Large deviations for random walks under subexponentiality: the big-jump domain

Transcription

1 Large deviations under subexponentiality p. Large deviations for random walks under subexponentiality: the big-jump domain Ton Dieker, IBM Watson Research Center joint work with D. Denisov (Heriot-Watt, UK) and V. Shneer (EURANDOM, The Netherlands)

2 Large deviations under subexponentiality p. Random walks and LD theory Let {S n = ξ ξ n } be a random walk, with i.i.d. step sizes ξ 1, ξ 2,... having distribution F. We often assume Eξ = 0. We are interested in asymptotics of P(S n > x), for large n and x. Light-tailed case: Bahadur-Ranga Rao asymptotics for P(S n > an) Logarithmic asymptotics on abstract spaces (Cramér s theorem), assuming the existence of exponential moments e.g. books by Varadhan, Dembo/Zeitouni, Deuschel/Stroock, etc. If Cramér s condition fails to hold, then we are in the so-called heavy-tailed framework.

3 Large deviations under subexponentiality p. Subexponentiality The subexponential distributions form a widely used family of heavy-tailed distributions F is subexponential if, with F(x) = 1 F(x), for all y R, lim x F(x + y) F(x) = 1 P(S 2 > x) 2F(x) as x Intuition: either ξ 1 is very large and ξ 2 moderate, or vice versa For subexponential F, P(S n > x) nf(x) as x for any n 1 single big jump principle

4 Large deviations under subexponentiality p. Subexponentiality: applications Internet traffic modeling; e.g., session durations Long sessions can explain long-term correlations Some models make self-similarity plausible by superposition Insurance mathematics; e.g., claim sizes Theory of ruin for insurance companies One big claim causes ruin Call durations in certain call centers

5 Large deviations under subexponentiality p. Motivation from queueing theory The step size ξ i is the difference between the service time of the i-th customer and its inter-arrival time (so Eξ i < 0) Let τ be the length of the busy period Baltrunas/Daley/Klüppelberg, Denisov/Shneer: For heavy-tailed service distributions, we have for some explicit constant C P(τ > n) C n P(S n > 0)

6 Large deviations under subexponentiality p. Subexponentiality (3) Recall: P(S n > x) nf(x) as x Main question: for which {x n } do we have P(S n > x n ) nf(x n )? We say that {x n } is a big-jump sequence. This is a LD theory for heavy tails, but we ll see an interesting connection with LD theory for light-tailed distributions at the end

7 Large deviations under subexponentiality p. Two important examples F with F(x) x α, α > 0, is subexponential (more generally, F(x) = l(x)x α for some slowly varying l) a finite mean/variance is not required if α > 2, known that for any ɛ > 0, x n = (1 + ɛ) (α 2)n log n is a big-jump sequence (A. Nagaev) F with F(x) e xβ, β (0, 1), is subexponential known that any {x n } with x n n 1 2(1 β) is a big-jump sequence. in particular x n = an is big-jump sequence only if β < 1/2

8 Large deviations under subexponentiality p. Two important examples F with F(x) x α, α > 0, is subexponential (more generally, F(x) = l(x)x α for some slowly varying l) a finite mean/variance is not required if α > 2, known that for any ɛ > 0, x n = (1 + ɛ) (α 2)n log n is a big-jump sequence (A. Nagaev) F with F(x) e xβ, β (0, 1), is subexponential known that any {x n } with x n n 1 2(1 β) is a big-jump sequence. in particular x n = an is big-jump sequence only if β < 1/2 Are these special cases of a general theory?

9 Large deviations under subexponentiality p. Related work A. V. and S. V. Nagaev, A. A. Borovkov, R. Doney, C. C. Heyde, I. Pinelis, L. V. Rozovskii Post-2003 contributions: Baltrunas/Daley/Klüppelberg, Borovkov/Mogulskii, Hult/Lindskog/Mikosch/Samorodnitsky, Jelenkovic/Momcilovic, Konstantinides/Mikosch, Ng/Tang/Yan/Yang, Tang

10 Large deviations under subexponentiality p. Four sequences {b n } is a natural-scale sequence if {S n /b n } is tight

11 Large deviations under subexponentiality p. Four sequences {b n } is a natural-scale sequence if {S n /b n } is tight {I n } is an insentitivity sequence if I n b n and lim sup n x I n F(x b n ) F(x) 1 = 0

12 Large deviations under subexponentiality p. Four sequences {b n } is a natural-scale sequence if {S n /b n } is tight {I n } is an insentitivity sequence if I n b n and lim sup n x I n {h n } is a truncation sequence if F(x b n ) F(x) 1 = 0 lim sup np(s 2 > x, ξ 1 > h n, ξ 2 > h n ) n x h n F(x) = 0

13 Large deviations under subexponentiality p. Four sequences {b n } is a natural-scale sequence if {S n /b n } is tight {I n } is an insentitivity sequence if I n b n and lim sup n x I n {h n } is a truncation sequence if F(x b n ) F(x) 1 = 0 lim sup np(s 2 > x, ξ 1 > h n, ξ 2 > h n ) n x h n F(x) = 0 {J n } is an h-small steps sequence if lim sup P(S n > x, ξ 1 h n,..., ξ n h n ) n x J n nf(x) = 0

14 Main result A lemma shows that these sequences exist if and only if F is subexponential

15 Main result A lemma shows that these sequences exist if and only if F is subexponential Theorem If h n = O(b n ) and h n J n, then P(S n > I n + J n ) np(s 1 > I n + J n ). It remains to find good sequences {I n } and {J n } otherwise result still valid, but weak statement

16 Choosing {h n } Recall: {h n } is a truncation sequence if lim sup np(s 2 > x, ξ 1 > h n, ξ 2 > h n ) n x h n F(x) = 0 If F(x) x α, then any {h n } satisfying nf(h n ) 0 is a truncation sequence If x r F(x) is subexponential for some r > 0, then any {h n } with lim sup n nh r n < is a truncation sequence

17 Choosing {J n }: a heuristic Recall: {J n } is an h-small steps sequence if lim sup P(S n > x, ξ 1 h n,..., ξ n h n ) n x J n nf(x) = 0 Suppose Eξ 2 <, and let J n satisfy J n 2n log[nf(j n)]. For any ɛ > 0, J n = (1 + ɛ)j n is a good choice, provided {n/j n } is a truncation sequence

18 The polynomial-tail example Here F(x) x α, for some α > 2. Suppose that Eξ = 0, Eξ 2 = 1. By the CLT we may set b n = n

19 The polynomial-tail example Here F(x) x α, for some α > 2. Suppose that Eξ = 0, Eξ 2 = 1. By the CLT we may set b n = n Any I n b n suffices: lim sup n x I n (x b n ) α x α 1 = 0

20 The polynomial-tail example Here F(x) x α, for some α > 2. Suppose that Eξ = 0, Eξ 2 = 1. By the CLT we may set b n = n Any I n b n suffices: lim sup n x I n (x b n ) α x α 1 = 0 Any {h n } with nf(h n ) 0 is a truncation sequence

21 The polynomial-tail example Here F(x) x α, for some α > 2. Suppose that Eξ = 0, Eξ 2 = 1. By the CLT we may set b n = n Any I n b n suffices: lim sup n x I n (x b n ) α x α 1 = 0 Any {h n } with nf(h n ) 0 is a truncation sequence Use the heuristic for {J n }: set (α 2)n log n and note that 2n log[nf(j n)] = 2αn log[n 1/α J n] (α 2)n log n = J n 2αn log[n 1/2 1/α ]

22 The Weibull example Here F(x) e xβ, for some β (0, 1). By the CLT we may set b n = n

23 The Weibull example Here F(x) e xβ, for some β (0, 1). By the CLT we may set b n = n To find I n we study sup x I n e (x n) β +x β 1 = 0. Since (x n) β + x β β nx β 1, we need I n n 1 2(1 β)

24 The Weibull example Here F(x) e xβ, for some β (0, 1). By the CLT we may set b n = n To find I n we study sup x I n e (x n) β +x β 1 = 0. Since (x n) β + x β β nx β 1, we need I n n 1 2(1 β) For any r > 0, x r F(x) is the tail of a subexponential distribution, so any {h n } with h n n 1/r is a truncation sequence

25 The Weibull example Here F(x) e xβ, for some β (0, 1). By the CLT we may set b n = n To find I n we study sup x I n e (x n) β +x β 1 = 0. Since (x n) β + x β β nx β 1, we need I n n 1 2(1 β) For any r > 0, x r F(x) is the tail of a subexponential distribution, so any {h n } with h n n 1/r is a truncation sequence The heuristic can be used for {J n }, yielding {J n } with J n I n

26 Local analogues We also study for T (0, ) P(S n (x n, x n + T]) np(s 1 (x n, x n + T]) Statements very similar, parts of the proof a lot harder!!! Use in the context of light tails: given γ > 0, subexponential F with L(γ) = e γy F(dy) <, define the RW on P through P (S 1 dx) = e γx F(dx)/L(γ). The local case allows us to conclude that P (S n > x n ) nl(γ) 1 n P (S 1 > x n )

27 Further examples We worked out examples with infinite variance or mean We always recovered the sharpest known big-jump sequences... but we also found big-jump sequences in new examples