Relative entropy and waiting times for continuous-time Markov processes

Transcription

1 Relative entropy and waiting times for continuous-time Markov processes arxiv:math/052386v [math.pr] 6 Dec 2005 J.-R. Chazottes C. Giardina F. Redig February 2, 2008 Abstract For discrete-time stochastic processes, there is a close connection between return/waiting times and entropy. Such a connection cannot be straightforwardly extended to the continuous-time setting. Contrarily to the discrete-time case one does need a reference measure and so the natural object is relative entropy rather than entropy. In this paper we elaborate on this in the case of continuous-time Markov processes with finite state space. A reference measure of special interest is the one associated to the time-reversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almost-sure convergence to relative entropy of suitable waiting-times and their fluctuation properties central it theorem and large deviation principle. Introduction Many it theorems in the theory of stochastic processes have a version for discretetime as well as for continuous-time processes. The ergodic theory of Markov chains e.g. is more or less identical in discrete and in continuous time. The same holds for the Ergodic Theorem, martingale convergence theorems, central it theorems and large deviations for additive functionals, etc. Usually, one obtains the same results with some additional effort in the continuous-time setting, where e.g. extra measurability issues can show up. Key-words: continuous-time Markov chain, law of large numbers, central it theorem, large deviations, entropy production, time-reversed process CPhT, CNRS-Ecole polytechnique, 928 Palaiseau Cedex, France, and CMM, UMI CNRS 2807, Universidad de Chile, Av. Blanco Encalada 220, Santiago, Chile, jeanrene@cpht.polytechnique.fr Eurandom, Postbus 53, 5600 MB Eindhoven, The Netherlands, giardina@eurandom.tue.nl Mathematisch Instituut Universiteit Leiden, Niels Bohrweg, 2333 CA Leiden, The Netherlands, redig@math.leidenuniv.nl

2 For discrete-time ergodic processes, there is a remarkable theorem connecting recurrence times and entropy [9]. In words, it states that the logarithm of the first time the process repeats its first n symbols typically behaves like n times the entropy of the process. This provides a way to sample entropy observing a single, typical trajectory of the process. This result seems a natural candidate to transport to a continuoustime setting. The relation between entropy and return times is sufficiently intuitive so that one would not expect major obstacles on the road toward such a result for continuous-time ergodic processes. There is however one serious problem. On the path space of continuous-time processes on a finite state space, say, there is no natural flat measure. In the discrete-time setting one cannot distinguish between entropy of a process and relative entropy between the process and the uniform measure on trajectories. These only differ by a constant and a minus sign. As we shall see below, this difference between relative entropy and entropy does play an important role in turning to continuous-time processes. Therefore, there will be no continuous-time analogue of the relation between return times and entropy. In fact, the logarithm of return times turns out to have no suitable way of being normalized, even for very simple processes in continuous time such as Markov chains. To circumvent this drawback, we propose here is to consider differences of the logarithm of suitable waiting times and relate them to relative entropy. Another aspect of our approach is to first discretize time, and show that the relation between waiting times and relative entropy persists in the it of vanishing discrete time-step. From the physical point of view, the time-step of the discretization is the acquisition frequency of the device one uses to sample the process. We can also think of numerical simulations for which the discretization of time is unavoidable. Of course, the natural issue is to verify if the results obtained with the discretized process give the correct ones for the original process, after a suitable rescaling and by letting the time-step go to zero. This will be done in the present context. In this paper, we will restrict ourselves to continuous-time Markov chains with finite-state space for the sake of simplicity and also because the aforementioned problem already appears in this special, yet fundamental, setting. The main body of this paper is: a law of large numbers for the difference of the logarithm of certain waiting times giving a suitable relative entropy, a large deviation result and a central it theorem. One possible application is the estimation of the relative entropy density between the forward and the backward process which is physically interpreted as the mean entropy production, and which is strictly positive if and only if the process is reversible i.e., in detailed balance, or equilibrium. Our paper is organized as follows. In Section 2, we show why the naive generalization of the Ornstein-Weiss theorem fails. Section 3 contains the main results about law of large numbers, large deviations and central it theorem for the logarithm of ratios of waiting times. In the final section we consider the problem of shadowing a given continuous-time trajectory drawn from an ergodic distribution on path space. 2

3 2 Naive approach In this section we start with an informal discussion motivating the quantities which we will consider in what follows. Let {X t, t 0} be a continuous-time Markov chain with state space A, with stationary measure µ, and with generator Lfx = y A cxpx, yfy fx where px, y is a transition probability of a discrete-time irreducible Markov chain on A, with px, x = 0, and where the escape rates cx are strictly positive. Given a time-step, we can discretize the Markov chain to obtain its -discretization {X i, i = 0,, 2,...}. Next we define the first time the -discretized process repeats its first n symbols via the random variable R n X := inf{k : X 0,...,X n = X k,...,x k+n }. 2. The analogue of the Ornstein-Weiss theorem [9] for this continuous-time process would be a it theorem for a suitably normalized version of log Rn for n, 0. However, for > 0 fixed, the ergodicity of the -discretization {X 0, X, X 2,...,X n,...}, Ornstein-Weiss and Shannon-McMillan-Breiman theorems [9] yield n log [ RnXPX n ] = o eventually a.s. as n. Using the fact that X is an ergodic Markov chain we obtain n log R n X = n log µx 0 + n log p X n X i, X i+ + o = E [ log p X X 0, X ] + o eventually a.s. as n 2.2 where E denotes expectation in the Markov chain started from its stationary distribution and where p X denotes the transition probability of the -discretized Markov chain {X i, i = 0,, 2,...}, i.e., Therefore, x A n p X x, y = e L xy = I xy cx + cxpx, y + O 2. n log R n X = µx cx log cx + µxcxpx, y logcxpx, y + O 2. x,y A In this expression we see that the first term is of order whereas the second one is of order log. Therefore, this expression does not seem to have a natural way to be normalized. This is a typical phenomenon for continuous-time processes: we need a suitable reference process in order to define entropy as relative entropy with respect to this reference process. Indeed, as we will see in the next sections, by considering differences of waiting times one is able to cancel the log term in order to obtain expression that makes sense in the it 0. 3

4 3 Main results: waiting times and relative entropy We consider continuous-time Markov chains with a finite state-space A. We will always work with irreducible Markov chains with a unique stationary distribution. The process is denoted by {X t : t 0}. The associated measure on path space starting from X 0 = x is denoted by P x and by P we denote the path space measure of the process started from its unique stationary distribution. For t 0, F t denotes the sigma-field generated by X s, s t, and P [0,t] denotes the measure P restricted to F t. 3. Relative entropy: comparing two Markov chains Consider two continuous-time Markov chains, one denoted by {X t : t 0} with generator Lfx = y A cxpx, yfy fx 3. and the other denoted by {Y t : t 0} with generator Lfx = y A cxpx, yfy fx where px, y is the Markov transition function of an irreducible discrete-time Markov chain. We further assume that px, x = 0, and cx > 0 for all x A. We suppose that X 0, resp. Y 0, is distributed according to the unique stationary measure µ, resp µ so that both processes are stationary and ergodic. Remark. The fact that the Markov transition function px, y is the same for both processes is only for the sake of simplicity. All our results can be reformulated in the case that the Markov transition functions would be different. We recall Girsanov s formula [5]: dp [0,t] ω = µω t 0 d P [0,t] µ ω 0 exp log cω s cω s dn sω 0 t 0 cω s cω s ds 3.3 where N s ω is the number of jumps of the path ω up to time s. The relative entropy of P w.r.t. P up to time t is defined as dp [0,t] s t P P = dpω log ω. 3.4 d P [0,t] Using 3.3 and stationarity, we obtain s t P P t t = x A =: sp P µxcx log cx cx µxcx cx 3.5 x A 4

5 where sp P is the relative entropy per unit time of P with respect to P. We refer to [4], [] for more details on relative entropy for continuous-time Markov chains. Notice also that, by ergodicity, dp[0,t] log ω = sp P P a.s.. t t d P [0,t] In the case {Y t : t 0} is Markov chain with generator Lfx = y A cx px, yfy fx 3.5 generalizes to sp P = x,y A µxcxpx, y log cxpx, y cx px, y µxcx cx 3.6 x A A important particular case is met when {Y t : t 0} is the time-reversed process of {X t : t 0}, i.e., Y t 0 t T = X T t 0 t T in distribution. This is a Markov chain with transition rates In that particular situation, the random variable cypy, xµy cx, y = cx. 3.7 cxµx S T ω = log d P [0,T] dp [0,T] 3.8 has the interpretation of entropy production, and the relative entropy density sp P has the interpretation of mean entropy production per unit time. see e.g. [6, 7]. 3.2 Law of large numbers For > 0, we define the discrete-time Markov chain X := {X 0, X, X 2,...}. This Markov chain has transition probabilities p X x, y = el xy = I xy cx + cxpx, y + O where I is the identity matrix. Similarly we define another Markov chain Y with transition probabilities p Y x, y = e L xy = I xy cx + cxpx, y + O

6 The path-space measure on A N of X, resp. Y, is denoted by P, resp. P. From now on, we will write P X n instead of P X, X 2..., X n to alleviate notations. We define waiting times, which are random variables defined on A N A N, by setting Wn X Y = inf{k : X,...,X n = Y k+,...,y k+n } 3. where we make the convention inf =. In words, this is the first time that in a realization of the process Y that one observes the first n symbols of a realization of the process X. Similarly, if X is an independent copy of the process X, we define Wn X X = inf{k : X,..., X n = X k+,...,x k+n }. 3.2 We then have the following law of large numbers. Theorem. P P P-almost surely: 0 n n log W n X Y WnX X = sp P. 3.4 Before proving this theorem, we state a theorem about the exponential approximation for the hitting-time law, which will be the crucial ingredient throughout this paper. For a n-block x n := x,..., x n A n and a discrete-time trajectory ω A N, we define the hitting time T x n ω = inf{k : X k+ = ω,...,x n+k+ = ω n+ }. 3.5 We then have the following result, see []. Theorem 2. For all > 0, there exist η, η 2, C, c, β, κ ]0, [ such that for all n N and for all x n An, there exists η = ηx n, with 0 < η η η 2 < such that for all t > 0 P Tx ω > n t e ηt P X n = x n The same theorem holds with P replaced by P. Ce ct P X n = xn κ Ce ct e βn. 3.7 The constants appearing in Theorem 2 except C depend on, and more precisely we have β = β 0, η = η 0 as 0. This is important in applications, since one wants to choose a certain discretization and then a corresponding word-length n for the waiting times, or vice-versa. From Theorem 2 we derive see [2]: Proposition. For all > 0, there exist κ, κ 2 > 0 such that κ log n log W n X Y P X n loglog n κ 2 P P eventually a.s. and κ log n log W n X X P X n loglog n κ 2 P P eventually a.s. 6

7 With these ingredients we can now give the proof of Theorem. Proof of Theorem. From Proposition it follows that, for all > 0, P P P almost surely log Wn n n X Y log W n n X X + log p Y X n i, X i+ log p X X i, X i+ = 0. i=0 i=0 3.9 By ergodicity of the continuous-time Markov chain {X t : t 0}, the discrete Markov chains X, Y are also ergodic and therefore we obtain log Wn n n X Y log W n X X + p µxp X Y x, y log x, y = 0. p X x,y A x, y 3.20 Using 3.9, 3.0 and px, x = 0, this gives log W n n n X Y log Wn X X = µx cx log cx µxcxpx, y log cxpx, y x A x,y A + µx cx log cx + µxcxpx, y logcxpx, y + O 2 x A x,y A = x,y A µxcxpx, y log cx cx + µx cx cx x A + O 2 = sp P + O Combining this with 3.5 concludes the proof of Theorem. Let us now specify the dependence on of the various constants appearing in Theorem 2. For the lower bound on the parameter we have see [], section 5 η C + K where C is a positive number independent of and K = 2 αl + l= n/2 k= sup {x n k } P X n k = x n k. Here αl denotes the classical α-mixing coefficient: αl = sup sup P S S 2 P S P S 2 j S F j 0,S 2 Fj+l 3.22 where F n m is the Borel sigma-field on A N generated by X n m 0 m n. By the assumption of ergodicity of the continuous Markov chain, the generator L resp. L 7

8 has an eigenvalue 0, the largest real part of the other eigenvalues is strictly negative and denoted by λ < 0, and one has Using 3.9 there exists λ 2 > 0 such that αl exp λ l P X n k = x n k exp λ 2 n/2 for k =,...,n/2. Therefore, there exists ĉ > 0 such that η > ĉ Similarly, from the proof of Theorem 2. in [2] one obtains easily the dependence on of the constants appearing in the error term of 3.7. c = c > γ, β = β > γ for some γ, γ 2 > 0. In applications, e.g., the estimation of the relative entropy from a sample path, one would like to choose the word-length n and the discretization = n together. This possibility is precisely provided by the estimates 3.24 and 3.25, as the following analogue of Proposition shows. Proposition 2. Let n 0 as n, then there exists κ, κ 2 > 0 log n κ log Wn n X Y P n X n κ2 log n log P n P eventually a.s. n 3.27 and log n κ log Wn n X X P n X n κ2 log n log P P eventually a.s.. n n Proof. The proof is analogous to the proof of Theorem 2.4 in [2]. For the sake of completeness, we prove the upper bound We can assume that n. By the exponential approximation 3.7 we have, for all t > 0, n, the estimates P P log WnX Y P X n log t e ηnt + Ce βnn e cnt e η nt + Ce γ nn e γ 2 nt Choosing t = t n = κ 2 log n n, with κ 2 > 0 large enough makes the rhs of 3.28 summable and hence a Borel-Cantelli argument gives the upper bound. Of course, whether this proposition is still useful, i.e., whether it still gives the law of large numbers with = n depends on the behavior of ergodic sums fx i 8

9 under the measure P n, i.e., the behavior of fx in under P. This is made precise in the following theorem: Theorem 3. Suppose that n 0 as n such that log n nn 2 probability: 0 then in P P P n Proof. By proposition 2 we can write log W n nx Y n n Wn n X X = sp P = log Wn n X Y log Wn n X X I Xi =X i+ log ncx i n cx i + I Xi X i+ log cx i cx i + Olog n/ n.3.3 The sum on the right hand site of 3.3 is of the form with F n X in, X i+n 3.32 EF n EF n 2 C n 3.33 where C > 0 is some constant. Now, using ergodicity of the continuous-time Markov chain {X t, t 0}, we have the estimate E F n X in, X i+n EF n F n X jn, X j+n EF n F n 2 2 e nλ i j with λ > 0 independent of n. Combining these estimates gives V ar F n X in, X i+n Cn n + j {,...,n}\{i} 3.34 n e nλ i j Cn n +C n n n 3.35 where C > 0 is some constant. Therefore, V ar F n 2 n 2 n X in, X i+n = O/nn Combining 3.3 and 3.36 with the assumption log n n 2 n 0 concludes the proof. 9

10 3.3 Large deviations In this subsection, we study the large deviations of W n log n X Y Wn X X More precisely, we compute the large deviation generating function F p in the it 0 and show that it coincides with the large deviation generating function for the Radon-Nikodym derivatives dp [0,t] /d P [0,t]. As in the case of waiting times for discretetime processes, see e.g. [3], the scaled-cumulant generating function is only finite in the interval,. For the sake of convenience we introduce the function Ep := E dp [0,t] p p = 0 t t log E P = t t t log E P exp p log cω s 0 cω s dn sω t 0 d P [0,t] cω s cω s ds By standard large deviation theory for continuous-time Markov chains see e.g. [0] this function exists and is the scaled-cumulant generating function for the large deviations of t log cω t s cω s dn sω cω s cω s ds as t. 0 We can now formulate the following large deviation theorem. Theorem 4. For all p R and > 0 the function F p := n n log E P P P exists, is finite in p, whereas F p = for p. Moreover, as 0, we have, for all p, : Fp := 0 F p = Ep. 0 W n X Y W n X X p 3.39 The following notion of logarithmic equivalence will be convenient later on. Definition. Two non-negative sequences a n, b n are called logarithmically equivalent notation a n b n if n n log a n log b n = 0. 0

11 Proof. To prove Theorem 4, we start with the following lemma. Lemma.. For all > 0 and for p <, W p E n X Y n P P P E Wn X X P exp p log 2. For p >, i=0 p X X i, X i+ p Y X i, X i+ W p n n log E n X Y P P P = WnX X Proof. The proof is similar to that of Theorem 3 in [3]. p W E n X Y P P P Wn X X = P P X n = xn X n = x n x,...,x n P Y n = x n p Tx n Y P Y n = xn E P P T x n X P X n = x n = P X n = x n +p P Y n = x n p ξn E P P ζ x,...,x n n where ξ n = T x n Y P Y n = x n and ζ n = T x n X P X n = xn. The random variables ξ n, ζ n have approximately an exponential distribution in the sense of Theorem 2 and are independent. Using this, we can repeat the arguments of the proof of Theorem 3 in [3] -which uses the exponential law with the error-bound given by Theorem 2- to prove that for p, p ξn 0 < C E P P C 2 < ζ n where C, C 2 do not depend on n, whereas for p >, p ξn E P P = ζ n Therefore, with the notation of Definition, for p < W E n X Y P P P WnX X p P X = x,...,x n = x n +p P Y = x,...,y n = x n p x,...,x n p X = E P exp p log X i, X i+ p Y X 3.45 i, X i+ p p

12 and for p > we obtain 3.43 from This proves the existence of F p. Indeed, the it F p = n n log E P exp p log p X X i, X i+ p Y X 3.46 i, X i+ exists by standard large deviation theory of discrete-time, finite state space Markov chains since > 0 is fixed. In order to deal with the it 0 of F p, we expand the expression in the rhs of 3.42, up to order 2. This gives p X E P exp p log X i, X i+ p Y X i, X i+ = e On2 E P exp p log I X i,x i+ + cx i px i, X i+ cx i I Xi,X i+ + cx i px i, X i+ cx i = e On2 E P [ exp p IX i = X i+ cx i cx i + p IX i X i+ log cx ] i cx i [ = e On2 E P exp p IX i = X i+ cx i cx i + p IX i X i+ log cx ] i cx i Next we prove that for all K R log E P exp K n IXi = X i+ cx i cx i + IX i X i+ log cx i cx i exp K n 0 cx s cx s ds + K n 0 log cxs cx dn s s 3.47 = On 2. This implies the result of the theorem by a standard application of Hölder s inequality, see e.g., [4]. We first consider the difference An, := IX i X i+ log cx n i cx i log cx s 0 cx s dn s. 2

13 If there does not exist an interval [i, i + [, i {0,...,n } where at least two jumps of the Poisson process {N t, t 0} occur, then An, = 0. Indeed, if there is no jump in [i, i + [, both IX i X i+ log cx i and i+ cx i log cxs dn i cx s s are zero and if there is precisely one jump, then they are equal. Therefore, using the independent increment property of the Poisson process, and the strict positivity of the rates, we have the bound An, C Iχ i 2 where the χ i s, i =,...,n, form a collection of independent Poisson random variables with parameter, and C is some positive constant. This gives Next, we tackle Bn, := E P e 2KAn, = O 2 e 2K + O n = Oe n IX i = X i+ cx i cx i n 0 cx s cx s ds If there is no jump in any of the intervals [i, i + [, this term is zero. Therefore is is bounded by Bn, C Iχ i where the χ i s, i =,...,n, form once more a collection of independent Poisson random variables with parameter, and C is some positive constant. This gives E P e 2KBn, Oe C + n = Oe n2 where C is some positive constant. Hence, 3.47 follows by combining 3.48 and 3.49 and using Cauchy-Schwarz inequality. The following propoisition is a straightforward application of Theorem 4 and [8]. Proposition 3. For all > 0, F is real-analytic and convex, and the sequence {log W nx Y log W nx X : n N} satisfies the following large large deviation principle: Define the open interval c, c +, with de c ± := p ± dp < 0 Then, for every interval J such that J c, c + { } n n log P P W P n log n X Y J = inf Wn X X q J c,c + I q where I is the Legendre transform of F. 3

14 Remark 2. In the case {Y t : t 0} is the time reversed process of {X t : t 0}, the cumulant generating function function Ep satisfies the so-called fluctuation theorem symmetry Ep = E p. The large deviation result of Theorem 4 then gives that the entropy production estimated via waiting times of a discretized version of the process has the same symmetry in its cumulant generating function for p [0, ]. 3.4 Central it theorem Theorem 5. For all > 0, n log W n X Y nsp P Wn X X converges in distribution to a normal law N0, σ 2, where σ 2 = P n n Var X n log P X n Moreover where 0 2σ2 = θ2 θ 2 dp [0,t] = t t Var log Proof. First we claim that for all > 0 W n n E P P P log n X Y Wn X X d P [0,t] log px X 2 i, X i+ p Y X = i, X i+ This follows from the exponential law, as is shown in [3], proof of Theorem 2. Equation 3.53 implies that a CLT for log Wn X Y log W n X X is equivalent to a CLT for n log px X i,x i+ p Y X i,x i+ and the variances of the asymptotic normals are equal. For fixed, n log px X i,x i+ satisfies the CLT for > 0 fixed, X p Y X i,x i+ i is a discrete-time ergodic Markov chain, so the only thing left is the claimed iting behavior for the variance, as 0. As in the proof of the large deviation theorem, we first develop up to order : = =: log px X i, X i+ p Y X i, X i+ IX i = X i+ cx i cx i + ξi + ζi. 4 IX i X i+ log cx i cx i

15 It is then sufficient to verify that 2 i+ i+ E ξi 0 n n cx s cx s ds + ζi log cx 2 s cx s dn s = 0 i which is an analogous computation with Poisson random variables as the one used in the proof of Theorem 4. i 4 Shadowing a given trajectory Let γ D[0,, X be a given trajectory. The jump process associated to γ is defined by N t γ = Iγ s γ s +. 0 s t For a given > 0, define the jump times of the -discretization of γ: Σ n γ = {i {,..., n} : γ i γ i }. For the Markov process {X t, t 0} with generator Lfx = cxpx, yfy fx y A define the hitting time T n γ X = inf{k 0 : X k,..., X k+n = γ,...,n}. In words this is the first time after which the -discretization of the process imitates the -discretization of the given trajectory γ during n time-steps. For fixed > 0, the process {X n ; n N} is an ergodic discrete-time Markov chain for which we can apply the results of [] for hitting times. More precisely there exist 0 < Λ < Λ 2 < and C, c, α > 0 such that for all γ, n N, there exists Λ < λ γ n < Λ 2 such that P Tn γ XPX i = γ i, i =,...,n + > t e λγ nt Ce ct e αn. 4. As a consequence of 4. we have Proposition 4. For all > 0, there exist κ, κ 2 > 0 such that for all γ D[0,, X, P P eventually almost surely κ log n log Tn γ Y PY = γ,...,y n = γ n loglog n κ 2 and κ log n log T n γ XPX = γ,...,x n = γ n loglog n κ 2. 5

16 Therefore, for > 0 fixed, we arrive at i Σ nγ log Tn γ X = 4.3 logcγ i pγ i, γ i + log cγ i + on. i {,...,n}\σ nγ The presence of the log term in the rhs of 4.3 causes the same problem as we have encountered in Section 2. Therefore, we have to subtract another quantity such that the log term is canceled. In the spirit of what we did with the waiting times, we subtract log T n γ Y, where {Y t : t 0} is another independent Markov process with generator Lfx = y A cx px, yfy fx. We then arrive at log T n γ X Tnγ X = log cγ i pγ i, γ i cγ i pγ i, γ i + log cγ i cγ i +on i Σ n γ i {,...,n}\σ n γ We then have the following law of large numbers 4.4 Theorem 6. Let P resp. P denote the stationary path space measure of {X t : t 0} resp. {Y t : t 0} and let γ D[0,, X be a fixed trajectory. We then have P Palmost surely: 0 n = 0. n log T n γ Y n Tnγ X log cγ spγ s, γ s + 0 cγ s pγ s, γ s + dn sγ n 0 cγ s cγ s ds Moreover, if γ is chosen according to a stationary ergodic measure Q on path-space, then Q-almost surely 4.6 log T 0 n n n γ Y log Tn γ X = cxpx, y qx, y log cx px, y + qx cx cx x,y A x A 4.7 where qx, y = t E Q qx = Qγ 0 = x N xy t t and where N xy t γ denotes the number of jumps from x to y of the trajectory γ in the time-interval [0, t]. Proof. Using proposition 4, we use the same proof as that of Theorem, and use that the sums in the rhs of 4.4 is up to order 2 equal to the integrals appearing in the lhs of 4.6. The other assertions of the theorem follow from the ergodic theorem. 6

17 Remark 3. If we choose γ according to the path space measure P, i.e., γ is a typical trajectory of the process {X t : t 0}, and choose px, y = px, y, then we recover the it of the law of large numbers for waiting times Theorem : log T 0 n n n γ Y log Tnγ X = x µxcx log cx cx + x µx cx cx = sp P References [] M. Abadi, Exponential approximation for hitting times in mixing processes, Math. Phys. Electron. J [2] M. Abadi, J.-R. Chazottes F. Redig and E. Verbitskiy, Exponential distribution for the occurrence of rare patterns in Gibbsian random fields, Commun. Math. Phys. 246 no , [3] J.-R. Chazottes and F. Redig, Testing the irreversibility of a Gibbsian process via hitting and return times, Nonlinearity 2005, 8, [4] A. Dembo and O. Zeitouni, Large deviation techniques and applications, Springer, 998. [5] I.I. Gihman, A.V. Skorohod, The theory of stochastic processes. II. Die Grundlehren der Mathematischen Wissenschaften 28, Springer, 975. [6] D.-Q. Jiang, M. Qian, M.-P. Qian, Mathematical theory of nonequilibrium steady states. On the frontier of probability and dynamical systems. Lecture Notes in Mathematics 833, Springer, [7] C. Maes, The fluctuation theorem as a Gibbs property, J. Stat. Phys. 95, , 999. [8] D. Plachky, J. Steinebach, A theorem about probabilities of large deviations with an application to queuing theory, Period. Math. Hungar , no. 4, [9] P.C. Shields, The ergodic theory of discrete sample paths. Graduate Studies in Mathematics 3, American Mathematical Society, Providence, RI, 996. [0] D. Stroock, An introduction to Markov processes. Graduate Texts in Mathematics 230, Springer, [] S.R.S. Varadhan, Large deviations and applications. Philadelphia: Society for Industrial and Applied Mathematics,