Large deviations for martingales

Transcription

1 Stochastic Processes and their Applications ) Large deviations for martingales Emmanuel Lesigne a, Dalibor Volny b; a Laboratoire de Mathematiques et Physique Theorique, UMR 6083 CNRS, Universite Francois Rabelais, Parc de Grandmont F Tours, France b Laboratoire Raphael Salem, UMR 6085 CNRS, Universite de Rouen, F Mont-Saint-Aignan Cedex, France Received 28 December 2000; received in revised form 25 May 2001; accepted 3 June 2001 Abstract Let X i) be a martingale dierence sequence and S n = n Xi. We prove that if sup i Ee X i ) then there exists c 0 such that S n n) 6 e cn1=3 ; this bound is optimal for the class of martingale dierence sequences which are also strictly stationary and ergodic. If the sequence X i) is bounded in L p ; 2 6 p, then we get the estimation S n n) 6 cn p=2 which is again optimal for strictly stationary and ergodic sequences of martingale dierences. These estimations can be extended to martingale dierence elds. The results are also compared with those for iid sequences; we give a simple proof that the estimate of Nagaev, Baum and Katz, S n n)=on 1 p ) for X i L p ; 1 6 p, cannot be improved and that, reciprocally, it implies the integrability of X i p for all 0. c 2001 Elsevier Science B.V. All rights reserved. MSC: 60F10; 60G42; 60G50; 28D05; 60G10 Keywords: Large deviations; Large deviations for sums of random variables and random elds; Independent random variables; Martingale dierence sequence; Martingale dierence eld; Stationary random variables; Stationary random eld; Measure preserving dynamical system 1. Introduction Let X i ) be a sequence of real integrable random variables with zero means, dened on a probability space ; A;) and S n = n X i. We shall study the asymptotic behaviour of the probabilities S n nx); x 0; n : 1) If the sequence X i ) is independent and identically distributed iid), the wea law of large numbers asserts that S n nx) 0 when n. More generally, if the sequence X i ) is stationary in the strong sense), then the ergodic theorem asserts that the result is still true. If the sequence X i ) is a sequence of martingale dierences bounded in L 2, then the convergence still holds. Corresponding author. Fax: addresses: lesigne@univ-tours.fr E. Lesigne), dalibor.volny@univ-rouen.fr D. Volny) /01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved. PII: S )

2 144 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) In this paper, we study the speed of convergence of 1) with x as constant when X i ) is a sequence of martingale dierences. In the iid case, the study of these so-called large deviations estimates is a widely studied subject. It is well nown cf. e.g. Hall and Heyde, 1980) that for strictly stationary and ergodic martingale dierence sequences classical limit theorems lie the central limit theorem, the law of iterated logarithm, or the invariance principle remain valid in the same form as for iid sequences. As we shall see, this is not the case for large deviations estimates, except for bounded martingale dierences where a bound for probabilities 1) is given by Azuma s 1967) theorem. We give new results under nite exponential moments or nite pth moments 1 6 p ) hypothesis. Every estimation which is given is shown to be optimal even in the restricted class of strictly stationary sequences of martingale dierences. We remar that the hypothesis of stationarity by itself cannot give any estimation of large deviation. Indeed, if the ergodic theorem implies that S n nx) 0, the speed of convergence can be arbitrarily slow. We refer to Lesigne and Volny 2000) for a study of this subject. In order to facilitate the comparison with the martingale case, we rst briey recall some results in the iid case. 2. Independent random variables Let X i ) be a sequence of iid real random variables, integrable and with zero mean: EX i ) = 0. We denote S n = X 1 + X X n. By the wea law of large numbers we have, for any x 0, lim S n nx)=0: n The speed of convergence has been studied by many authors. The classical result concerning the case when random variables X i have nite exponential moments is attached to the name of Cramer. Results under L p conditions, for p 1, have been obtained by Spitzer 1956), Hsu and Robbins 1947), Erdos 1949, 1950), Baum and Katz 1965), Nagaev 1965). The three following theorems are classical results and can be found, for instance, in the boos of Petrov 1995) or Chow and Teicher 1978). Theorem 2.1. There is equivalence between the following. i) There exists c 0 such that Ee c X1 ). ii) There exists d 0 such that S n n) = oe dn ). iii) For every x 0 there exists c x 0 such that S n nx) = oe cxn ). Theorem 2.2. Let p [1; + ). There is equivalence between the following. i) E X 1 p ). ii) n=1 np 2 S n n). iii) For every x 0; n=1 np 2 S n nx). Theorem 2.3. Let p [1; + ). If E X 1 p ) ; then S n n)=on 1 p ).

3 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) Remar 1. The following lemma, which can be used to prove that ii) implies i) in Theorem 2.1, is very similar to an exercise from Durrett 1991, Chapter 1, Section 9), and its proof is left to the reader. Lemma 2.4. lim inf n S n n) n X 1 2n) 1: Remar 2. There exists a partial converse to Theorem 2.3: Proposition 2.5. Let p 1. If there exists x 0 such that S n nx)=on 1 p ); then 0; E X 1 p ) : This proposition is an easy consequence of Theorem 2.2 or Lemma 2.4. The detailed proof is again left to the reader, who will also remar that the claim E X p ) can be strengthened; we can, e.g. get E X p =log + X ) 1+ ), where log + x is de- ned as logmaxx; e)). Remar 3. Lemma 2.4 also allows us to prove that the estimate of Theorem 2.3 cannot be essentially improved, as it is stated in the following proposition. Proposition 2.6. Let p 1 and c n ) be a real positive sequence approaching zero. There exists a probability law on the real line such that for iid random variables X i with the distribution we have E X i p ) ; EX i =0 and lim sup n n p 1 S n n)= : c n More precisely; for any increasing sequence of integers n ) with =1 c n ; the law can be chosen so that n p 1 =c n ) S n n ) as. ProofofProposition 2.6. By Lemma 2.4 it suces to prove the existence of a probability law with zero mean and nite pth moment, for which 1 lim n p c [2n ; + [) = : n Let ) be a sequence of positive numbers such that c n 0 and and dene a = c n n p ; a = c n 2a n p ; = a 2n + 2n )

4 146 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) x denotes the Dirac measure at point x). By its denition, is a symmetric probability law and, since a n p, has a nite pth moment. But we have 1 n p c [2n ; + [) 1 n p n c a : n Remar 4. The theorems which have been briey recalled in this section have various generalizations, in particular to the case of variable x s and to the case of mixing sequences of random variables, see e.g. Peligrad 1989). 3. Martingale dierence sequences By X i ) we denote a not necessarily stationary) martingale dierence sequence and we set n S n = X i : We study the speed of convergence of S n nx) to zero in three dierent cases. The case when variables X i are uniformly bounded is well nown: Azuma s inequality gives a speed similar to the one in Cramer s theorem on iid sequences. But in the case of exponential nite moments, the results are dierent from those in the iid case, and we prove that there exists a constant c such that S n nx) 6 exp cn 1=3 ): In the case of nite pth moments, we also obtain a new estimate of the type S n nx) 6 cn p=2 : One can show that these inequalities tae place even for max 166n S nx). For Azuma s inequality it was proved by Laib cf. Laib, 1999), the other inequalities can be generalized using Laib s result and standard techniques cf. Hall and Heyde, 1980, Theorem 2:10). In the three cases, we show that these estimates are optimal, even in the restricted class of strictly stationary and ergodic sequences of martingale dierences. Strictly stationary sequences will be always represented under the form X i )=f T i ), where T is a measurable and measure preserving transformation of a standard probability space ; A; ). Theorem 3.1. Azuma 1967)). Let X i ) 16i6n be a nite sequence of martingale dierences. If X i a for all i; then we have S n nx) 6 e nx2 =2a 2 : 2) The fact that this estimation is optimal is already nown for iid sequences. Theorem 3.2. For any positive numbers K; x; ; there exists a positive integer n 0 such that; if n n 0 ; if X i ) 16i6n is a nite sequence of martingale dierences and if Ee Xi 6 K for all i; then S n nx) exp )x2=3 n 1=3) : 3)

5 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) If for a 0, Ee Xi are uniformly bounded, then we can apply 3) to the sequence of X ;i = X i and obtain S n nx)= S ;n nx) exp )2=3 x 2=3 n 1=3 ) 3 ) where S ;n = n X ;i = S n. As shown in de la Peña 1999), assumptions on EXi 2 F i 1 ) where F i ) is a ltration for the martingale dierence sequence X i )) can guarantee better estimates. In particular, if EXi 2 F i 1 ) are uniformly bounded, then we get an estimation S n n) 6 e cn for some c 0. The next result shows that without more assumptions, even for the class of ergodic stationary sequences, the estimate of Theorem 3.2 cannot be essentially improved. Theorem 3.3. In every ergodic dynamical system of positive entropy there exists c 0 and a function f; Ee f ; such that f T i ) is a martingale dierence sequence and S n f) n) e cn1=3 4) for innitely many n S n f):= n f T i ). The proof of Theorem 3.2 is based on Azuma s inequality and truncation arguments. In this proof, we shall give a precise and nonasymptotic upper bound for S n nx). In the proof of the second part, we use an estimation of moderate deviations for sums of iid random variables which is due to Cramer. We give this estimation as it appears in Feller s boo Feller, 1971, 2nd Edition, Vol. II, p. XVI.7). Theorem 3.4. Cramer, 1938). Let Y i ) be a sequence of iid random variables; with EY i =0 and Ee tyi ) for t real in a neighbourhood of zero. Denote 2 := EYi 2 ); and R n := n Y i. There exists a function dened and analytic on a neighbourhood of zero such that; for any real sequence x n ); if x n and x n =o n); then R n x n n) 1 = 2 + x n ) ) )) )) exp u2 x du exp n n xn xn 1+O : n n Corollary 3.5. Under the hypotheses of Theorem 3:4; for any 0; for all large enough n; exp 1 ) )x2 n R n x n n) exp 1 ) )x2 n : Let us now consider the case of martingale dierence sequences with nite moment of order p.

6 148 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) Theorem 3.6. Let X i ) 16i6n be a nite sequence of martingale dierences where X i L p ; 2 6 p ; X i p M for all i. Let x 0. Then S n nx) 6 18pq 1=2 ) p M p 1 x p n p=2 5) where q is the real number for which 1=p +1=q =1. Theorem 3.7. In every ergodic dynamical system ; A; ; T) of positive entropy and for every sequence b n ) of positive numbers approaching zero there exists f L p 0 such that f T i ) is a martingale dierence sequence and lim sup n n p=2 b n S n f) n)= : 6) The proof of Theorem 3.6 is based on classical norm estimates for martingale difference sequences. In the proof of Theorem 3.7, we use the central limit theorem for iid sequences and Azuma s inequality. Via the so-called Gordin s decomposition, Theorem 3.6 has wide applications to the study of speed of convergence in ergodic theorem see Corollary 4.4 below). Remar. Proposition 2.6 shows that for p = 1 there exist iid sequences for which the convergence S n f) n) 0 can be arbitrarily slow from the point of view of subsequences). We rst prove Theorems 3.2 and 3.6. ProofofTheorem 3.2. For a martingale dierence sequence X i ) 16i6n with X i 6 1, the Azuma s inequality gives, for any x 0, S n nx) 6 2 exp nx 2 =2): 7) Let X i ) 16i6n be a martingale dierence sequence such that Ee Xi 6 K and denote F i ) 16i6n as its ltration. Let us x a 0. For 1 6 i 6 n dene Y i = X i Xi 6an 1=3 ) EX i Xi 6an 1=3 ) F i 1 ); Z i = X i Xi an 1=3 ) EX i Xi an 1=3 ) F i 1 ); S = S = Y i ; Z i : Y i ) and Z i ) are martingale dierence sequences and, because X i ) is a martingale dierence sequence, X i = Y i + Z i 1 6 i 6 n). Let us x t 0; 1). For every x 0, S n nx) 6 S n nxt)+ S n nx1 t)): 8)

7 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) We have Y i 6 2an 1=3 for 1 6 i 6 n, hence by using 7) we get S S n nxt)= n 2an 1=3 nxt ) 2an 1=3 6 2 exp t2 x 2 ) 8a 2 n1=3 : 9) Let F i x)= X i x). From Ee Xi 6 K it follows that F i x) 6 Ke x for all x 0. Then, EZi 2 = EX i Xi an )) 2 ) EEX 1=3 i Xi 6an ) F 1=3 i 1 )) 2 ) 6 EX i Xi an )) 2 )= x 2 df 1=3 i x) an 1=3 ;+ ) = lim M 2 F i M) a 2 n 2=3 F i an 1=3 ) M an 1=3 ;M] ) 2xF i x)dx 6 Ka 2 n 2=3 e an1=3 +2K xe x dx = Ka 2 n 2=3 +2an 1=3 + 2)e an an 1=3 We have ES n ) 2 6 nka 2 n 2=3 +2an 1=3 + 2)e an1=3. From this we get S n nx1 t)) 6 1 n 2 x 2 1 t) 2 nka2 n 2=3 +2an 1=3 + 2)e an 1=3 K = x 2 1 t) 2 a2 n 1=3 +2an 2=3 +2n 1 )e an 1=3 : 10) In view of 9) and 10) we choose a so that t 2 x 2 =8a 2 = a = 1 2 tx)2=3. We obtain S n nx1 t)) 6 K ) tx) 4=3 x 2 1 t) 2 n 1=3 +tx) 2=3 n 2=3 +2n 1 4 1=3 exp 12 ) tx)2=3 n 1=3 : From 8), 9), and 11) we deduce that for every x 0 and every t 0; 1), S n nx) 2+ K 1 t) 2 : 11) )) 1 4 t4=3 x 2=3 n 1=3 + t 2=3 x 4=3 n 2=3 +2x 2 n 1 e 1=2)t2=3 x 2=3 n 1=3 : We can choose the parameter t arbitrarily close to 1, and 3) follows. ProofofTheorem 3.6. By Burholder s inequality cf. Hall and Heyde, 1980, Theorem 2:10), we have n p p=2 E X i 6 18pq 1=2 ) p n E Xi 2 : Set Y i = Xi 2. By a convexity inequality, we have n n ) 2=p Y i!) 6 n 1 2=p Y p=2 i!)

8 150 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) therefore, n p n E X i 6 18pq 1=2 ) p n p=2 1 E X i p 6 n p=2 18pq 1=2 ) p M p : It follows that Sn p S n nx) 6 x p n p d 6 18pq1=2 ) p M p 1 x p n p=2 : This completes the proof of 5). Remar that for p = 2 we can only use, instead of Burholder s inequality, the orthogonality of X i. It is not dicult to construct an example of a stationary martingale dierence sequence X i ) with nite exponential moments nite pth moments) for which 4) 6), respectively) is satised. It suces to tae an iid sequence Y i ) with Y 1 bounded and EY 1 ) = 0, to nd a suitable random variable Z independent of the sequence Y i ), and to substitute X i = Y i Z. For proving 4) we can use the inequality ) n S n n) Y i n 2=3 Z n 1=3 ) and Corollary 3.5, for proving 6) we can use ) n S n n) Y i n 1=2 Z n 1=2 ) and the central limit theorem the details are left to the reader). These examples, however, are based on nonergodicity of the process X i ). The study of ergodic sequences is often a more important one, while the construction of an ergodic example will be much more complicated. In the proofs of Theorems 3.3 and 3.7 we shall use the following lemma whose proof is a straightforward application of Sinai s theorem. For the convenience of a reader not used to woring in ergodic theory we present a proof here; similarly, we shall present a proof of Lemma 3.9. Lemma 3.8. Let ; A; ; T) be an ergodic probability measure preserving dynamical system of positive entropy. There exist two T-invariant sub--algebras B and C of A and a function g on such that: the -algebras B and C are independent; the function g is B-measurable; taes values 1; 0 and 1; has zero mean and the process g T n ) is independent; the dynamical system ; C; ; T) is aperiodic. Proof. Let h be the entropy of ; A;;T). Let a 0; 1] be such that h := 1 a) log 2 1 a) a log 2 a=2) h; and let us consider the full shift on three letters 1; 0; 1) with the innite product measure a=2; 1 a; a=2) Z. This dynamical system will be denoted by S 1. It is a Bernoulli

9 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) measure preserving system of entropy h. Consider another Bernoulli measure preserving system of entropy smaller than h h, and denote it by S 2. The product system S 1 S 2 is Bernoulli and its entropy is smaller than h. Thus, by Sinai s Theorem see e.g. Shields, 1973), it is a factor of the system ; A;;T). So we have in this system a copy of S 1 and a copy of S 2 which are independent. This gives the -algebras B and C. The function g is obtained by lifting on the zero coordinate function of the shift S 1. ProofofTheorem 3.3. Let n ) 1 be an increasing sequence of integers satisfying some quic growth condition that will be specied in the sequel. We shall use the following result, which is a direct application of Rohlin s lemma. Lemma 3.9. Let ; C; ; T) be an aperiodic probability measure preserving dynamical system. Let n ) 1 be an increasing sequence of integers satisfying some quic growth condition lie 2 exp n 1=3 ) 1 2 exp n 1=3 ): 16n There exists a sequence A ) 1 of pairwise disjoint elements of C such that and A ) 2 exp n 1=3 ); 12) n T j A exp n 1=3 ): 13) Proofofthe Lemma. For each, by Rohlin Lemma, there exists a set B in C such that 3 2 exp n 1=3 ) B ) 1 2 exp n 1=3 ); 16n 4n and T 1 B ;T 2 B ;:::;T 4n B are pairwise disjoint. We x such sets B, and we consider C = 4n T j B. We have thus, C ) 2 exp n 1=3 ) and 4n j=n +1 n T j C exp n 1=3 ): n T j B T j C ; Finally, we dene A = C \ C ). By construction, the sets A are pairwise disjoint and they satisfy A ) 2 exp n 1=3 ). Moreover, since n n ) n T j C T j A T j C ;

10 152 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) we have n n T j A T j C n C ): Hence, n T j A exp n 1=3 ) n 2 exp n 1=3 ); which gives the nown result under the right growth condition on the sequence n ). Following Lemma 3.8 we consider two sub--algebras B and C and a function g, and apply Lemma 3.9. We dene f = g =1 and we claim that n 1=3 A ; Ee f ) ; 14) Ef T n F n 1 ) = 0 15) where F is the -algebra generated by f T i ;i6, and S n f) n ) exp cn 1=3 ) 16) for some constant c. From the denition of f and from 12) we get Ee f ) 6 1+ =1 expn 1=3 ) 2 exp n 1=3 ): which proves 14). Denote f = =1 n1=3 A. By the assumptions, the random variable g T n is independent of the -algebra generated by C and the functions g T i ;i n, hence, Ef T n C F n 1 )=f T n )Eg T n C F n 1 ) = 0. This implies 15). In order to prove 16) we dene We have f = gn 1=3 A ; f + = g n 1=3 j Aj and f j = g j n 1=3 j Aj : S n f) n ) S n f ) 2n ) S n f + ) 0) S n f ) n ):

11 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) From S n f ) n 1=3 n S n g) T j A we deduce, by using independence of the -algebras B and C, that n S n f ) 2n ) T j A S n g) 2n 2=3 ): We denote 2 as the variance of g. From 13) and Corollary 3.5 applied with x n =2n 1=6 and = 1), we obtain S n f ) 2n ) exp n1=3 ) exp 4 ) 2 n1=3 : 17) Let us x a positive number c 1+4= 2. Now, we impose two conditions on the growth of the numbers n : n 2n 2=3 cn 1=3 ; 18) 1 j=+1 j 2 exp n 1=3 j ) 1 n exp cn 1=3 ): 19) By noticing that f 6 n1=3 1 and using Azuma s inequality and 18), we obtain S n f ) n ) 6 2 exp n =2n 2=3 1 ) 6 2 exp cn1=3 ): 20) We have S n f + ) 0)6n f + 0)6n A j ) and thans to 19), this implies that j S n f + ) 0) exp cn1=3 ): 21) From 17), 20) and 21) we conclude that, for all large enough, S n f) n ) exp cn 1=3 ); which concludes the proof of 4). ProofofTheorem 3.7. As in the preceding proof, we use Lemma 3.8 and consider -algebras B; C and a function g. Let us x a sequence c ) 1 of positive numbers such that 1 c1=p. By the central limit theorem we now that there exists c 0 such that, for all large enough n, S n g) 2 n) c: 22) We consider an increasing sequence of integers n ) 1 satisfying the following growth conditions: 2 exp 1 2 n nj =oc n p=2 ); 23) j

12 154 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) j c j n p=2 j =oc n 1 p=2 ); 24) b n =oc ): Similarly, as in Lemma 3.9, the aperiodicity of ; C; ; T) guarantees the existence of sets A C such that Denote A ) c n p=2 f = gn 1=2 A ; f = f : =1 ; n ) T i A 1 2 A ): We have f p 6 c 1=p hence, f L p. The same argument as in the proof of Theorem 3.3 shows that f T i ) is a martingale dierence sequence. We have S n f ) n n S n g) T i A : The independence of C and process g T i ) implies that S n f ) 2n ) S n g) 2 n ) n ) T i A : Under condition 22) we conclude that S n f ) 2n ) c c 2n p=2 25) : 26) The sequence 1 gf j ) T i ) i 0 is a martingale dierence sequence and 1 gf j 6 1 nj. By Azuma s inequality we have 1 2 S n gf j n 6 2 exp n nj ; and, by 23), we conclude that 1 S n gf j n =oc n p=2 ): 27) We have S n gf j 0 6n A j )=oc n p=2 ); 28) by 24). j=+1 j=+1

13 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) From 26), 27) and 28) we deduce that lim inf n p=2 c S n f) n ) 0: As a result of 25), this implies that n p=2 S n f) n ) b n and this concludes the proof of 6). 4. Some extensions and applications The notion of a martingale dierence sequence can be generalized to multiparameter processes. One possibility is to use the lexicographic ordering: Let ; denote indices from Z d ;d 1; we shall write 6 if = 1 ;:::; d ) precedes = 1 ;:::; d )inthe lexicographic order, and similarly we dene.x ) Z d is then called a martingale dierence eld if EX X j : j ))=0 a:s: for all Z d. Another and more strict denition was used by Nahapetian and Petrosian 1992): X ) Z d is called a martingale dierence eld if for all Z d, EX X j : j ))=0 a:s: The next two theorems are valid for both denitions. Let d 1; X ) Z d a martingale dierence eld, and V n ) a sequence of subsets of Z d ; V n, where V n denotes the number of elements of V n. We denote S n := i V n X i : By a dynamical system we mean a probability space ; A;) with a measure preserving action T of Z d, i.e. for any ; Z d ;T ;T are measurable and measure preserving transformations of onto itself, and T T = T +. Theorem 4.1. Let X i ) be a martingale dierence eld such that Ee Xi 6 K for all i Z d. For every x; 0, for any large enough n, S n V n x) exp )x2=3 V n 1=3 ): 3 ) In every ergodic dynamical system with a Z d -action T of positive entropy there exists c 0 and a function f, Ee f, such that f T i ) is a martingale dierence eld and S n f) V n ) e c Vn 1=3 4 ) for innitely many n.

14 156 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) Theorem 4.2. Let X i ) be a martingale dierence eld where X i L p ; 2 6 p ; X i p M for all i Z d. Let x 0. Then, S n V n x) 6 18pq 1=2 ) p M p 1 x p V n p=2 5 ) where q is the real number for which 1=p +1=q =1. In every ergodic dynamical system with a Z d -action T of positive entropy and for every sequence b n ) of positive numbers approaching zero there exists f L 2 0 such that f T i ) is a martingale dierence eld and lim sup n V n p=2 b n S n f) V n )= : 6 ) The elements of V n written in a lexicographic order in the case of the Nahapetian and Petrosian s denition, in any order) form a martingale dierence sequence. Therefore, 3 ) and 5 ) are direct consequences of, respectively, 3) and 5). In order to prove 4 ) and 6 ), we only need the multidimensional version of the Rohlin Lemma cf. Conze, 1972 or Katznelson and Weiss, 1972) and we follow the proofs of 4) and 6). Details are left to the reader. Now, we explain how, via martingale approximation, Theorem 3.6 has applications to the study of large deviations for some classes of stationary processes. This type of martingale approximation rst appears in Gordin 1969), and the following theorem can be found in Volny 1993). Let ; A; ) be a probability space and T a one-to-one, bimeasurable and measure preserving transformation of this space. Let M be a sub--algebra such that M T 1 M. Let us denote M = T i M and M = T i M: i Z i z Let 1 6 p and f L p ) be such that f is M -measurable and Ef M ) = 0. Remar that if ; A;;T) is a K-system and if M is well chosen, then any f L p 0 ) satises these two conditions.) Theorem 4.3. The condition Ef T n M) and n=0 [f T n Ef T n M)] converge in L p 29) n=0 is equivalent to the existence of u; m L p such that m T i ) is a martingale dierence sequence and f = m + u u T: 30) In Volny 1993), the proof is formulated for p =1; 2 but in fact it wors for all p.) From Theorems 3.6 and 4.3 we deduce the following result. Corollary 4.4. Let 1 6 p. Under condition 29) or 30); we have ) 1 S n f) n)=o : n p=2

15 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) ProofofCorollary 4.4. We start from the decomposition f = m + u u T given by Theorem 4.3. From the integrability of u p, we deduce that lim x x u p x)=0; hence ) 1 u n=3)=o n p : Theorem 3.6 gives us 1 S n m) n=3)=o n p=2 ) : But we have S n f) 6 S n m) + u T n + u, hence S n f) n) 6 S n m) n=3)+2 u n=3): This concludes the proof of Corollary 4.4. Let h T i ) be a stationary sequence of martingale dierences, a i real numbers, i= a2 i. The process f T = X = i= a ih T i is a stationary linear process. Corollary 4.5. Suppose that h L p ;p 2; and that i= i a i. Then ) 1 S n f) n)=o : n p=2 Proof. Let T i M) be the ltration of h T i ), i.e. h = Eh M) Eh TM), M T 1 M. For g L p, we dene Ug = g T, P i g = Eg T i M) Eg T i+1 M). Notice that UP i g = P i+1 Ug, hence P i+1 Ug p = P i g p. We have P i f = a i h T i Ef M)= i=0 P if, and f Ef M)= P if. Therefore, Ef T n M) p + f T n Ef T n M) p ) n=1 ) 6 P i U n f p + P i U n f p 6 6 n=1 i=0 P i f p + n=1 i=n ) P i f p i=n+1 i a i h p : i= From 29) and Corollary 4.4 we now get the result. As another special case let us consider an -mixing sequence f T i ), f L p, p 1 for the denition, see e.g. Ibragimov and Linni 1971) or Hall and Heyde 1980)). Here, we do not intend to give optimal estimations of large deviations for stationary

16 158 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) mixing sequences, but only to show a direct application of our previous estimates. We shall denote M = f T i : i 6 ). Corollary 4.6. Let f T i ) be an -mixing sequence; Ef=0. If f L p+h for some h 0 and [)] 1=p 1=p+h) ; then =1 ) 1 S n f) n)=o n p=2 : Proof. Dene M = f T i : i 6 0). For all n 0, f T n = Ef T n M) and by McLeish 1975), for h 0 we have Ef M ) p =p +1) f p+h [)] 1=p 1=p+h) : This implies 29) and the result follows by Theorem 4.3. For more results on large deviations for -mixing sequences the reader can consult e.g. Rio 2000a) or Douhan 1994). For -mixing random variables with exponential moments and exponential decay of n) one can get better estimates than for a uniformly bounded martingale dierence sequence cf. Rio, 2000b). We can also get estimations of large deviations for -mixing sequences but in that case, using other methods, one can get almost the same results as for independent sequences cf. Peligrad, 1989). 5. Uncited References Yoshihara, References Azuma, K., Weighted sums of certain dependent random variables. Tôhou Math. J. 19, Baum, L.E., Katz, M., Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, Chow, Y.S., Teicher, H., Probability Theory: Independence, Interchangeability, Martingales. Springer Texts in Statistics. Springer, Berlin. Conze, J.-P., Entropie d un groupe abelien de transformations. Z. Wahrscheinlicheitstheorie Verw. Gebiete 25, Cramer, H., Sur un nouveau theoreme-limite de la theorie des probabilites. Actualites Sci. Indust. Hermann, Paris) 736, de la Peña, V.H., A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27, Douhan, P., Mixing: Properties and Examples. Lecture Notes in Statistics, Vol. 85. Springer, Berlin. Durrett, R., Probability: Theory and Examples. Wadsworth and Broos=Cole, Pacic Grove, CA. Erdos, P., 1949,1950. On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 21, , 138. Feller, W., An Introduction to Probability Theory and its Applications, Vol. II, 2nd Edition. Wiley, New Yor.

17 E. Lesigne, D. Volny / Stochastic Processes and their Applications ) Gordin, M.I., The central limit theorem for stationary processes. Soviet Math. Dol. 10, Hall, P., Heyde, C.C., Martingale Limit Theory and its Applications. Academic Press, New Yor. Hsu, P.L., Robbins, H., Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33, Ibragimov, I.A., Linni, Y.V., Independent and Stationary Sequences of Random Variables. Wolters-Noordho, Groningen. Katznelson, I., Weiss, B., Commuting measure-preserving transformations. Israel J. Math. 12, Laib, N., Exponential-type inequalities for martingale dierence sequences. Application to nonparametric regression estimation. Commun. Statist.-Theory. Methods 28, Lesigne, E., Volny, D., Large deviations for generic stationary processes. Coll. Math. 84=85, McLeish, D.L., A maximal inequality and dependent strong laws. Ann. Probab. 3, Nagaev, S.V., Some limit theorems for large deviations. Theory Probab. Appl. 10, Nahapetian, B., Petrosian, A.N., Martingale-dierence Gibbs random elds and central limit theorem. Ann. Acad. Sci. Fenn., Series A-I Math. 17, Peligrad, M., The r-quic version of the strong law for stationary -mixing sequences. Almost Everywhere Convergence, Proceedings, Columbus, OH, Academic Press, Boston, MA. pp Petrov, V.V., Limit Theorems of Probability Theory. Oxford Science Public., Oxford. Rio, E., 2000a. Theorie Asymptotique des Processus Aleatoires Faiblement Dependants. Mathematiques & Applications, Vol. 31. Springer, Berlin. Rio, E., 2000b. Personal communication. Shields, P., The Theory of Bernoulli Shifts. University of Chicago Press, Chicago. Spitzer, F., A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82, Volny, D., Approximating martingales and the central limit theorem for strictly stationary processes. Stochastic Processes Appl. 44, Yoshihara, K., Summation Theory for Wealy Dependent Sequences. Sanseido, Toyo.