S n = X X n of heavy-tailed risks X 1,..., X n 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 ( F 2 (X 2 ) > u F1 (X 1 ) > u ) u 1 and it is noted that the asymptotics of P(S n > x) is the same as in the independent case when λ =. For general discussion of bounds, see further Denuit et al. [15, Cossette et al. [13 and Embrechts & Puccetti [17. Finally Wüthrich [25 and Alink et al. [3 gave sharp asymptotics of the tail of S n 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 X k = e Y k where the random vector (Y1,..., Y n ) has a multivariate Gaussian distribution with E Y k = µ k, Var Y k = σk 2 and Cov(Y k, Y l ) = σ kl (here σ kk = σk 2). The marginal density of X k is 1 x 2πσ k exp { (log x µ k) 2 } Our investigations go in two directions, asymptotics of P(S n > 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 Asymptotic Results The asymptotics of P(S n > x) is a problem with a well-known solution when the X k are independent. More precisely, let σ 2 = 2σ 2 k max k=1,...,n σ2 k, µ = max µ k, m n = # { k : σ k: σk 2 k 2 = σ 2, µ k = µ }. (1) =σ2 2

3 Since the tail behaviour is well-known to be P(X k > x) σ { k 2π log(x µk ) exp (log x µ k) 2 } 2σk 2 it follows that the lognormal distribution F µ,σ with parameters µ, σ given by (1) is the heaviest among the marginal distributions of the X k. Therefore by standard subexponential theory P(S n > x) mf µ,σ σm { n (log x } µ)2 exp 2π(log x µ) 2σ 2 where F µ,σ is the heaviest tail and m n the number of summands with this tail. Our main asymptotic result on P(S n > x) states that this remains true in the dependent setting under consideration: Theorem 2.1. Let X k = e Y k where the Yk s have multivariate normal distribution with µ k = E[Y k and σk 2 = Var[Y k. Let S n = X X n and let σ 2, µ, m n be defined by (1). Then P(S n > x) m n F µ,σ (2) Remark 2.1. As will be seen, our proof that m n F µ,σ is an asymptotic lower bound for P(S n > 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. 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. X i with tails being only slightly lighter than the heaviest one. This suggests considering the adjusted approximation P(S n > x) n P(X i > x) (3) which obviously has the same asymptotics. 3 Simulation Algorithms We next consider simulation of P(S n > x). By an estimator, we understand a r.v. Z that can be generated by simulation and is unbiased, i.e. E Z = P(S n > x). The standard efficiency concepts in rare event simulation are bounded relative error, meaning lim sup Var Z P(S n > x) 2 < 3

4 and the slightly weaker concept of logarithmic efficiency, where one only requires lim sup Var Z P(S n > x) 2 ɛ < for all ɛ > ; 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 X 1,..., X n, forms the order statistics X (1) < < X (n), discards the largest and returns the conditional Monte Carlo estimator Z 1 = P ( S n > n X (1),..., X (n 1) ) = F ((x S (n 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 Fk c denote the conditional distribution of X k given the X l with l k. Well-known formulas for conditioning in the normal distribution show that Fk c is a lognormal distribution with parameters µc k, σ2 kc where µ c 1 = µ 1 + ( ) ( ) σ σ 1n Σ 1 t 1 log X2 µ 2... log X n µ n σ1 2 c = σ 2 1 ( σ σ 1n ) Σ 1 1 σ 12. σ 1n where Σ 1 = and similar formulas hold for k > 1. Let K be the k with X k = X (n). σ 22 σ 2n..... σ n2 σ nn Theorem 3.1. Let X 1,.., X n be lognormal random variables with Gaussian copula, X (1),..., X (n) the corresponding order statistics, S (n 1) = X (1) X (n 1). Then = F c K((x S (n 1) ) X (n 1) ) F c K(X (n 1) ) (4) is an unbiased and logarithmically efficient estimator of P(S n > 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 X 1,..., X n 1, takes the maximum M n 1 and returns the estimator nf (M n 1 (x S n 1 )). The next modification provides a first approach to the exchangeable case: Z 3 = nf c (M n 1 (x S n 1 )) (5) where F c = F c 1 =... = F c n. 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

5 Theorem 3.2. The estimator Z 3 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 {X j,i : j i} with the same distribution as {X j : j i}, let M n,i = max{x j,i : j i}, S n,i = j i X j,i and return = n F c i(m n,i (x S n,i )) (6) A faster and simpler version of this algorithm is obtained when simulating just the set {X 1,..., X n } and considering M n,i = max{x j : j i} and S n,i = j i X j. In practice, the first version showed smaller variances. Theorem 3.3. Both versions of the estimator 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 be an unbiased estimator of z with bounded relative error (logarithmically efficient). If g is a density such that: f(t) g(t) < t Then the estimator given by the importance sampling method by making the change of measure from f to g has bounded relative error (logarithmically efficient). 5

6 4 Numerical Examples Example 1 We considered 1 lognormal r.v. with multivariate Gaussian copula and parameters µ = i 1, σi 2 = i and σ ij =.4σ i σ j. The first graph in figure 1 contains the approximations in Proposition 2.1 and Remark 2.2 and the simulated values for the tail probability P(S 1 > x) with the associated 95% confidence interval corresponding to R = 5 replications using the first version of estimator. The second graph shows how fast the relative difference (difference among the two approximations divided by the largest one) goes to. Point estimate Confidence interval Approximation Theorem 1.1 Approximation Remark x Relative Difference Figure 1: Tail probability P(S 1 > x). Comparison of approximations and simulations. x 1 4 6

7 Example 2 We use the same n = 8 and the same parameters as in Example 1. Figure 2 shows the comparison of the estimator and the two versions of the estimator using R = 5 replications (no importance sampling technique used). A general algorithm by Glasserman, Heidelberger and Shahabuddin [21 was used for verification purposes using R = 5 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 Version 1 Version 2 GHS 1 Variances 1 2 Version 1 Version 2 GHS Times elapsed Version 1 Version 2 GHS 1 5 Times*variances Version 1 Version 2 GHS Figure 2: Tail probability P(S 1 > x). Comparison of estimators, and GHS. 7

8 Example 3 In the third example we consider 1 lognormal r.v. with multivariate gaussian copula. Let F i N(i 9, i + 1) with i =,..., 9 and X 1i+1,..., X 1(i+1) with common distribution F i. All correlations equal to.4. Figure 3 provides a comparison of the estimators and the two versions of using R = 1, replications. Although the estimator has a bigger variance, it provides a considerable faster algorithm than any version of, and for this example it is the most efficient in terms of time-variance comparisons. 1 2 Point Estimates Version 1 Version Variances Version 1 Version Times elapsed x 1 5 Version 1 Version Times*variances x 1 5 Version 1 Version x x 1 5 Figure 3: Tail probability P(S 1 > x). Comparison of estimators and. 8

9 Example 4 Consider two lognormal r.v. with multivariate gaussian copula and parameters µ 1 = µ 2 =, σ1 2 = σ2 2 = 1 and σ 12 =.7. We approximate the tail probability P(S 2 > x) using the estimator plus importance sampling. The new measure employed for sampling was a lognormal r.v. with parameters µ = σ 12 x and σ 2 = 1. Results with R = 5 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 GHS with IS without IS 1 4 Variances GHS with IS without IS Times elapsed GHS with IS without IS Variance Time Comparison (var*time) GHS with IS without IS Figure 4: Tail probability P(S 2 > x) 9

10 5 Proof of Asymptotic Results Proof of Theorem 2.1. W.l.o.g. consider that X 1,..., X n are ordered such that X 1 F µ,σ and µ = (otherwise replace X i and x by X i 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 < β < 1. Then as x. P(S n 1 > x x β ) P(S n 1 > x) Proof. By the hypothesis of the induction, we have Now, just note that P(S n 1 > x x β ) σ 1 m n 1 (x x β ) 2π log(x xβ ) exp{ log2 } log(x x β ) = log x + log(1 1/x 1 β ) log x 2σ 2 1 log 2 (x x β ) = log 2 x + log 2 (1 1/x 1 β ) + 2 log x log(1 1/x 1 β ) = log 2 x + o(1) 2 log x x 1 β = log 2 x + o(1). Lemma 5.2. There exists < β < 1 such that P(S n 1 > x β, X n > x β ) is asymptotically dominated by P(X 1 > x) as x. Proof. If σ 1 > σ n choose σ n σ1 < β < 1. Then P(S n > x β, X n > x β ) P(X n > x β ), and this has lighter tail than P(X 1 > x) since β > σ n If σ 1 = σ n choose β such that β 2 > max{1/2, γ} and (β γ β )2 + β 2 > 1 where γ = max{ σ kn } (observe that this is possible since γ ( 1, 1)). Let α = 1/β and σn 2 consider σ1. P(S n 1 > x β, X n > x β ) = P(S n 1 > x β, x α > X n > x β ) + P(S n 1 > x β, X n > x α ) P(S n 1 > x β, x α > X n > x β ) + P(X n > x α ) It follows that P(X n > x α ) is asymptotic dominated by P(X 1 > x) since α > 1 and σ 1 = σ n. Denote Y c (y) = (Y 1,..., Y n 1 Y n = y) and the corresponding definitions for X c (y) and S c n 1(y) and consider P(S n 1 > x β, x α > X n > x β ) = α log x P(S c n 1(y) > x β )f Yn (y)dy β log x 1

11 Using the standard fact that Y c (y) N({µ i + σ in (y µ σn 2 n )} i, {σ ij b ij } ij ), where b ij = σ inσ jn, we can bound the last by σn 2 α log x P(Sn 1()e c γy > x β )f Yn (y)dy (7) β log x Let K be the index of the Yk c (y) with the heaviest tail. If γ <, then last expression is bounded by P(Sn 1() c > x β )P(X n > x β ) which is asymptotically bounded by m c P(XK c () > xβ )P(X n > x β ) (hypothesis of induction), and this is asymptotically bounded by m c P 2 (X 1 > x β ) (because XK c and X n do not have heavier tails than X 1 ). It follows that m c P 2 (X 1 > x β ) σ 1m c log 2 x log 2 x β exp{ 2β2 } 2σ1 2 is asymptotically dominated by P(X 1 > x) since 2β 2 > 1 Now, consider the case γ >, then the expression (7) is bounded by: P(S c n 1()e γ log xα > x β )P(X n > x β ) = P(S c n 1() > x β γα )P(X n > x β ) By the hypothesis of induction last expression is asymptotically equivalent to m c P(X c K() > x β γα )P(X n > x β ) which is asymptotically bounded above by m c P(X 1 > x β γα )P(X 1 > x β ) Observe that β γα = β γ β > since we choose β2 > γ. It follows that m c P(X n > x β γα )P(X n > x β ) σ1m 2 c 2π log x β γα log x exp{ [(β γα)2 + β 2 log 2 x } β is asymptotically dominated by P(X 1 > x) since (β γ/β) 2 + β 2 > 1. 2σ 2 1 For an asymptotic lower bound of P(S n > x), consider P(S n > x) i = i i = i = i P(X i > x, X j < x; j i) P(X i > x) P( j i{x i > x, X j > x}) P(X i > x) P(X i > x, X j > x) j i P(X i > x) P(X j > x X i > x)p(x i > x) j i P(X i > x) o(1)p(x i > x) m n P(X 1 > x) j i 11

12 where we use the fact that in the multivariate case X i and X j are tail independent. For an asymptotic upper bound, choose β as in the hypotesis of Lemma 5.2 and consider P(S n > x) P(S n > x, S n 1 < x β ) + P(S n > x, S n 1 > x β, X n < x β ) + P(S n > x, S n 1 > x β, X n > x β ) P(X n > x x β ) + P(S n 1 > x x β ) + P(S n 1 > x β, X n > x β ) We use succesively Lemma 5.1 twice, the hypotesis of induction and Lemma 5.2 to get the asymptotic upper bound completing the proof. P(X n > x) + P(S n 1 > x) + P(S n 1 > x β, X n > x β ) P(X n > x) + m n 1 P(X 1 > x) + P(S n 1 > x β, X n > x β ) P(X n > x) + m n 1 P(X 1 > x) m n P(X 1 > x) 6 Proofs of Efficiency of the Simulation Algorithms Proof of Theorem 3.1. We start by proving that estimator is unbiased. Let F (n 1) = σ(k, X i : i K), hence P(S n > x) = E[P(S n > x F (n 1) ) = E[P(X (n) + S (n 1) > x F (n 1) ) = E[P(X K > x S (n) F (n 1) ) [ c F = E K((x S (n 1) ) X (n 1) ) F c K(X (n 1) ) To prove that is logarithmically efficient we need the following lemmas: Lemma 6.1. Let F 1 and F 2 lognormal distributions such that F 2 has a heavier tail than F 1. Then, there exists c R such that for all y x. F 1 F 1 (y) cf 2 F 2 (y) Proof. We will prove the case where σ 1 < σ 2. Let λ 1, λ 2 the corresponding failure rate functions and consider the next function [λ 1 (t) λ 2 (t) + = λ 1 (t) λ 2 (t) + [λ 2 (t) λ 1 (t) I {t:λ1 (t)<λ 2 (t)}(t) λ 1 (t) λ 2 (t) + λ 2 (t) I {t:λ1 (t)<λ 2 (t)}(t) 12

13 We use L Hopital rule to check λ 1 lim λ 2 = lim log x σ 2 1 log x σ 2 2 = σ2 2 σ 2 1 proving that {t : λ 1 (t) < λ 2 (t)} [, y for some y R +. Now, λ 2 (t) is real-valued on the closed interval [, y and hence bounded by continuity. So, we get the next inequality λ 1 (t) λ 2 (t) + c 1 I [,y (t) Then { F 1 x F 1 (y) = exp y } { x λ 1 (t)dt exp { exp y x y > 1 λ 2 (t)dt + λ 2 (t)dt + x y y { = exp log F } 2 F 2 (y) + c 2 } c 1 I [,y (t)dt } c 1 dt = c F 2 F 2 (y) Let s turn to the case σ 1 = σ 2 and µ 1 < µ 2. It will be enough to prove that λ 2 λ 1. This is equivalent to prove that λ(x, µ) is a decreasing of µ. λ (x, µ) = = log x µ f(x, µ)f (x, µ) f(x, µ) σ 2 x x log t µ f(t, µ)dt σ 2 F 2 (t, µ) log xf(x, µ)f (x, µ) f(x, µ) log tf(t, µ)dt σ 2 F 2 (t, µ) The last expression is negative since log tf(t, µ)dt > log x f(t, µ)dt = log xf x x Lemma 6.2. Let X 1, X 2 lognormal random variables with Gaussian copula. Then is bounded. h = P(X 1 > x, X 2 > x) P(X 1 > x)p(x 2 > x) 13

14 Proof. h is a continuous function. The result follows by observing that h() = 1 and lim h = 1: lim P(X 1 > x, X 2 > x) P(X 1 > x)p(x 2 > x) P(X 1 > X 2 > x) + P(X 2 > X 1 > x) = lim P(X 1 > x)p(x 2 > x) P(X 1 > t X 2 = t)f 2 (t)dt + P(X 2 > t X 1 = t)f 1 (t)dt x x = lim P(X 1 > x)p(x 2 > x) Applying L Hopital s rule we obtain: = lim P(X 1 > x X 2 = x)f 2 + P(X 2 > x X 1 = x)f 1 P(X 1 > x)f 2 + P(X 2 > x)f 1 Recall that F µ,σ is the distribution of the random variable with the heaviest tail. Lemma 6.3. The estimator satisfies with c 2, c 3 positive constants. Proof. Consider: [ c 2 F K ((x Sn 1 ) X (n 1) ) E F c 2 K (X(n 1) ) Var( ) F 2 µ,σ(x/n) [ c 2 log F µ,σ (x/n) + c 3 [ c F = E 2 K ((x S(n 1) ) X (n 1) ) F c 2 ; X (n 1) < x K (X(n 1) ) n 2 K ((x S(n 1) ) X (n 1) ) F c 2 ; X (n 1) > x K (X(n 1) ) n [ c F + E If X (n 1) < x/n then F c K(x S (n 1) ) < F c K(x/n), so the last expression can be bounded by [ c 2 F K (x/n) E 2 (X(n 1) ) ; X (n 1) < x [ + E 1; X (n 1) > x n n F c K By Lemma 6.1 this can be bounded by [ 2 F µ,σ(x/n) c 1 E F 2 µ,σ(x (n 1) ) ; X (n 1) < x [ + E 1; X (n 1) > x n n = c 1 F 2 µ,σ(x/n) x/n f (n 1) (y) F 2 µ,σ(y) dy + F (n 1)(x/n) 14

15 Using partial integration we can write this as [ c 1 F 2 µ,σ(x/n) [ c 1 F 2 µ,σ(x/n) F (n 1)(y) F 2 µ,σ(y) x/n Next, we use Lemma 6.2 to get x/n + 2 x/n F (n 1) (y)f(y) F 3 µ,σ(y) F (n 1) (y)f(y) F 3 µ,σ(y) dy + F (n 1) (x/n) dy + F (n 1) (x/n) (8) F (n 1) i>j P(X i > x, X j > x) c i>j P(X i > x)p(x j > x) cn(n 1)F 2 µ,σ hence, equation (8) is bounded by [ c 1 F 2 µ,σ(x/n) 1 + 2cn(n 1) x/n = F 2 µ,σ(x/n) [ c 2 log F µ,σ (x/n) + c 3 f µ,σ (y) F µ,σ (y) dy + cn(n 1)F 2 µ,σ(x/n) By Lemma 6.3: lim inf log Var( ) log P 2 (S n > x) lim inf = lim inf log(f 2 µ,σ(x/n) [ c 2 log F µ,σ (x/n) + c 3 ) log(f µ,σ (x/n)) log(f µ,σ ) log(f 2 µ,σ) (9) since lim inf log( [ c 2 log F µ,σ (x/n) + c 3 ) 2 log(f µ,σ ) Using L Hopital rule on expression (9) we obtain: = lim inf log(f µ,σ (x/n)) log(f µ,σ ) = lim inf f µ,σ (x/n) F µ,σ (x/n) f µ,σ F µ,σ = lim inf λ(x/n) λ where λ is the failure rate of a lognormal random variable and is well known to be asymptotically equivalent to log x. So the last limit is equal to σ 2 x lim inf log(x/n) σ 2 x/n log x σ 2 x = n lim inf log(x/n) log x = n 15

16 Proof of Theorem 3.2. lim sup Var(Z 3 ) P 2 (S n > x) lim sup lim sup E( 3) P 2 (X 1 > x) (n + 1) 2 E(F 2 (M n (x S n ))) F 2 (1) If M n < x/2n, then M n < x/2 < x nm n < x S n, so E[F 2 (M n (x S n )) F 2 = E[F 2 (M n (x S n )); M n < x/2n + E[F 2 (M n (x S n )); M n > x/2n F 2 [ 2 F (x nmn ) [ 2 F (Mn ) E F 2 ; M n < x/2n + E F 2 ; M n > x/2n (11) We consider these two terms separately. First Integral: The first term in (11) is equivalent to x/2 F 2 (x y) F 2 f Mn (y/n)dy = nf 2 (x y)f Mn (y/n) F 2 x/2 + 2n < nf 2 nf 2 (x/2)f Mn (x/2n) F 2 < n + 2n 2 x/2 F 2 (x y) f(x y) F 2 F (x y) x/2 + 2n 2 F (x y)f(x y) F 2 F Mn (y/n)dy x/2 F (y/n) f(y/n) f(y/n)dy F (x y)f(x y) F 2 F (y/n)dy where we used F Mn < nf. For large x the last expression is bounded by n + 2n 2 c log(x/2) x/2 = n + 2nc log(x/2) log(x/2n) x/2n log(x/2n) x/2 x/2 F 2 (x y) F 2 f(y/n)dy F 2 (x y) F 2 f(y/n)dy This expression remains bounded due to the following Lemma. 16

17 Lemma 6.4. Proof. Consider x/2 lim F 2 (x y) F 2 f(y/n)dy < { x F = exp } λ(t)dt where λ(t) is the failure rate of the lognormal distribution and by standard subexponential theory we know that λ(t) is asymptotically equivalent to log Let y = sup{y} By choosing c > 1 σ bound for λ(t). 2 we obtain that c log t t σ 2 x. is an asymptotic upper F (x y) F { x = exp = exp x y } { x λ(t)dt < exp c log x x y { } c log x(log x log(x y)) 1 } t dt Using a first order Taylor expansion of log around (x y) and the fact that it is a concave function we have that log x < log(x y) + y, so the last expression is x y bounded by exp{c log x y }. x y We claim that log(2y) > log x y when y (y x y, x/2). (This is true since both functions are increasing and equal when y = x/2, but log(2y) is concave and log x y y x is convex). So, we have proved that F (x y) F < c 1 exp{c log y} Using this result and considering x > 2y we get x/2 y F 2 (x y) F 2 f(y/n)dy < For y (, y ) we simply use the bound: c 3 exp{c 4 log y log2 y 2π y 2 }dy < y F 2 (x y) F 2 f(y/n)dy < F 2 (x y ) F

18 Second Integral: x/2 The second term in (11) is equivalent to: F 2 (y) F 2 f M n (y/n)dy = nf 2 (y)f Mn (y/n) F 2 which goes as x. < nf 2 (x/2)f Mn (x/2n) F 2 x/2 2n x/2 F (y)f(y) F 2 F Mn (y/n)dy < n2 F 2 (x/2)f (x/2n) F 2 Proof of Theorem 3.2 (Version 1). First, we prove that the estimator is unbiased. Let E i the expectation taken over the set of r.v. s {X j,i : j i}, then P(S n+1 > x) = i P(S n > x, X i = M n ) = i P[X i > (M n,i (x S n,i )) = i E i [F c i(m n,i (x S n,i )) (12) Next, we will prove that it has bounded relative error. lim sup Var( ) P 2 (S n > x) = lim sup < lim sup Var( n F c i(m n,i (x S n,i ))) P 2 (S n > x) n E(F c 2 i (Mn,i (x S n,i ))) P 2 (S n > x) n Var(F c i(m n,i (x S n,i ))) = lim sup P 2 (S n > x) c n E(F 2 µ,σ(m n,i (x S n,i ))) < lim sup F 2 µ,σ where we used that the sets {X i j : j i} where simulated independently. Last expression has the same form than expression (1). Proof is completed in the same way but checking that F Mn,i < nf µ,σ. Proof of Theorem 3.3 (Version 2). Consider E i the expectation taken over the set of r. v. {X j : j i}. P(S n > x) = i P(S n > x, X i = M n ) = i P[X i > (M n,i (x S n,i )) = E i [F c i(m n,i (x S n,i )) = i i [ = E F c i(m n,i (x S n,i )) i E[F c i(m n,i (x S n,i )) 18

19 proving that it is an unbiased estimator of P(S n > x). Now, for the bounded relative error: lim sup Var( ) P 2 (S n > x) Var( n = lim sup lim sup lim sup + j,k F c i(m n,i (x S n,i ))) P 2 (S n > x) [( n ) E F c 2 i(m n,i (x S n,i )) P 2 (S n > x) [ n 1 P 2 (S n > x) E F c 2 i (Mn,i (x S n,i )) F c j(m n,j (x S n,j ))F c k(m n,k (x S n,k )) we use Cauchy-Schwarz inequality: lim sup + j,k lim sup lim sup [ n 1 EF c 2 i (Mn,i (x S n,i )) P(S n > x) [ c EF j2 (Mn,j (x S n,j ))EF c 2 k (Mn,k (x S n,k )) 1 2 c n E(F c 2 i (Mn,i (x S n,i ))) P(S n > x) c n E(F 2 µ,σ(m n,i (x S n,i ))) F 2 µ,σ The last expression has the same form than the expression in (1). Proof follows in the same way but checking that F Mn,i < nf µ,σ. Proof of Proposition 3.1. Take n N such that f(t) g(t) < m. Then: [ Var g Z f(x) [( = E g Z f(x) ) 2 z 2 g(x) g(x) [ me g f(z) z 2 g(z) = me g [ z 2 = mvar g [Z + (m 1)z 2 Dividing by z 2, taking limits and using the fact that Z has bounded relative error we obtain the desired result. 19

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