Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications

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1 Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications Remigijus Leipus (with Yang Yang, Yuebao Wang, Jonas Šiaulys) CIRM, Luminy, April 26-30, 2010

2 1. Preliminaries 2. Negative dependence 3. Some heavy-tailed distribution classes 4. Asymptotics of random sums 4.1 P(S N > x) is dominated by P(N > x) 4.2 P(S N > x) is dominated by P(X 1 > x) 4.3 P(S N > x) and P(X 1 > x) are comparable 5. Applications to some dependent risk models 6. Application to the Galton-Watson branching process 1

3 1. Preliminaries X 1, X 2,... (claim sizes) sequence of dependent identically distributed nonnegative r.v.s with distribution F and finite mean µ = EX 1 N (claim number) nonnegative integer-valued r.v. with distribution H, which is not necessarily independent of {X 1, X 2,... } Let S n = X X n We are interested in behavior of P(S N > x) as x 2

4 X 1, X 2,... are i.i.d. and independent of N; F and N are both light-tailed (that is Ee δx 1 < and Ee δn < for some δ > 0). Then (under some additional assumptions) P(S N > x) c 1 e c 2x X 1, X 2,... are i.i.d. and independent of N; F heavy-tailed, N light-tailed. Then P(S N > x) EN F (x) 3

5 X 1, X 2,... are negatively dependent, independent of N; F C (consistent variation), EX1 r < for some r > 1, N light-tailed. Then P ( N (Chen and Zhang (2007)) (X k µ) > x k=1 ) EN F (x) Behavior of P(S N > x) is dominated by the heavier tail: F (x) = P(X 1 > x) or H(x) = P(N > x) 4

6 2. Negative dependence a) ξ 1, ξ 2,... are Upper Negatively Dependent (UND), if for each n 1 and y 1,..., y n P ( n {ξ k > y k } k=1 ) n k=1 P(ξ k > y k ) b) ξ 1, ξ 2,... are Lower Negatively Dependent (LND), if for each n 1 and y 1,, y n P ( n {ξ k y k } k=1 ) n k=1 P(ξ k y k ) 5

7 c) Negative Dependence (ND) = UND+LND d) If n = 2, then UND, LND, and ND structures are equivalent. In this case r.v.s ξ 1 and ξ 2 are called Negative Quadrant Dependent (NQD) e) ξ 1, ξ 2,... are pairwise NQD, if for all i j, ξ i and ξ j are NQD. Clearly, if r.v.s ξ 1, ξ 2,... are either UND or LND, they are also pairwise NQD 6

8 UND, LND: Ebrahimi and Ghosh (1981) and Block, Savits and Shaked (1982) NQD: Lehmann (1966) Negative Association: Alam and Saxena (1981), Joag-Dev and Proschan (1983) D.D. Mari, S. Kotz. Correlation and Dependence Imperial College Press. M. Denuit, J. Dhaene, M. Goovaerts, R. Kaas. Actuarial Theory for Dependent Risks Wiley. 7

9 Lemma (i) if {ξ k, k 1} are UND (LND) and {f k, k 1} are strictly increasing functions, then {f k (ξ k ), k 1} are still UND (LND, respectively); (ii) if {ξ k, k 1} are UND (LND) and {f k, k 1} are strictly decreasing functions, then {f k (ξ k ), k 1} are LND (UND, respectively); (iii) if {ξ k, k 1} are ND and {f k, k 1} are strictly increasing or strictly decreasing functions, then {f k (ξ k ), k 1} are still ND; (iv) if {ξ k, k 1} are nonnegative and UND, then for each n 1 E ( n k=1 ξ k ) n k=1 Eξ k 8

10 An example of r.v.s which are pairwise ND but not independent: n-dimensional Farlie Gumbel Morgenstern distribution F X1,...,X n (x 1,..., x n ) = ( n i=1 ) ( F i (x i ) 1+ a jk F j (x j )F k (x k ) 1 j<k n F i = L(X i ), the a jk 0 fulfill certain requirements so that F X1,...,X n is a proper n-dimensional distribution. ), Ali Mikhail Haq copula: C(u 1, u 2 ; θ) = u 1 u 2 1 θ(1 u 1 )(1 u 2 ), 1 θ < 1 This copula can generate both positive and negative quadrant dependence structures depending on the sign of θ 9

11 Negative association A finite family of random variables ξ 1,..., ξ n is said to be negatively associated (NA) if for every pair of disjoint subsets A 1 and A 2 of {1, 2,..., n}, Cov(f 1 (ξ i, i A 1 ), f 2 (ξ j, j A 2 )) 0 whenever f 1 and f 2 are coordinatewise increasing. An infinite family is negatively associated if every finite subfamily is negatively associated (Joag-Dev and Proschan (1983)). A number of well known multivariate distributions possess the NA property 10

12 NA ND UND LND In fact, ND is much weaker than NA. Because of the wide applications of ND random variables, it is highly desirable and of considerable significance to extend the limit theorems of independent or NA random variables to the case of ND. Asymptotic properties of random sums is one of important topics in probability theory 11

13 3. Some heavy-tailed distribution classes Dominatedly varying tail (V D): lim sup x Consistently varying tail (V C ): Long tail (V L ): V (xy)/v (x) < for any 0 < y < 1 lim lim sup V (xy)/v (x) = 1 y 1 x lim V (x + y)/v (x) = 1 for any y > 0 x Subexponential tail (V S ): V n (x) nv (x) for any n 2 12

14 It holds: C L D S L Upper and lower Matuszewska indices: J + V = lim y J V = lim y log V (y) log y log V (y) log y with with V (y) := lim inf x V (y) := lim sup x V (xy) V (x), V (xy) V (x), y > 1 Let L V = lim y 1 V (y). Then V D V (y) > 0 for some y > 1 L V > 0 J + V < V C L V = 1. 13

15 4. Asymptotics of random sums 4.1 P(S N > x) is dominated by P(N > x) Theorem 1 Assume that the claim number N has distribution H D, and the claim sizes X 1, X 2,... are ND r.v.s with common distribution F. If EX1 r < for some r > 1 and then Here, L H = lim lim inf y 1 x xf (x) = o(h(x)), (1) L H H(µ 1 x) P(S N > x) L 1 H H(µ 1 x) (2) H(xy) H(x) u(x) v(x) lim sup u(x) v(x) 1 Robert and Segers (2008): H C (L H = 1), independent structure 14

16 If N is independent of the claim sizes, we can weaken assumption (1): Theorem 2 Assume that X 1, X 2,... have with common d.f. F, 0 < µ = EX 1 < and the claim number N is independent of the claim sizes with d.f. H D. Assume that one of the following two conditions holds: (i) {X k, k 1} are UND, N satisfies EN <, F (x) = o(h(x)) (ii) {X k, k 1} are ND r.v.s and EX1 r < for some r > 1. N satisfies EN = and there exists a constant q [1, r) such that EN1 {N x} = O(1)x q H(x) Then L H H(µ 1 x) P(S N > x) L 1 H H(µ 1 x) 15

17 4.2 P(S N > x) is dominated by P(X 1 > x) Theorem 3 Suppose X 1, X 2,... are ND r.v.s with common d.f. F D, such that F ( x) = o(f (x)) and 0 < µ = EX 1 <. Let N be independent of the sequence X 1, X 2,.... If H(x) = o(f (x)), then ν EN < and L F νf (x) P(S N > x) L 1 νf (x) F 16

18 4.3 P(S N > x) and P(X 1 > x) are comparable Theorem 4 Suppose X 1, X 2,... are ND r.v.s with common d.f. F D, such that F ( x) = o(f (x)) and 0 < µ = EX 1 <. Let N be independent of the sequence X 1, X 2,.... If H(x) F (x), then H D, ν EN < and P(S N > x) L 1 F νf (x) + L 1 H H(xµ 1 ), P(S N > x) L F νf (x) + L H H(xµ 1 ) 17

19 Corollary 1 If the conditions of Theorem 3 are satisfied and H(x) c F (x) for some constant c > 0, then L F = L H and, therefore, L H (νf (x) + H(xµ 1 )) P(S N > x) L 1 H (νf (x) + H(xµ 1 )) Corollary 2 If the conditions of Theorem 3 are satisfied, F C and H(x) c F (x) for some c > 0, then L F = L H = 1 and, therefore, P(S N > x) νf (x) + H(xµ 1 ) 18

20 5. Applications to some dependent risk models Dependent compound renewal risk model Assumption H 1 The individual claim sizes {X k, k 1} form a sequence of UND and identically distributed nonnegative r.v.s with common distribution F and finite mean µ. Assumption H 2 The inter-arrival times {θ k, k 1} are LND and identically distributed nonnegative r.v.s, which are independent of {X k, k 1}. 19

21 Assumption H 3 Individual claim sizes and the claim number caused by the n-th accident at the time τ n = n θ k k=1 {X (n) k, k 1} and N n, respectively. Here, {X (n) k, k 1} are independent copies of {X k, k 1} and {N k, k 1} are i.i.d. positive integer-valued r.v.s with common distribution H and finite mean EN 1. In addition, random sequences {N k, k 1}, {θ k, k 1} and {X (n) k, k 1}, n 1 are mutually independent. Tang, Su, Jiang and Zhang (2001): Independent compound renewal risk model. are N 1 = N 2 = = 1: Classical renewal risk model 20

22 Number of accidents in the interval [0, t]: τ(t) = sup{n 0 : τ n t}, λ(t) := Eτ(t) Total claim amount at time τ n : S (n) N n = N n k=1 X (n) k Total claim amount up to time t 0: τ(t) n=1 S (n) N n = τ(t) N n n=1 k=1 X (n) k 21

23 The total amount of premiums accumulated up to time t 0, denoted by C(t) with C(0) = 0 and C(t) < almost surely for every t > 0, is a nonnegative and nondecreasing stochastic process, which is independent of {θ k, k 1} and {X (n) k, k 1}, n 1. Let r > 0 be the constant interest rate, x > 0 initial capital reserve Discounted surplus process of the insurance company: R(t) = x + t 0 e rs C(ds) τ(t) n=1 S (n) N n e rτ n 22

24 Sundt, Teugels (1995), Klüppelberg and Stadtmüller (1998), Tang (2005), Wang (2008) (classical risk model, independent structure), Kong and Zong (2008) (Poisson risk model, ND structure), Finite-time ruin probability: Ψ(x, t) := P = P ( inf R(v) < 0 R(0) = x 0<v t sup 0<v t τ(v) n=1 S (n) N n e rτn ) v 0 e rs C(ds) > x 23

25 Theorem 5 Assume that assumptions H 1, H 2, H 3 are satisfied. (i) If H D, F (x) = o(h(x)), then for any fixed t > 0 t L 4 H 0 H(xµ 1 e rs )dλ(s) Ψ(x, t) L 7 H In particular, if H C, then Ψ(x, t) t t 0 H(xµ 1 e rs )dλ(s). 0 H(xµ 1 e rs )dλ(s). (ii) If H C, H(x) c F (x) for some c [0, ), then for any fixed t > 0 Ψ(x, t) t 0 ( EN1 F (xe rs ) + H(xµ 1 e rs ) ) dλ(s). 24

26 Since the total (aggregated) claim amount at time τ n is S (n) N n = X (n) 1 + X (n) X (n) N n, it is important to know the closeness properties for random sums S N = X X N. If X 1, X 2,... are i.i.d., in C, or D, N is independent of X 1, X 2,..., EN < and either F (x) H(x), or F (x) H(x), or F (x) H(x), then X 1 + X X N is in C, or D. 25

27 Conjecture: If X 1, X 2,... are i.i.d. nonnegative random variables with d.f. in L (or L D), N is independent of X s, EN <. Then P(X X N x) L. 26

28 6. Application to the Galton-Watson branching process Dependent Galton-Watson process {Y n, n 0}: Y 0 = 1 and Y n+1 = Y n k=1 X (n+1) k, where {X (2) k, k 1}, {X (3) k, k 1},... are independent copies of {X (1) k, k 1}, and {X (1) k, k 1} is a sequence of ND nonnegative integer-valued r.v.s with common distribution F and finite mean µ. Here, Y n represents the number of items in the nth generation. Such dependence structure shows that the numbers of the descendants in a same generation are negatively dependent. We obtain the following application of Corollary 1 with H = F. 27

29 Y 2 = Y 1 k=1 X (2) k = X (1) 1 k=1 X (2) k. Theorem 6 In such a ND Galton-Watson process, if F D, then L F ( µf (x) + F (xµ 1 ) ) P(Y 2 > x) L 1 F ( µf (x) + F (xµ 1 ) ). Furthermore, if F C and the branching process is critical, i.e. µ = 1, then for each fixed n 1 P(Y n > x) nf (x)

30 Main references Yang, Y., Wang, Y., Leipus, R. and Šiaulys, J Asymptotics for the tail probability of total claim amounts with negatively dependent claim sizes and its applications, Lithuanian Mathematical Journal 49, Leipus, R., Šiaulys, J. and Yang, Y Asymptotics of random sums of negatively dependent random variables in the presence of dominatedly varying tails. Preprint. Other references Aleškevičienė, A., Leipus, R. and Šiaulys, J., Tail behavior of random sums under consistent variation with applications to the compound renewal risk model. Extremes 11, Chen, Y. and Zhang, W., Large deviations for random sums of negatively dependent random variables with consistently varying tails. Statist. and Probab. Lett. 77, Embrechts, P., Klüppelberg, C. and Mikosch, T., Modelling Extremal Events for Insurance and Finance. Springer, Berlin. 28

31 Faÿ, G., González-Arévalo, B., Mikosch, T. and Samorodnitsky, G., Modelling teletraffic arrivals by a Poisson cluster process. Queueing Syst.: Theor. Appl. 54, Kong, F. and Zong, G., The finite-time ruin probability for ND claims with constant interest force. Statist. Probab. Lett. 78, Robert, C. and Segers, J., Tails of random sums of a heavy-tailed number of light-tailed terms. Insurance: Mathematics & Econom. 43, Tang, Q., Insensitivity to negative dependence of the asymptotic behavior of precise deviations. Electron. J. Probab. 11, Tang, Q., Su, C., Jiang, T. and Zhang, J., Large deviations for heavytailed random sums in compound renewal model. Statist. and Probab. Lett. 52, Wang, D., Finite-time ruin probability with heavy-tailed claims and constant interest rate. Stochastic Models 24,