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)
Large deviations under subexponentiality p. Random walks and LD theory Let {S n = ξ 1 +... + 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.
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
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
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)
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
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
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?
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
Large deviations under subexponentiality p. Four sequences {b n } is a natural-scale sequence if {S n /b n } is tight
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
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
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
Main result A lemma shows that these sequences exist if and only if F is subexponential
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
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
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
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
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
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
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 ]
The Weibull example Here F(x) e xβ, for some β (0, 1). By the CLT we may set b n = n
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)
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 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
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 )
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