EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS. O.V.Gulinskii*, and R.S.Liptser**

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1 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS O.V.Gulinskii*, and R.S.Liptser** *Institute for Problems of Information Transmission Moscow, RUSSIA **Department of Electrical Engineering-Systems Tel Aviv University, Tel Aviv, ISRAEL Abstract. We give an example of the large deviations for the family Xt ) t, > with Ẋt = R ax t ) + bx t )η t/, where η t is stationary process obeying the Wold decomposition: t η t = ht s)dn s w.r.t. homogeneous process N t with independent square integrable increments. Key words: Large deviation, Skorokhod space, Wold decomposition 1. Introduction and main result 1. It is well known see [3]) that a family X = Xt ) t, > of diffusion processes: dxt = axt )dt + bxt )dw t 1.1) subject to fixed X, where W t is a Wiener process and ax), bx) are Lipschitz continuous and b 2 x) >, obeys the large deviation principle l.d.p.) in the space of continuous function C [,T ] and the corresponding rate function is given by the formula: { 1 ϕt aϕ t )) 2dt, 2 I T ϕ) = bϕ t ) ϕ = X, dϕ dt, otherwise, 1.2) where the notation ϕ = X, dϕ dt is used for designating ϕ t = X + ϕ sds, t T, and where ϕ t is the Radon-Nikodym derivative of ϕ t. Evidently the l.d.p. for X t ) also holds in the metric space C, ρ) with C = C [, ), ρx, Y ) = n 1 2 n sup t n X t Y t 1 + sup t n X t Y t This work is supported by Russian Found of Fundamental Researches and in part by AT&T Bell Lab Typeset by AMS-TEX 1

2 2 O.V.GULINSKII*, AND R.S.LIPTSER** and the rate function Iϕ) = sup I T ϕ) see, e.g. [9]). T In the contrast to 1.1), in this paper we formulate the l.d.p. for family of processes Xt ) t, > defined by an ordinary differential equation Ẋ t = ax t ) + bx t )η t/, 1.3) subject to fixed X, where η t ) t R is the restricted sense stationary and ergodic process with Eη =. Since the Markovian property for η t ) is not to be assumed the general approach consists in checking the l.d.p. for the occupation measures of η t ) for, so called, third level of the Donsker and Varadhan theory [2] and applying the contraction principle of Varadhan [11] to get the l.d.p. for X. To avoid an application of the third level we restrict ourselves by the consideration of η t ) obeying the Wold decomposition 1 η t = ht s)dn s, 1.4) w.r.t. right continuous having limits from the left homogeneous process with independent increments N = N t ) t R such that N = and for any s, t R E N t N s = k 2 t s, EN t N s ) = and R h2 t)dt < so the integral in 1.4) is understood as Ito s stochastic integral). For such process η t ) a simple proof of the l.d.p. for X is found, the explicit formula for the rate function takes place and what is more this model serves different applications. 2. Formally, letting Wt = 1/ η s/ ds, 1.5) 1.3) can be rewritten to the similar form as 1.1): dxt = axt )dt+ bxt )dwt. But the l.d.p. even for Wt ) would be difficult to obtain by the method from [3]. Nevertheless, it is possible by different method and by virtue of 1.4). Two reasons are for Wt ) to obey the l.d.p.: 1. By virtue of the Birkhoff - Khinchin theorem for any fixed T > P a.s.) lim Wt =. sup 2. Under weak dependence assumptions ht) dt < h 2 s)dsdt <. 1.6) t the functional central limit theorem for Wt ) holds, i.e. L limwt ) = ΣW t ), where W t ) is Wiener process, L means convergence in the distribution sense, and Σ = 1 for discrete time case the corresponding result can be found in [4] ht)dt 1.7)

3 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS 3 see [8, Ch.9,Sec.2,Ex. 2]). Along with 1.6) assume that 2dt hs)ds) <. 1.8) t Also the Cramer type conditions for jumps N t = N t lim s t N s ) of N t are required for getting the l.d.p. of Wt/. As any process with square integrable and independent increments N is characterized by its continuous Gaussian component N c = Nt c ) t R with EYt c ) 2 = σ 2 t and the Levy measure Kdx)dt, x R = R\{}, t R, x 2 Kdx) <. R Taking into account that for any λ R E e λ N t ) 1 λ N t = e λx 1 λx ) Kdx) t 1 R E e λht) N t ) 1 λht) N t = e λht)x 1 λht)x ) Kdx)dt t R assume A.1) for any λ R A.2) for any λ R R e λx 1 λx ) Kdx) < ; R [e λht)x 1 λht)x]kdx)dt <. Under A.1) the cumulant function is defined as: Gλ) = λ2 σ e λx 1 λx)kdx), λ R. 1.9) R Due to A.1) Gλ) is twice continuous differentiable with G λ) = λσ 2 + xe λx 1)Kdx) R G λ) = σ 2 + x 2 e λx Kdx). 1.1) R and so it is nonnegative G) = ) and convex function. The Legendre-Fenchel transformation of Gλ) is defined in the usual way see [1]): [ ] H σ 2y) = sup λy Gλ), y R. 1.11) λ R As Gλ) is continuous sup in 1.11) may be taken over rational λ and so H σ 2y) is measurable function in y. In the notation H σ 2y) a dependence on σ 2 is emphasized. In particularly, property H σ 2y) H y), σ 2 is used in proving the l.d.p. below.

4 4 O.V.GULINSKII*, AND R.S.LIPTSER** 3. For any ϕ C, letting =, put { H σ 2 ϕ t Σ Iϕ) = )dt, ϕ =, dϕ dt, otherwise, 1.12) where ϕ =, dϕ dt is used for designating ϕ t = ϕ sds, t, and where ϕ t is the Radon-Nikodym derivative of ϕ t. Evidently, if Σ = then {, ϕt Iϕ) =, otherwise. Now, we are in the position to formulate the l.d.p. for the family W t ). Theorem ) Let assumptions 1.6), 1.8), and A.1), A.2) be fulfilled. Then the family W t ) obeys the l.d.p. in C, ρ) with the rate function Iϕ) is given by 1.12). The main result is implied by Theorem 1.1 and the contraction principle of Varadhan [11]. Theorem 1.2. Let functions ax) and bx) be Lipschitz continuous and there exist constants c and C such that < c bx) C. Then under assumptions of Theorem 1.1 the family Xt ) defined by 1.3) obeys the l.d.p. in C, ρ) with the rate function Jϕ), ϕ C defined as: { H ϕt aϕ t )) σ 2 Σbϕ Jϕ) = t ) dt, ϕ = X, dϕ dt, otherwise. The proof of these theorems are situated in Sections 4 and 5. In Section 3 we formulate the l.d.p. for N t/. Section 2 contains an auxiliary result. In the last Section an example is considered having an independent interest. 2. Properties of η sds 1. Taking into account 1.4) and well known property of the stochastic Ito integral we find Eη t η s, s ) = ht s)dn s P-a.s. On the other hand, by 1.6) h 2 s)dsdt <. Both facts imply t E Eη t η s, s ) ) 2 dt <. Then by [8, Lemma 9.2.1] there exists a semimartingale V t ) t, w.r.t. F = F t ) t R generated by η t ), satisfying the general conditions, such that V t = V + the filtration η s ds M t, 2.1)

5 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS 5 where M t ) is a square integrable martingale with in the restricted sense ergodic stationary increments and V t, η t ) t forms in the restricted sense stationary process. For η t obeying the Wold decomposition 1.4), we have with Σ given at 1.7)) M t = ΣN t V t = t hu s)dn s )du 2.2) for more details see [8, Ch.9, Sec.2, Ex.2]. The aim of this Section is to show an exponential integrability of sup V t. Lemma 2.1. Let 1.6), 1.8), and A.1), A.2) be fulfilled. Then for any λ > and T > E exp { λ sup V t } <. Proof. Assume there exists some a positive martingale L t ) w.r.t. F such that V t L t, t T E exp { λl T } <, λ >. 2.3) Then the result holds. In fact, using Jensen s, Cauchy-Schwartz s, and Doob s inequalities we get E exp { λ sup V t } E exp { } λ sup L t = E sup exp { } λl t = E sup exp { EλL T F t ) }. E sup E exp { } ) λl T Ft Jensen) E sup E exp { } ) 2 λl T Ft Cauchy-Schwartz) 2 E exp { } 2λL T Doob) <. Thus, it remains to find L t ) satisfying 2.3). Two facts are used here: decomposition 2.1) and F t -measurability of V t. From 2.1) we find V t = V T η sds+σn T N t ), t T and what follows from martingale property of N t ) that is V t = E V T η ) sds F t. So V t E V T + η ) s ds F t, i.e. Lt = E V T + η s ds ) Ft. Thereby, it remains to show only that E exp { λ[ V T + η s ds] } <. 2.4)

6 6 O.V.GULINSKII*, AND R.S.LIPTSER** We examine 2.4) by using Cauchy-Schwartz s and Jensen s inequalities: E exp { λ[ V T + η s ds] }) 2 E exp { 2λ V T } E exp { 2λ = E exp { 2λ V } E exp { 1 T E exp { 2λ V } E 1 T = E exp { 2λ V } E exp { 2T λ η }. η s ds } Cauchy-Schwartz) 2T λ η s ds } exp { 2T λ η s } ds Jensen) Thus 2.4) holds if for any positive λ E exp { λ V } < and E exp { λ η } <. The direct proof for the validity of these inequalities would be difficult. It is more convenient to use instead of V and η random values V = hu)dudn s s V is defined by virtue of 1.8)) and η = hs)dn s which coincide with V and η in the distributions and furthermore, using the estimate e x e x + e x, to examine only E exp { λv } < E exp { λη } <, λ R. 2.5) Denote by Ht) any of functions ht) or hu)du. From the definition o V t and η it follows that { V η = Ht)dN t Z ). Thereby, the validity of E exp { λz } <, λ R 2.6) has to be checked. To this end define a square integrable martingale Z t = Hs)dN s having as the limit point Z = lim Z t. By the Fatou lemma t E exp { } λz lim sup E exp { } λz t t and so it is sufficient to show that for any λ R there exists constant Cλ) depending on λ only such that E exp { λz t } Cλ). 2.7) For finding Cλ) we use the fact that Z t ) is the process with independent increments. Namely, the Levy measure Kdx)dt is a compensator for the measure µdt, dx) of jumps of N t ) w.r.t. a filtration F N = Ft N ) t generated by N t ). Then the pour discontinuous part Nt d ) of N t ) can be represented as Ito s integral w.r.t. the martingale measure µdt, dx) Kdx)dt : Nt d = x[µds, dx) Kdx)ds] R

7 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS 7 and so Then N t = N c t + R x[µds, dx) Kdx)ds]. 2.8) Z t = = Hs)dN s Hs)dN c s + and by [5,Ch.II ] for any λ R we find Ee λz t R Hs)x[µds, dx) Kdx)ds] { λ 2 σ 2 = exp H 2 s)ds + e λhs)x 1 λhs)x ) } Kdx)ds. 2.9) 2 R The right hand side of this inequality growth in t to { λ 2 σ 2 Cλ) = exp H 2 s)ds + e λhs)x 1 λhs)x ) } Kdx)ds 2 R 2.1) and only it remains to show that this Cλ) is finite. In case Ht) ht) it holds by A.2). In case Ht) hs)ds it it is implied by 1.8) and A.1) since λ t = λ sup Ht) ) t e λht)x 1 λht)x λ 2 H 2 t)e λ x and for any λ R R x 2 e λx Kdx) <. λ 2 H 2 t) e λ x + e λ x ) 3. The l.d.p. for N t/ In this Section the l.d.p. is established for family of homogeneous processes with independent increments h.i.i) what kind is N t/. Since N t has paths in the Skorokhod space D = D [, ) of the right continuous having limits from the left functions we formulate the l.d.p. in the metric space D, d) with the Skorokhod-Lindvall metric d see [6] or e.g. [8, Ch. 6]). The metric d is equivalent to some sense to the metric ρ see Section 1) and is defined in the following way. For each X = X t ) t put Xt n = X t min1, n t), t n. If X, Y D and X n = Xt n ) t n, Y n = Yt n ) t n defined as it was mentioned above we have dx, Y ) = n 1 2 n d n X n, Y n ) 1 + d n X n, Y n ), where d n X n, Y n ) is the Skorokhod distance between X n, Y n see [1]).

8 8 O.V.GULINSKII*, AND R.S.LIPTSER** Theorem 3.1. Let assumption A.1) be fulfilled. The family N t/ obeys the l.d.p. in the metric space D, d) with rate function { I hii H σ 2 ϕ t )dt, ϕ =, dϕ << dt ϕ) =, otherwise with H σ 2y) from 1.11). Formally Theorem 3.1 follows from general result of the l.d.p. for Markovian processes [12] or semimartingales [7]. We give here the direct proof. The reason for this is that the process N t has simpler structure than Markovian process or semimartingale considered in [12] and [7] and so suggested proof is simpler and could be interested by itself. Nevertheless, we use the method of proving from [7] which is based on the such notions as the exponential tightness, the partial l.d.p., and the Pukhalskii theorem [9]. Following it only two sets of conditions have to be checked. C.1) C-exponential tightness: for any T >, γ > lim lim log P sup N t/ > c ) = c lim lim sup log P sup N t+τ)/ N τ/ > γ ) =, δ t δ where sup is taken over all stopping times τ w.r.t. F N t/ ) t ) which is bounded by T. C.2) C-local l.d.p.: for any T > and ϕ C Î T ϕ) = lim δ lim log P sup N t/ ϕ t δ ) = lim δ lim log P sup N t/ ϕ t δ ). If C.1) and C.2) are fulfilled then the l.d.p. in D, d) holds with rate function { I hii supt ϕ) = > Î T ϕ), ϕ C, ϕ D \ C. 3.1) Conditions C.1) and C.2) are examined below in three lemmas. Lamma 3.1. Let assumption A.1) be fulfilled. Then C.1) holds. Proof. For checking the first condition in C.1) Chernoff s, Jensen s,

9 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS 9 Cauchy-Schwartz s, and Doob s inequalities are applied: P sup N t/ > c ) e c/ E exp { sup N t/ } Chernoff) = e c/ E sup exp { N t/ } = e c/ E sup exp { EN T/ Ft/ N } e c/ E sup E [ e c/ E sup exp { N T/ } F N t/ ) Jensen) E exp { N T/ } 2Cauchy-Schwartz) Ft/)] N 2e c/ E exp { 2 N T/ } Doob) 2e c/ E { e 2N T / + e 2N T / }. 3.2) Since N t ) is the martingale with independent increments by virtue of 2.8) and [5, Ch.II ] we get { Ee ±2N T / T [ = exp 2σ 2 + e ±2x 1 2x ) } Kdx) R exp { T l}, 3.3) where l = 2σ 2 + R e 2x 1 2x ) Kdx) + R e 2x 1 + 2x ) Kdx)< ) see Cr.1)). Then, as it follows from 3.2) and 3.3), log P sup N t/ > c ) log 2 c + const.t,, c. For checking the second condition in C.1) note that by virtue of strong Markovian property N t+τ)/ N τ/ ) t coinsides in the distribution with N t/ ) t. So by the same way as 3.2) and 3.3) have been obtained, we find for any λ > P sup N t+τ)/ N τ/ > γ ) 2e E λγ/ t δ [e 2λN δ/ + e 2λN δ/ ] 2 exp{ λγ/ δlλ)/}, where lλ) = 2λ 2 σ 2 + e 2λx 1 2λx)Kdx) + e 2λx 1 + 2λx)Kdx). R R Then lim log P sup N t+τ)/ N τ/ > γ ) λγ + 1 δlλ). 3.4) t δ 2 The function lλ), λ is nonnegative, continuous, and increasing. If it is bounded then the result evidently holds. If it is increasing to then taking λ δ such that lλ δ ) = 2/δ we arrive to upper bound in 3.4): λ δ γ + 1 which decreases to in δ.

10 1 O.V.GULINSKII*, AND R.S.LIPTSER** Lemma 3.2. Let assumption A.1) be fulfilled. Then for any T > and ϕ C where H σ 2 ϕ t is defined in 1.11). lim δ lim log P sup N t/ ϕ t δ ) { H σ 2 ϕ t)dt, ϕ =, dϕ << dt, otherwise, Proof. Let νt) be a simple function of the form νt) = i ν ii ]ti 1,t i ]t). Put ZT ν) = exp { νt)dn t/ 1 where Gλ) is the cumulant given at 1.9). By [5,Ch.II] we have which implies Gνt))dt }, 3.5) EZ T ν) = 1 3.6) 1 EIsup N t/ ϕ t δ)zt ν). 3.7) Inequality 3.7) is the general tool in proving the upper bound. It is naturally to evaluate ZT ν) from below on the set {sup N t/ ϕ t δ}. Put by definition νt)dϕ t = ν i [ϕ T ti ϕ T ti 1 ]. Then i { 1 ZT [ ν) exp { exp νt)dϕ t Gνt))dt ]} νt)[dn t/ 1 } dϕ t]. 3.8) Taking into account that on the set {sup N t/ ϕ t δ} the following estimate holds: νt)[dn t/ 1 dϕ t] const. δ, where const. depends only on ν i, we derive from 3.7) and 3.8) the following upper bound: lim δ lim log P sup N t/ ϕ t δ ) [ ] νt)dϕ t Gνt))dt. The equality [ ] sup νt)dϕ t Gνt))dt = { H σ 2 ϕ t)dt, ϕ =, dϕ dt, otherwise, where sup is taken over all simple functions νt), follows from [7, Lemma 6].

11 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS 11 Lemma 3.3. Let assumption A.1) be fulfilled. Then for any T > and ϕ C where H σ 2 ϕ t is defined in 1.11). lim δ lim log P sup N t/ ϕ t δ ) { H σ 2 ϕ t)dt, ϕ =, dϕ dt, otherwise, Proof. Since the announced lower bound coincides with the upper one only the validity of finite bound have to be checked, or it is sufficient to consider only the case ϕ =, dϕ dt. Assume at first that σ 2 > ϕ t N. 3.9) Under these assumptions there exists a measurable bounded function νt) satisfying for a.s. w.r.t. Lebesgue measure) t the equality: G νt)) = see 1.9)). Then we have H σ 2 ϕ t ) = νt) ϕ t Gνt)), a.s. 3.1) For fixed T > put ν T t) = IT t)νt) and define the process { Zt ν T ) = exp ν T s)dn s/ 1 } Gν T t))dt. 3.11) Since G) = lim t Z t ν T ) = Z T ν T ) and EZ T ν T ) = ) Noticing that ZT ν T ) > P-a.s. and letting dp T = Z T ν T )dp we get the probability measure P dp T, which is equivalent to P and dp = Z T ν T ) ) 1 the expectation w.r.t. P T T is denoted by E T ). From identity P sup with see 3.1) and 3.11)) N t/ ϕ t δ ) = EIsup N t/ ϕ t δ)zt ν T ) ZT ν T ) ) 1 Z T ν T ) ) { 1 = exp = E T Isup N t/ ϕ t δ) ZT ν T ) ) 1 νt) [ dn t/ 1 ϕ tdt ] } H σ 2 ϕ t )dt

12 12 O.V.GULINSKII*, AND R.S.LIPTSER** we find lim log P sup N t/ ϕ t δ ) H σ 2 ϕ t )dt { lim log E T Isup N t/ ϕ t δ) exp νt) [ dn t/ 1 ϕ tdt ]}) H σ 2 ϕ t )dt γ lim log P T I sup N t/ ϕ t δ, νt) [ dn t/ 1 ϕ tdt ] γ ). Thereby by the arbitrariness of γ the desired lower bound holds if for any ζ > lim log P T I sup N t/ ϕ t δ, νt) [ dn t/ ϕ t dt ] ζ ) =. 3.13) To this end we examine some properties of the process Z t ν T ). Taking into account 2.8) and applying Ito s formula to the right hand side of 3.11) we find that dzt ν T ) = Zt ν T )ν T t)dn t/ + Zt ν T ) e ν T t)x 1 ν T t)x ) [µdt/), dx) Kdx)dt/)] R and so Zt ν T ) is a martingale w.r.t. P, Ft/ N ) t ). Furthermore, it is a square integrable martingale since E Zt ν T ) ) { 2 = exp 1/ [ ] } G2νt)) 2Gνt)) dt. Also note that the mutually quadratic variation for pair of martingales Zt ν T ) and N t/ is defined as: [ Z ν T ), N./ ] t = 1 + Z sν T )σ 2 ν T s)ds Zsν T ) x [ e ν T s)x 1 ] µds/), dx) R and consequently, the mutually predictable quadratic variation, being the compensator for it, is given by the formula: Define new process N T, t Z ν T ), N./ t = 1 [ Z sν T )σ 2 ν T s)ds + Zsν T ) x [ e ν T s)x 1 ] ] Kdx)ds. R N T, t = N t/ 1 ϕ t T. It is easy to check that = N t/ 1 Z s ν T ) ) 1 Z ν T ), N./ s.

13 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS 13 Then by [8, Ch.4] process N T, t is a martingale w.r.t. P T, F t/ N ) t ) and what is more continuous component N T,,c t of it is Gaussian, E T N T,,c) 2 t = σ 2 t/, and the Levy measure of it is: KT dt, dx) = eν T t)x Kdx)dt/). So it is a square integrable martingale whose predictable quadratic variation is given by the formula: N T,,c t = 1 [ t σ 2 t + R x 2 e ν T t)x Kdx)dt ]. Consequently, by Doob s inequality see e.g. [8, Ch.I]) we have E T sup N T,,c t 2) 4 [ σ 2 T + x 2 e ν T t)x Kdx)dt ]. R = const. 3.14) Analogously we get E T sup ν T s)dn T,,c s ) 2) = const. 3.15) Since for t T we have N T, t = N t/ ϕ t 3.13) is implied by 3.14) and 3.15). In the next step we show the validity of the lower bound under weaker conditions than 3.9). Namely, taking into account that for dϕ dt we have ϕ s ds < t >, only σ 2 > 3.16) has to be assumed. For N 1, taking ϕ N t = ϕ t I ϕ t N), put ϕ N t = ϕn s ds. Evidently ϕ N t satisfies 3.9) and so by the obtained above result for any δ > we have lim sup log P sup N t/ ϕ N t δ ) H σ 2 ϕ N t )dt. On the other hand H σ 2) = and so H σ 2 ϕ N t ) = H σ 2 ϕ t )I ϕ t N) what implies inequality H σ 2 ϕn t )dt H σ 2 ϕ t)dt. Then lim log P sup N t/ ϕ t δ sup ϕ t ϕ N t ) lim log P sup N t/ ϕ N t δ ) H σ 2 ϕ t )dt. The desired lower bound evidently holds since ϕ t ϕ N t ϕ ti ϕ t > N)dt, N.

14 14 O.V.GULINSKII*, AND R.S.LIPTSER** Thus, it remains to check only the validity of the lower bound under Due to σ 2 = the Gaussian component of N t equals zero. Put σ 2 =. 3.17) N γ t = N t + γw t, where W t is Wiener process which is to be assumed independent of N t and γ >. By the obtained above result we get lim log P sup N γ t/ ϕ t δ ) H γ 2 ϕ t )dt. As it was mentioned in Section 1 function H σ 2y) is increasing to H y) in σ 2 and so H γ 2 ϕ t)dt H ϕ t )dt, i.e. lim log P sup N γ t/ ϕ t δ ) Now we use the following chain of the lower estimates: { 2 max P sup P sup P sup N t/ ϕ t 2δ ), P γ sup W t/ > δ )} N t/ ϕ t 2δ ) + P γ sup W t/ > δ ) N t/ ϕ t δ + γ sup W t/ ) H ϕ t )dt. 3.18) P sup N γ t/ ϕ t δ ) 3.19) From 3.19) and 3.18) it follows that for any δ >, γ > { lim log P sup N t/ ϕ t 2δ ) and what follows from Lemma 3.1 max lim log P ) sup W t/ > 2δ γ H ϕ t )dt Thus, the desired lower bound holds. lim lim log P sup W t/ > 2δ ) =. γ γ

15 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS Proof of Theorem 1.1 Decomposition 2.1), inlet M t = ΣN t see 2.2)), implies W t = V t/ V ) + ΣN t/. 4.1) The first step consists in showing of exponential negligibility of the process V t/ V ) in a sense given below. Lemma 4.1. Let Cramer s conditions A.1) and A.2) be fulfilled. Then for any T >, γ > lim log P sup V t/ V γ ) =. Proof. Evidently, only lim log P sup V t/ γ ) = has to be checked. Let α k = sup T k 1) k V t, k 1. Noticing that α k ) k 1 forms in the restricted sense stationary sequence and using the Chernoff inequality with parameter λ > ), we find P sup V t/ γ ) = P sup V t γ/ ) / P max k γ/ ) k 1/ P α k γ/ ) Then k 1/ 1/)P α 1 γ/ ) 1/)e λγ/ Ee λα 1 Chernoff) = 1/)e λγ/ Ee λ sup V t. log P sup V t/ γ ) log λγ + log Ee λ sup Vt. By virtue of Lemma 2.1 the right hand side of last inequality goes to as limit lim lim is taken. λ In the second step the fact that the family N t/, as well as ΣN t/, is satisfied conditions C.1) and C.2) and Lemma 4.1 are used. Following them for any ϕ C and T > we get lim δ lim log P sup ΣN t/ ϕ t δ ) = lim δ lim log P sup W t/ ϕ t δ ) ; = ÎT ϕ)

16 16 O.V.GULINSKII*, AND R.S.LIPTSER** and lim δ lim log P sup ΣN t/ ϕ t δ ) = lim δ lim log P sup W t/ ϕ t δ ). = ÎT ϕ), where, letting =, Î T ϕ) = { H σ 2 ϕ t Σ )dt, ϕ = dϕ << dt, otherwise. 4.2) Thus by the method which has been used for proving Theorem 3.1 we get the l.d.p. for family Wt in the metric space D, d) with the rate function Iϕ) = sup Î T ϕ) T which coincides as such at 1.12). On the other hand, since for each the paths of the process Wt are continuous also the l.d.p. for this family holds in the metric space C, ρ) with the same rate function. 5. Proof of Theorem To apply the contraction principle of Varadhan [11] a continuous mapping, serving W s ) t = X t ) t, has to be constructed. Due to 1.3) consider differential equation Ẋ t = ax t ) + bx t )Ẏt, 5.1) subject to X, where Ẏt is the Radon-Nikodym derivative of some absolutely continuous function Y t from C. We show that the mapping Y t ) t = X t ) t defined by 5.1) can be extend for any function Y t from C. Lemma 5.1. Let functions ax) and bx) be Lipschitz continuous and there exist constant c and C such that < c bx) C. Then for fixed X there exists continuous in the metric ρ mapping Y t ) t = X t ) t, X t ), Y t ) C

17 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS 17 such that for any absolutely continuous function Y t ) this mapping is defined by differential equation 5.1). Proof. Let F x) = x dy by). 5.2) By virtue of assumptions making function F x) is continuous differentiable having inverse F 1 x) which is continuous differentiable too and both F x) and F 1 x) satisfy the linear growth condition: there exists positive constant l such that { F x) F 1 l1 + x ). x) Put gx) = af 1 x)) bf 1 x)). 5.3) Let Y t be absolutely continuous function and X t be a solution of 5.1). Put θ t = F X t ). It is easy to check that θ t is a solution of a differential equation θ t = gθ t ) + Ẏt subject to θ = F X ) which is unique by virtue of the local lipschitzianity of gx). Then we arrive to a mapping defined by X t = F 1 θ t ) θ t = F X ) + gθ s )ds + [Y t Y ]. 5.4) Evidently this mapping takes place not only for absolutely continuous but also for continuous function Y t. In fact gx), involving in the integral equation from 5.4), is satisfied the linear growth and the local Lipschitz conditions and so this integral equation obeys the unique solution also in case of continuous function Y t. Thus it remains to show that this mapping is continuous in the metric ρ. Note that the continuity in the metric ρ is equivalent to the following implication: for any T > { limn sup Y n t Y t = Y n t ), Y t ) C, n 1 { limn sup Xt n X t = = Xt n ), X t ) C, n 1, 5.5) where Xt n, n 1 and X t are solution of 5.4) corresponding to Yt n, n 1 and Y t respectively. The validity of 5.5) also is implied by the local Lipschitz condition for function gx). 2. Taking Y t Wt we obtain X t Xt, i.e the contraction principle is applicable. Then the family Xt obeys the l.d.p. in the metric space C, ρ) with rate function { inf Iψ), ϕ = X, dψ dt Jϕ) =, otherwise,

18 18 O.V.GULINSKII*, AND R.S.LIPTSER** where Iψ) is the rate function corresponding to the l.d.p. for W t ) and inf is taken over all absolutely continuous functions from C with ϕ = such that see 5.1)) ϕ t = aϕ t ) + bx t ) ψ t. Since this equation has the unique solution the inf is attained on ψ t = ϕ s aϕ s ) ds. bϕ s ) 6. Application From application point of view more realistic model than 1.4) for process η t, involving in 1.3), is η t = ht s)dn s, 6.1) where N t and ht) are the same as in 1.4) note that η t is not stationary process). Below we consider a such kind of process and show that nevertheless X obeys the l.d.p. with the same rate function. Let ξ t = ξ 1 t, ξ 2 t,..., ξ n t ) be vector-column Ito s process dξ t = Aξ t dt + BdN t, t R, 6.2) where A and B are matrices of dimensions n n and n 1 respectively and also the eigenvalues of A belongs to the left half of the plane. Under assumption making ξ t = = e At ξ + e At s) dn s E At s) dn s = e At ξ + ξ t. 6.3) As a process η t take ξ t 1 η t ξ t 1 ). Following 6.3) we have for η t decomposition 6.1) type with n { ht) = e At } 1,j B j, 6.4) j=1 where { e At} 1,j, 1 j n are elements of the first row of matrix eat and B j, 1 j n are elements of B. Put W t = 1 η s/ ds 6.5)

19 EXAMPLE OF LARGE DEVIATIONS FOR STATIONARY PROCESS 19 comp. 1.5)). We show that the family W t obeys the l.d.p. in the metric space C, ρ) with the rate function Iϕ) given in 1.12) with Σ = n { } j=1 e At B 1,j j. To this end introduce Wt = 1 ξ 1 sds. 6.6) Due to 6.3) ξs 1 = s hs u)dn u with ht) from 6.4). Under assumptions making there exist positive constants c 1 and c 2 such that ht) c 1 e c 2 and so Theorem 1.1 is applicable, i.e. Wt obeys the l.d.p. with the mentioned above rate function. Thus, the desired result for W t holds if for any T > and γ > lim log P sup Wt W t ) > γ ) =. 6.7) To check 6.7) note that under assumptions making W t W t ) = = c / n j=1 n j=1 ξ 1 s ξ 1 s)ds ξ 1 s ξ 1 s ds n { e As} 1,j ξj ds j=1 The, using the Chernoff inequality, we find for any λ > : P sup Wt W t ) > γ ) P n ξ j < γ ) c { e As} 1,j ds = c <. Then ξ j. 6.8) j=1 n P ξ j < j=1 n max 1 j N n max 1 j N γ ) nc { P ξ j < γ )} nc { e λγ/nc) Ee } λ ξj. In accordance to the last inequality the follwing upper estimate holds: log P sup Wt W t ) > γ ) { log n λγ/nc) + max log Ee λ ξ j } 1 j n n log n λγ/nc) + log Ee λ ξj. 6.9) j=1

20 2 O.V.GULINSKII*, AND R.S.LIPTSER** Assume that Ee λξj <, λ R, j = 1, 2,..., n. 6.1) Then, due to inequality e x e x + e x, we have for each j = 1, 2,..., n Ee λ ξj < for all λ >. Consequently under assumption 6.1) the right hand side of 6.9) goes to if limit lim is taken and so the desired property 6.7) takes place. lim λ Therefore, only 6.1) has to be checked. From 6.3) it follows that ξ = e As BdN s and by the homogeneity of increments of N t random vector ξ coinsides in the distribution with ξ = e As BdN s. 6.11) Let Λ = λ 1, λ 2,..., λ n ) be a vector-row with λ j R, j = 1, 2,..., n. Putting analogously of proving 2.6) we get Hs) = Λe As B Ee Λξ = Ee Λξ = E exp { } Hs)dN s <, i.e. 6.1) holds. Thus, W t obeys the l.d.p. The l.d.p. for X t follows from this result due to the contraction principle of Varadhan [11] and Lemma 5.1. References [1] P.Billingsley, Convergence of Probability Measures. J.Wiley, New York, [2] M.D.Donsker and S.R.S.Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, IV Communication of Pure and Applied Mathematics. 36, 1983, pp [3] M.I.Freidlin and A.D.Wentzell Random Perturbations of Dynamic Systems. Springer, Berlin. [4] O.V.Gulinsky and A.Yu.Veretennikov Large deviations for discrete-time Processes with averaging. VSP, Utrecht, Nehterland, [5] J.Jacod and A.N.Shiryayev Limit Theorem of Stochastic Processes. Springer, Berlin, [6] T.Lindvall, Weak convergence of probability measures and random function in the functional space D[, ) J. Appl. Probab. 1, 1973, pp [7] R.S.Liptser and A.A.Pukhalskii, Limit theorems on large deviations for semimartingales Stochastic and Stochastics Reports. 38, 1992, pp [8] R.S.Liptser and A.N.Shiryayev Theory of Martingales. Nauka in Russian) Kluwer, Moscow) Dordrecht 1986) [9] A.A.Puhalskii, On functional principle of large deviations, in: New trends in Probability and Statistics. V.Sazonov and Shervashidze eds.) Vilnius, Lithuania, VSP/Mokslas, 1991, pp [1] R.T.Rockafellar Convex Analysis. Princeton University Press, Princeton, 197. [11] S.R.S. Varadhan Large Deviations and Applications. SIAM, Philadelphia, [12] A.D.Wentzell Limit Theorems in Large Deviations for Markov Random Processes. Reidel, Dordrecht, 1984.