Sample path large deviations of a Gaussian process with stationary increments and regularily varying variance

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1 Sample path large deviations of a Gaussian process with stationary increments and regularily varying variance Tommi Sottinen Department of Mathematics P. O. Box 4 FIN-0004 University of Helsinki Finland tommi.sottinen@helsinki.fi A joint work with Yu. Kozachenko and O. Vasylyk

2 The setting Consider a fluid queue serving at unit rate. The input process Z = Z t : t IR) is centred Gaussian with stationary increments Z 0 = 0 and the variance function σ 2 is regularly varying at infinities with index H 0, ), i.e. for all t IR lim α σ 2 αt) σ 2 α) = t 2H. The storage process V = V t : t IR) is V t = sup s t Z t Z s t s)). We are interested in the excursions of V busy periods) and V 0 queue length). The large deviations of these are known in the case of fractional Brownian motion. 2

3 Fractional Brownian motion fbm) The fractional Brownian motion B = B H can be characterised by the following properties: it is continuous, Gaussian, centred, of stationary increments and self-similar with a parameter Hurst index) H 0, ), i.e. B at : t IR) d = a H B t : t IR). Alternatively, function one can give the covariance CovB t, B s ) = 2 t 2H + s 2H + t s 2H). If H > /2 the increments of B are positively correlated, if H < /2 they are negatively correlated. The case H = /2 corresponds the standard Brownian motion. 3

4 Convergence to fbm Set Z α) t = σα) Z αt. Lemma Z α) converges to fbm in finite dimensional distributions. Proof CovZ α) s = 2 Obviously VarZ α) t, Z t α) ) VarZ s α) + VarZ α) t 2 t 2H + s 2H + t s 2H). t 2H. Hence + VarZ t s α) ) Since we are in the centred Gaussian case the claim follows. QED Define Ω, ) by Ω = { ω = sup t IR ω CIR) : ω 0 = 0 = ω t lim t } ω t + t + t. 4

5 Convergence to fbm, cont. Define a majorising variance and the associated metric entropy integral σ 2 αt) σt) = sup sup σ 2 α) Jk, T ) = α k 0 s t ln T σ ε) + )) 2 dε. Assumptions C: J σt ), T ) < for all T and B: there exists a sequence x k such that k= k=t imply more or less) IP sup Z t α) t s <ε IP + x k < J σ x k, x k ) + x k < sup t T Z α) s > δ exp δ 2 σ 2 ε) Z t α) + t > ε exp ε 2 T ). ) 5

6 Convergence to fbm, cont., cont. Lemma lim sup IP δ 0 α Z α) is tight in Ω, ) iff sup Z t α) t s <δ lim sup IP T α sup t T Z α) s > ε = 0 Z t α) + t > ε = 0. Theorem Ω, ). Z α) converges to fbm weakly in Example The input traffic is composed of n independent streams, each of which is a fbm, with different Hurst indexes, i.e. Z = n k= a k B H k. Counterexample? 6

7 Large deviations Definition A scaled family X α), vα) ) satisfies the large deviations principle LDP) in Ω, ) with rate function I : Ω [0, ] if for each closed F Ω and open G Ω lim sup α lim inf α vα) ln IP X α) F ) inf ω F vα) ln IP X α) G ) inf ω G Set vα) = α2 σ 2 α) and consider the family vα) Z α), vα) ). ) Lemma The family ) satisfies the LDP on Ω equipped with the topology of pointwise convergece with the rate function Ix) = sup p 2 Γ p px), px) 2) where p is a finite dimensional projection on Ω and Γ p is the covariance matrix of pfbm). 7

8 Large deviations, cont. Definition A scaled family X α), vα) ) is exponentially tight in Ω, ) if for each l > 0 there exists a compact set K l such that lim sup α vα) ln IP X α) ) K l l. Theorem The family ) satisfies the LDP on Ω, ) with the rate function 2). Proof Assumptions C and B imply the exponential tightness. The LPD can hence be lifted to the norm topology by means of the inverse contraction principle. QED 8

9 Large buffer and busy period asymptotics Set Q x = {V 0 x} and A = sup{t 0 : V t = 0}, B = inf{t 0 : V t = 0}, K T = {A < 0 < B} {B A > T }. Theorem lim x σ 2 x) x 2 ln IPZ Q x ) = inf ω Q Iω). Proof Since IPZ Q x ) = IPsupZ xt xt) x) t 0 = IPsup Z α) x 0 vα) t t) ) = IP Z α) Q ) vα) the claim follows from the LDP and the fact that inf ω Q Iω) = inf ω Q Iω). QED 9

10 Large buffer and busy period asymptotics, cont. Theorem lim T σ 2 T ) T 2 ln IPZ K T ) = inf ω K Iω). Proof Since IPZ K T ) = IP vα) Z α) K ) the claim follows from the LDP and the fact that inf ω K Iω) = inf ω K Iω). QED 0

11 References Billingsley, P. Convergence of Probability Measures. Second Edition. Wiley, 999. Buldygin, V. V. and Kozachenko, Yu. V. Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Providence, RI, Dembo, A. and Zeitouni, O. Large Deviations Techniques and Applications. Second Edition. Springer, 998. Deuschel, J.-D., Stroock, D. Large Deviations. Academic Press, 984. Kozachenko, Yu. and Vasilik, O. On the Distribution of Suprema of Sub ϕ Ω) Random Processes. Theory of Stochastic Processes, Vol. 420), no. 2, pp , 998. Norros, I. A storage model with self-similar input. Queueing Systems, Vol. 6, pp , 994. Norros, I. Busy Periods of Fractional Brownian Storage: A Large Deviations Approach. Advances in Performance Analysis, Vol. 2), pp. 9, 999.