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 e-mail: tommi.sottinen@helsinki.fi A joint work with Yu. Kozachenko and O. Vasylyk
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
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
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
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
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
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
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
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
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
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