The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-090 Wien, Austria Large Deviations for Independent, Stationary, and Martingale Dierence Sequences Emmanuel Lesigne Dalibor Volny Vienna, Preprint ESI 667 (999) February 9, 999 Supported by Federal Ministry of Science and Transport, Austria Available via http//

2 LARGE DEVIATIONS FOR INDEPENDENT, STATIONARY, AND MARTINGALE DIFFERENCE SEQUENCES Emmanuel Lesigne, Dalibor Volny 3 February, 999 P n Abstract. Let ( i ) be a martingale dierence sequence and S n = i= i. We prove that if sup i E(e jij ) < then there exists c > 0 such that (S n > n) e?cn=3 ; this bound is optimal for the class of martingale dierence sequences which are also strictly stationary and ergodic. If the sequence ( i ) is bounded in L p, 2 p <, we get the estimation (S n > n) c n?p=2 which is again optimal for strictly stationary and ergodic sequences of martingale dierences. These results are compared with those for iid sequences; we give a simple proof that the estimate of Nagaev, Baum and Katz, (S n > n) = o(n?p ) for i 2 L p, p <, cannot be improved and that, reciprocally, it implies the integrability of j i j p? for all > 0. For an ergodic automorphism T of the probability space (; A; ) we show that there exists a dense set of f 2 L 0 for which the probabilities of (S n > n) go to zero arbitrarily slowly when n tends to innity (here i = f T i ). We study the asymptotic behaviour of (S n > n) for a generic f 2 L p 0.. Introduction. Let ( i ) be a sequence of real integrable random P variables with n zero means, dened on a probability space (; A; ) and S n = i= i. We shall study the asymptotic behaviour of the probabilities () (S n > nx); x > 0; n! The classical Cramer's theorem (cf. [C]) states that for independent and identically distributed (iid) random variables with nite exponential bounds the probabilities () are exponentially small. More precisely, Theorem. (Cramer). Let ( i ) be a sequence of iid random variables, E i = 0. There is equivalence between (i) There exists c > 0 such that Ee cj j <. (ii) For every x > 0 there exists c x > 0 such that (js n j > nx) = o(e?c xn ).? ( S n If the variables i are bounded, the estimates remain of the same type. Under suitable conditions, the convergence p n < x) =? (x)! where x = n and is a normal distribution function, holds. A survey of such results for independent random variables can be found e.g. in [P]. A nice result of the same type for bounded martingale dierence sequences has been given by Rackauskas ([Ra]). Typeset by AMS-TE

3 In this paper, we study the asymptotic behaviour of () with x constant. We are mostly concerned in the case when ( i ) is a sequence of martingale dierences, and in the case when ( i ) is a strictly stationary and ergodic sequence. Remark that we do not use the rate function to express this behaviour (cf., e.g. [Dem-Z]). It is well known (cf. e.g. [Ha-He]) that for strictly stationary and ergodic martingale dierence sequences classical limit theorems like 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 () is given by Azuma's theorem ([A]). The case of unbounded martingale dierence sequences has remained up to now relatively neglected. Large deviations estimates for martingale dierence sequences are studied in the third section. We give new results under nite exponential moments or nite p-th moments ( p < ) hypothesis. In order to give a comparison we recall in the second section a classical result of Nagaev, Baum and Katz for iid random variables with nite p-th moments; we prove that their estimate cannot be substantially improved and that the asymptotic behaviour of probabilities () characterizes the integrability of j i j p?. In the third section, we also give large deviations estimates for stationary mixing sequences, which follow from martingale approximation. A strictly stationary and ergodic sequence of random variables ( i ) can always be represented under the form i = f T i where T is an ergodic bimeasurable and measure preserving automorphism of a probability space (; A; ), and f is a measurable function on. If f 2 L p (), then classical ergodic theorems of von P n? Neumann and Birkho say that the averages (=n)s n (f), S n (f) = f T i i=0, converge in L p and almost surely to E(f). U. Krengel ([Kr]) showed that the convergence can be arbitrarily slow and A. del Junco with J. Rosenblatt ([dj-ro]) showed that this event is generic. R. Burton and M. Denker ([Bu-Den], see also [V3]) proved that for a dense set of f 2 L 2 0 the central limit theorem holds; in [V], D. Volny showed that for any admissible sequence a n! there is a generic set of f 2 L p 0 such that the distributions of (=a n)s n (f) converge along subsequences to all probability laws. (We denote by L p 0 the space of zero mean functions in Lp ().) In the fourth section we deal with large deviations for ergodic sequences (f T i ) where f belongs to a generic subset of L p 0, p. 2

4 2. Independent random variables. Let ( i ) be a sequence of iid (independent identically distributed) random variables, E i = 0. We denote S n = n. By the weak law of large numbers we have, for any x > 0, lim n! (js nj > nx) = 0 The speed of convergence has been studied by many authors. The classical result concerning the case when random variables i have nite exponential moments has been recalled in Introduction. Here we shall consider the case when the random variables have a nite moment of order p. Theorem 2. (Nagaev, Baum and Katz). Assume E(j p j) < where p. Then (js n j > nx) = o(n?p ) This theorem can be found in [N], [Ba-Ka] and in the complements in Chapter 4 in [P] or in Chapter 2 of [Re]. Conversly, the speed of the convergence of (js n j > nx) characterizes the moments of i Theorem 2.2. Let p >. If there exists x > 0 such that (js n j > nx) = O(n?p ), then 8 > 0; E(j j p? ) < Theorem 2.2 is a strengthening of a result of P. Revesz who proved that if p > 2 and (js n j > nx) = O(n?p ) for some x > 0, then E(j j p?? ) < for all > 0. We shall also show that the estimate given in Theorem 2. cannot be essentially improved Theorem 2.3. Let p and (c n ) be a real positive sequence going to zero. For any increasing sequence of integers (n k ) with P k= c n k < ; there exists a probability law on the real line such that for iid random variables i with the distribution we have E(j i j p ) <, E i = 0 and as k!. n p? k c nk (js nk j > n k )! In the proofs of Theorem 2.2 and Theorem 2.3 we shall use the following lemma which occurs as an exercise in [Du]. Lemma 2.4. Let ( i ) be a sequence of integrable iid random variables with E i = 0. Then lim inf n! (js n j > n) n(j j > 2n) Proof. Let us denote A k = f k > 2n and S n? k >?ng. We have (S n > n) n k= n [ k= (A k )? A k jkn (A j \ A k ) = n(a )? 3 n(n? ) (A \ A 2 ) 2

5 We denote by a random variable distributed as the i 's. >From and (A ) = ( > 2n) (S n? >?n) ( > 2n) (js n? j < n) (A \ A 2 ) ( > 2n and 2 > 2n) = (( > 2n)) 2 ((jj > 2n)) 2 we deduce (js n j > n) n(jj > 2n)P (js n? j < n)? n 2 ((jj > 2n)) 2 The statement of the Lemma now follows from the last inequality, and lim n! (js n?j < n) = lim n(jj > 2n) = 0 n! Proof of Theorem 2.2. Let x > 0 be the x from the statement of the Theorem. By Lemma 2.4 we then have Therefore, for any > 0, n (jj > 2nx) = O(n?p ) n p?? P (jj > 2nx) < ; hence E(jj p? ) <. The proof shows that the the claim E(jj p? ) < can be strengthened; we can, e.g. get E(jj p =(log + jj) + ) <, where log + x is dened as log(max(x; e)). Proof of Theorem 2.3. By Lemma 2.4 it suces to prove the existence of a probability law with zero mean and nite p-th moment, for which lim k! c nk n p k ([2n k; +[) = Let ( k ) be a sequence of positive numbers such that c k! nk 0 and < k and dene k a = k c nk k n p k ; a k = c n k 2a k n p ; k = k a k ( 2nk +?2nk ) ( x denotes the Dirac measure P at point x). By its denition, is a symmetric probability law and, since k a kn p k <, has a nite p-th moment. But we have n p k c ([2n k; +[) > n p k nk c a k?! nk 4

6 3. Martingale dierence sequences. By ( i ) we denote a (not necessarily stationary) martingale dierence sequence and we set S n = n i= We study the speed of convergence of (S n nx) to zero in three dierent cases. The case when variables i are uniformly bounded is well known 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 i (S n nx) exp(?cn =3 ) In the case of nite p-th moments, we obtain also a new estimate of the type (S n nx) cn?p=2 In these three cases, we show that the 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 ( i ) = (f T i ) where T is a measurable and measure preserving transformation of onto itself. Theorem 3. (Azuma [A]). Let ( i ) be a sequence of martingale dierences. If j i j < a < for all i, we have (2) (S n nx) e?nx2 =2a 2 ; n = ; 2; For the reader's convenience, Azuma's proof of this useful inequality is recalled at the end of this section. The fact that this estimation is optimal is already known for iid sequences. Theorem 3.2. Let ( i ) be a martingale dierence sequence such that Ee j ij K < for all i. For every x; > 0, for any large enough n,? 2 (? )x2=3 n =3 (3) (js n j > nx) < exp In every ergodic dynamical system of positive entropy there exists c > 0 and a function f, Ee jfj <, such that (f T i ) is a martingale dierence sequence and (4) (js n (f)j > n) > e?cn=3 for innitely many n (S n (f) = P n i= f T i ). The proof of the rst part of this theorem is based on Azuma's inequality and truncation arguments. In this proof we shall give a precise and non asymptotic upper bound for (js n j > nx). In the proof of the second part we use an estimation of \moderate variation" for sums of iid random variables which is due to Cramer. We give this estimation as it appears in Feller's book ([F], second edition, volume II, VI.7). 5

7 Theorem 3.3 (Cramer). Let (Y i ) be sequence of iid random variables, with EY i = 0 and E(e P ty i ) < for t real in a neighbourhood of zero. Denote 2 = E(Yi 2 ), and n R n = Y i= 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( p n), then p (R n > x n n) = Z + p 2 x n exp? u2 2 2 du exp x 3 pn n xn xn pn + O pn Corollary 3.4. Under the hypotheses of Theorem 3.3, for any > 0, for all large enough n, exp?? 2 ( + p 2 )x2 n < R n > x n n < exp? 2 (? 2 )x2 n Let us consider now the case of martingale dierence sequences with nite moment of order p. Theorem 3.5. Let ( i ) be a martingale dierence sequence where i 2 L p, 2 p <, k i k p < M < for all i. Let x > 0. Then (5) (js n j > nx) (8pq =2 ) p M p x p n p=2 where q is the real number for which =p + =q =. In every ergodic dynamical system (; A; ; T ) of positive entropy and for every sequence (b n ) of positive numbers going to zero there exists f 2 L 2 0 such that (f T i ) is a martingale dierence sequence and (6) lim sup n! n p=2 b n (js n (f)j > n) = The proof of the rst part of this theorem is based on classical norm estimates for martingale dierence sequences. In the proof of the second part we use the Central Limit Theorem for iid sequences and Azuma's inequality. Via the so-called Gordin's decomposition, Theorem 3.5 have wide applications to the study of speed in Ergodic Theorem (see Corollary 3.8 below). Remark. Theorem 2.3 shows that for p = there exist iid sequences for which the convergence (js n (f)j > n)! 0 can be arbitrarily slow (from the point of view of subsequences). We rst prove (3) and (5). argument. In the proofs of (4) and (6) we use a common Proof of (3). For a martingale dierence sequence ( i ) with j i j, the Azuma inequality gives, for any x > 0, (7) (js n j > nx) 2 exp(?nx 2 =2) 6

8 Let ( i ) be a martingale dierence sequence satisfying the assumptions of Theorem 3.2 and denote by (F i ) its ltration. Let us x a > 0. For n = ; 2; and i n dene Y n;i = i (ji jan =3 )? E( i (ji jan =3 )jf i? ) ; Z n;i = i (ji j>an =3 )? E( i (ji j>an =3 )jf i? ) ; S 0 n = S 00 n = n n i= i= Y n;i ; Z n;i (Y n;i ) and (Z n;i ) are martingale dierence arrays and, because ( i ) is a martingale dierence sequence, i = Y n;i + Z n;i ( i n). Let us x t 2 (0; ). For every x > 0, (8) (js n j > nx) (js 0 nj > nxt) + (js 00 nj > nx(? t)) We have jy n;i j 2an =3 for i n, hence by using (7) we get (9) (jsnj 0 js 0 > nxt) = n j 2an > nxt 2 exp? t2 x 2 =3 2an =3 8a 2 n=3 Let F i (x) = (j i j > x). >From Ee j ij K it follows that F i (x) Ke?x for all x 0. Then? EZn;i 2 = E ( i (ji j>an )) 2?? =3 E (E( i (ji jan )jf =3 i? )) 2? E ( i (ji j>an )) 2 Z =? =3 x 2 df i (x) = (an =3 ;+) Z!? lim 2xF i (x) dx (an =3 ;M ] xe?x dx = K a 2 n 2=3 + 2an =3 + 2 e?an=3 an =3 M! M 2 F i (M)? a 2 n 2=3 F i (an =3 )? Ka 2 n 2=3 e?an=3 + 2K Z Thus we have E(Sn) 00 2? nk a 2 n 2=3 + 2an =3 + 2 e?an=3 and hence (0) (js 00 nj > nx(? t)) n 2 x 2 (? t) nk a 2 n 2=3 + 2an = e?an=3 = K a x 2 (? 2 n?=3 + 2an?2=3 + 2n? e?an=3 t) 2 In view of (9) and (0) we choose a so that t2 x 2 8a 2 () (js 00 nj > nx(? t)) K 4 x 2 (? t) 2 (tx) 4=3 n?=3 + (tx) 2=3 n?2=3 + 2n? exp 7 = a = 2 (tx)2=3. We obtain? 2 (tx)2=3 n =3

9 >From (8), (9), and () we deduce that for every x > 0 and every t 2 (0; ), (js n j > nx) < 2 + K (? t) ( 2 4 t4=3 x?2=3 n?=3 + t 2=3 x?4=3 n?2=3 + 2x?2 n? ) e? 2 t2=3 x 2=3 n =3 We can choose the parameter t arbitrarily close to, and (3) follows. Proof of (5). By Burkholder's inequality (cf. [Ha-He], Theorem 2.0), we have E i p (8pq =2 ) p E n? i=0 Set Y i = 2 i. By a convexity inequality we have therefore It follows E n? i=0 Y i (!) n? 2 p n? i n? p (8pq =2 ) p n p=2? n? i=0 (S n > nx) Z js n j>nx i=0 i=0 n? i 2 i=0 p=2 2=p Y p=2 i (!) Ej i j p n p=2 (8pq =2 ) p M p js n j p x p n p d (8pq=2 ) p M p x p n p=2 This nishes the proof of (5). Remark that for p = 2 we can use, instead of Burkholder's inequality, the only orthogonality of i. In the proof of (4) and (6) we shall use the following lemma. Lemma 3.6. 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, takes values -, 0 and, 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 2 (0; ] be such that h 0 =?(? a) log 2 (? a)? a log 2 (a=2) < h ; and let us consider the full shift on three letters (?; 0; ) with the innite product measure (a=2;? a; a=2) Z. This dynamical system will be denoted by S. It is a Bernoulli measure preserving system of entropy h 0. Consider another Bernoulli 8

10 measure preserving system of entropy smaller than h?h 0, and denote it by S 2. The product system S S 2 is Bernoulli and its entropy is smaller than h. Thus, by Sinai's Theorem (see e.g. [Sh]), it is a factor of the system (; A; ; T ). So we have in this system a copy of S 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. Proof of (4). Let (n k ) k be an increasing sequence of integers satisfying some quick growth condition that will be specied in the sequel. Following Lemma 3.6 we consider two sub--algebras B and C and a function g. The fact that the dynamical system (; C; ; T ) is aperiodic guarantees the existence of a sequence (A k ) k in C such that the A k 's are two by two disjoint, (2) (A k ) < k?2 exp(?n =3 k ); and (3) n k \ j= T?j (A k ) > 2 k?2 exp(?n =3 k ) (It is always possible to construct such A 0 ks, using for example thin vertical slices in some Rokhlin tower.) We dene and we claim that f = g k= n =3 k A k ; (4) E(e jfj ) < ; (5) E f T n jf n? = 0 where F k is the -algebra generated by f T i, i k, (6) (js nk (f)j > n k ) > exp(?cn =3 k ) for some constant c. >From the denition of f and from (2) we get E(e jfj ) k= exp(n =3 k )k?2 exp(?n =3 k ) which proves (4). 9

11 Denote f = P k= n=3 k A k. 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 E(f T n jc _ F n? ) = ( f T n ) E(g T n jc _ F n? ) = 0. This implies (5). In order to prove (6) we dene f k = gn =3 k A k ; f + k = g j>k n =3 j Aj and f k? = g n =3 j Aj j<k We have (js nk (f)j > n k ) (js nk (f k )j > 3n k )? (js nk (f + k )j > n k)? (js nk (f? k )j > n k) From js nk (f k )j n =3 k js n k (g)j Yn k j= T?j A k we deduce, by using independence of the -algebras B and C, that (js nk (f k )j > 3n k )? \ n k j= T?j A k jsnk (g)j > 3n 2=3 k We denote by 2 the variance of g. From (3) and Corollary 3.4 (applied with x n = 3n =6 and = ), we get (7) (js nk (f k )j > 3n k ) 2k 2 exp(?n=3 k ) exp? 9 2 n=3 k Let us x a positive number c > + 9= 2. Now, we impose two conditions on the growth of the numbers n k (8) (9) j=k+ n k 2n 2=3 k? > cn =3 k ; n =3 j j?2 exp(?n =3 j ) < exp(?cn =3 k ) By noticing that jf? k j n =3 k? and using Azuma's inequality and (8), we obtain (20) (js nk (f? k )j > n k) 2 exp(?n k =2n 2=3 k? ) 2 exp(?cn=3 k ) We have EjS nk (f + k )j n kejf + k j = n k and thanks to (9), this implies that j=k+ n =3 j j?2 exp(?n =3 j ) (2) (js nk (f + k )j > n k) EjS n k (f + k )j n k < exp(?cn =3 k ) 0

12 >From (7), (20) and (2) we conclude that, for all large enough k, which concludes the proof of (4). (js nk (f)j > n k ) exp(?cn =3 k ) ; Proof of (6). As in the preceding proof, we use Lemma 3.6 and consider -algebras B, C and a function g. Let us x a sequence (c k ) k of positive numbers such that Pk c=p k <. By the Central Limit Theorem we know that there exists c > 0 such that, for all large enough n, (22)? js n (g)j > 2 p n > c We consider an increasing sequence of integers (n k ) k satisfying the following growth conditions (23) exp? 2 n k j<k p nj?2 = o(c k n?p=2 k ) ; (24) j>k c j n?p=2 j = o(c k n??p=2 k ) ; (25) b nk = o(c k ) By the Rokhlin lemma there exists a sequence (F k ) of elements of C, such that, for each k, the sets F k ; T? F k ; ; T?2n k+ F k are two by two disjoints and (F k ) = c k 2n +p=2 k Denote A k = 2n [ k i= T?i F k ; f k = gn =2 k A k ; f = k= f k We have kf k k p c =p k hence f 2 L p. The same argument as in the proof of (4) shows that (f T i ) is a martingale dierence sequence. We have? \ n k i= T?i A k 2 (A k) = c k 2n p=2 k

13 and js nk (f k )j p n k js nk (g)j Yn k i= The independence of C and process (g T i ) implies T?i A k (js nk (f k )j > 2n k ) (js nk (g)j > 2 p n k )? \ n k i= T?i A k Under condition (22) we conclude that (26) (js nk (f k )j > 2n k ) c c k 2n p=2 k The sequence (( P k? j= gf j) T i ) i0 is a martingale dierence sequence and j P k? gf j= jj P k? p j= nj. By Azuma's inequality we have Snk k? k? j= and, by (23), we conclude that (27) We have (28) S nk gf j nk exp? 2 n k j= j= Snk k? gf j nk = o(c k n?p=2 k ) j=k+ gf j 6= 0 nk by (24). >From (26), (27) and (28) we deduce that lim inf k! Because of (25), this implies and this concludes the proof of (6). j=k+ n p=2 k c k (js nk (f)j > n k ) > 0 n p=2 k b nk (js nk (f)j > n k )! p nj?2 ; (A j ) = o(c k n?p=2 k ) ; We explain now how, via martingale approximation, Theorem 3.5 have applications to the study of large deviations for some classes of stationary processes. This type of martingale approximation rst appears in [G], and the following theorem can be found in [V2]. Let (; T ; ) 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? M. Let us denote M _ = T i M i2z T i M and M? = \ i2z Let p and f 2 L p () be such that f is M -measurable and E(fjM? ) = 0. (Remark that if (; T ; ; T ) is a K-system and if M is well chosen, then any f 2 L p 0 () satises these two conditions.) 2

14 Theorem 3.7. The condition (29) n=0 E(f T n jm) and n=0 [f T?n? E(f T?n jm)] converge in L p is equivalent to the existence of u; m 2 L p such that (m T i ) is a martingale dierence sequence and f = m + u? u T (In [V2], the proof is formulated for p = ; 2 but in fact it works for all p.) >From Theorem 3.5 and Theorem 3.7 we deduce the following result. Corollary 3.8. Let p <. Under condition (29), we have (js n (f)j > n) = O n p=2 Proof of Corollary 3.8. We start from the decomposition f = m + u? u T given by Theorem 3.7. >From the integrability of juj p we deduce that hence lim x! x(jujp > x) = 0 ; (juj > n=3) = o n p Theorem 3.5 gives us (js n (m)j > n=3) = O n p=2 But we have js n (f)j js n (m)j + juj + ju T n j, hence (js n (f)j > n) (js n (m)j > n=3) + 2(juj > n=3) This concludes the proof of Corollary 3.8. Remark. As a special case let us consider a -mixing or an -mixing sequence (f T i ), f 2 L p, p > (for the denition see e.g. [I-L] or [Ha-He] or [Y]). We shall denote M k = (f T i i k) and M k = (f T i i k). Let (ft i ) be -mixing. By [I-L] (cf. also [Ha-He] or [Y]), for q > 0, =p+=q =, g 2 L p 0 measurable w.r.t. M 0, and h 2 L q 0, measurable w.r.t. Mk, k, we have Notice that q = p Ejghj 2(k) =p kgk p khk q p? hence jfjp? 2 L q. Thus, for k we have ke(fjm?k )k p p = EjE(fjM?k )j p = E E(fjM?k ) f p? 3 2(k) =p ke(fjm?k )k p kf p? k q

15 hence Therefore, if f 2 L p and ke(fjm?k )k p C(f)(k) k= (k) p(p?) < ; p(p?) then (29) is guaranteed with M = M 0 and consequently (js n (f)j > n) = O n p=2 Let (f T i ) be an -mixing sequence. By e.g. [Hall-Heyde, Corollary A.2] or [Y] for q > 0, =p + =q <, there exists a c > 0 such that for g 2 L p 0 measurable w.r.t. M 0, and h 2 L q 0, measurable w.r.t. Mk, k, we have Ejghj c(k)? p? q kgk p khk q Suppose that f 2 L p+h, p 2, h > 0, and q = p+h. We then have p? ke(fjm?k )k p p = EjE(fjM?k )j p c ke(fjm?k )k p kjfj p? k q [(k)] h(p?) p(p+h) (note that h(p?) =?? p(p+h) p q ), hence ke(fjm?k )k p C(f)[(k)] Therefore, if f 2 L p+h, p 2 and h > 0, k= h [(k)] p(p+h) < h p(p+h) guarantees (29), hence (js n (f)j > n) = O n p=2 Two comments on related results.. In [V4] a nonstationary version of Theorem 3.7 is shown. As a consequence one can prove the preceding estimates for or mixing sequences ( i ) where the assumption of stationarity is replaced by uniform boundedness of the L p norms. 2. Let us notice that for any -mixing stationary sequence (f T i ) of random variables (with an arbitrarily slow decay of (k)) where f is integrable and bounded from above, an exponential bound for the probabilities (S n (f) > nx) was found by R. H. Schonmann [Sc] (cf. also [Br]). W. Bryc and A. Dembo [Br-Dem] proved the Varadhan's version of the large deviation principle for and -mixing sequences with \hyperexponential" decay of (k) and (k); a counter-example shows that an exponential decay is not sucient. In [Br-Dem], one can also nd more references on large deviations for mixing processes. Now we recall the proof of Azuma's inequality. 4

16 Proof of Theorem 3.. Let ( n ) be a martingale dierence sequence uniformly bounded by. If t 2 (?; +) et x 2 [?; ], we have tx = + x 2 t +? x (?t) 2 and, by convexity of the exponential function, this gives e tx cosh t + x sinh t Therefore E? e ts n E n Y k=(cosh t + k sinh t)! We develope the product, use the martingale property and naly observe that all the terms vanish except one and But cosh t e t2 =2 so we have Now, for any x > 0, P (S n nx) E E? e ts n (cosh t) n E? e ts n e nt 2 =2? e x(s n?nx) = e?nx2 E e n xs e?nx 2 e nx2 =2 = e?nx2 =2 4. Genericity in dynamical systems. Let (; A; ) be a Lebesgue probability space and T a bimeasurable and measure preserving one-to-one transformation of. We shall suppose that the dynamical system (; A; ; T ) is ergodic and aperiodic. For a measurable function f, the sequence (f T i ) is a strictly stationary process. We set S n (f) = n? i=0 f T i ; n = ; 2; If f is integrable and has zero mean, then, by the Ergodic Theorem, we know that n S n(f)! 0. For an arbitrary sequence of real numbers (b n ) going to innity, we shall study the behaviour of the sequence? b n (S n (f) > n). We shall see that, in a great variety of functional spaces, for a generic f, the sequence? b n (S n (f) > n) does not go to zero; and we shall see that this sequence goes to innity for a wide class of functions f. In Theorems 4. and 4.2 we recall some known results on a related problem, namely the behaviour of (S n (f) > nc n ) where c n! 0, for a generic function f. 5

17 Theorem 4.. Let (c n ) n>0 be a sequence of real numbers, c n! 0. Then for every p there exists a dense G set G L p 0 such that for every f 2 G lim sup (js n (f)j > nc n ) = n! Theorem 4. is well known and can be proved by the following argument denote by H n the set of functions f in L p 0 for which there exists k > n such that (js k(f)j > kc k ) >? n ; this set H n is open in L p 0, and by using Rokhlin lemma, it can be shown that this set is dense; the intersection of these sets is a dense G satisfying the announced property. Next theorem, from [V], is a renement of Theorem 4.. Theorem 4.2. Let c n! 0, nc n!. a dense G set G L p 0 Then for every p there exists such that for every f 2 G and every probability measure on the real line there exists a sequence n k! such that the distributions of (=n k c nk )S nk (f) weakly converge to. Let us now come back to our original question. P Theorem 4.3. Let (a n ) be a sequence of positive real numbers such that a n =. Let be a strictly positive continuous increasing function on ]0; ]. For every p < there exists a dense G set G L p 0 such that for every f 2 G This result implies n= a n ((S n (f) > n)) = Theorem 4.4. Let (b n ) n>0 be a sequence of positive real numbers, b n!. For every p < there exists a dense G set G L p 0 such that for every f 2 G lim sup n! b n (S n (f) > n) = Proof of Theorem 4.4. If b n!, there exists an increasing function satisfying the hypotheses of Theorem 4.3 and such that n= p bn < We choose a n =, n = ; 2;. By Theorem 4.3 there exists a dense G set G L p 0 such that for every f 2 G n= If f 2 G then, innitely often, (S n (f) > n) = ((S n (f) > n) > p bn 6

18 Lemma 4.5. For any real 2]0; [ 4 and any integer k > 0 there exists h 2 L 0 such that (S j (h) 2j; j k) and Z jhj p d 2 p+2 Proof of Lemma 4.5. We consider a Rokhlin tower A; T? A; ; T?2k+ A and a measurable subset B disjoint from the tower such that (A) = =k and (B) = 2. We consider the function h equal to 2 on the tower, to?2 on B and 0 everywhere else. The event fs j (h) 2j; j kg is contained in the union of the T?j A for k j < 2k. Proof of Theorem 4.3. There exists a sequence ( n ) of positive numbers such that n! 0 and By H n we denote the set of all f 2 L p 0 n= k? a j (S j (f) > j) > j= a n n = for which there exists a k > n such that k j= a j j Each set H n is open in L p 0. We shall prove that it is dense. The intersection of all the H n will be the set G satisfying the property described in the Theorem. The functions g? g T with g 2 L form a dense subset of L p 0. Let us show that H n intersects each neighbourhood of any of them. We consider an integer n, a bounded measurable function g and a positive real number. We x an integer ` such that ` > 2kgk and j > ` =) j < 2 () and an integer k such that k > n ; ` and ` a j j < k j= j=`+ a j j By Lemma 4.5 we know that there exists a function h such that Z jhj p d 2 p+2 ; and (S j (h) 2j) for all j k Let For j > ` we have f = h + g? g T? (S j (f) j)? (S j (h) 2j) () 7

19 Therefore k? a j (S j (f) j) j= k j=`+ a j () > 2 k j=`+ a j j > k j= a j j Thus f 2 H n and kf? (g? g T )k p = khk p (2 p+2 ) =p. It is an easy observation that Theorem 4.4 cannot be extended to L 0. However the two next theorems give results of the same type for bounded functions. Remark. In the proof of Theorem 4.3 we used the fact that in the spaces L p 0, p <, the coboundaries g? g T with g 2 L form a dense subset. This, however, is not true for p = in any ergodic and aperiodic dynamical system and for each p > 0 there exists a function f 2 L p 0 and > 0 such that for each coboundary g? g T 2 L 0 with g 2 L p we have kf? (g? g T )k > (cf., e.g., [K] or [V-W]). In [V-W] it is shown that for each p <, the closure (in L 0 ) of the set of all g? g T 2 L 0, g 2 L p, can be characterized by the rate of decay of the probabilities of (js n (f)j > nc). Let B be the set of real measurable functions f bounded by one and of zero mean, endowed with the distance d(f; g) = inffa > 0j(jf? gj > a) < ag The metric space (B; d) is complete. Let M be the set of elements of B taking only values? and. This set is closed in (B; d). Theorem 4.6. Let (b n ) n>0 be a sequence of positive real numbers, b n!. For every c <, in the space (B; d) there is a dense G set of functions f satisfying lim supb n (js n (f)j > cn) = n! In the space (M; d) there is a dense G set of functions f satisfying lim sup b n (js n (f)j = n) = n! The proof of Theorem 4.6 can be done by using the ideas of Theorem 4. or [V] and is left to the reader. Theorem 4.7. Let (b n ) n>0 be a sequence of positive real numbers, b n!. In the Banach space L 0, there exists a dense G subset G such that for every f 2 G (30) lim sup n! b n(js n (f)j > cn) = holds for some c > 0. Proof of Theorem 4.7. For c > 0 denote V c = ff lim sup n! b n(js n (f)j > c) = g ; V c;n = ff 9k n; b k (js k (f)j > c) > ng 8

20 We have V c = T V c;n and, every f 2 S n= `= S Each set V c;n is open in L 0. Therefore Let f 2 L 0 V =` satises (30). `= V =` is G. and > 0. Theorem 4.6 guarantees the existence of a function g in V 2 bounded by 3. If f =2 V, we have f + g 2 V. This proves that V is 3-dense S in L 0. Therefore V =` is dense. `= We now present the \liminf" type result. Theorem 4.8. Let (b n ) n>0 be a sequence of realr numbers, b n!. There exists a bounded measurable function f on such that f d = 0 and lim inf n! b n(s n (f) n) = Remarks.. The set of functions f satisfying the preceding property is in fact dense in every L p 0 (), p <. 2. However, in each L p (), there is a dense G 0 set of functions f such that lim inf n! b n(s n (f) n) = 0 Both statements follow from the fact that the set of coboundaries g?g T with g measurable and bounded is dense in each L p 0, p <. In the proof of the second remark we need a Baire argument like in the proof of Theorem 4. or Theorem 4.3. Lemma 4.9. Let n; m be positive integers with n < m. Let A; T A; ; T 4n? A and B; T B; ; T 2m? B be two Rokhlin towers. Denote A = A [ T A [ [ T 4n? A and B = B [ T B [ [ T 2m? B. There exists A 0 A such that and, for each n k < 3n, (A 0 ) (A) 4(B) (A) B \ T k A T k A 0 Lemma 4.9 says that the intersection of the tower B with the 2n middle oors of the tower A is contained in a subtower of A of measure smaller than 4(B). Here, by \subtower of A", we mean a tower A 0 ; T A 0 ; ; T 4n? A 0 with A 0 A. A proof of this lemma is given at the end of the paper. Proof of Theorem 4.8. The sequence (b n ) is given. Let (n k ) k be an increasing sequence of positive integers such that b n > 4 ; 00b nk < b nk+ and b nk = inf nn k b n To each k p we associate a Rokhlin tower A k ; T A k ; ; T 8n k+? A k such that (A p k ) = = b nk and a set B k in A S k disjoint from A j= j such that (B k ) = S = b nk. (As before we use the notation A k = T j A k ). 0j<8n k+ 9

21 and We recursively dene a sequence (f k ) of functions by f = f k = 8 >< > 8 >< > on S 4n 2? j=0 T j A ;? on S 8n 2? j=4n 2 T j A ; 0 otherwise ; on S 4n k+? j=0 T j A k ;? on S 8n k+? j=4n k+ T j A k ; f k? on (A k [ B k ) c ; c k on B k ; where the constant value c k of the function f k on B k is chosen so that the mean value of f k on is zero. Because (f k 6= f k? ) p 2= b nk is a summable sequence, the sequence (f k ) converges almost surely to a function f. Notice that all these functions are bounded by one. By construction (recall that, for ` > k; B` \ A k = ;), we have? ff 6= fk g \ A k [ Let k < `. By Lemma 4.9 we know that the trace of the tower A` on the 4n k+ middle oors of the tower A k is contained in a subtower of A k of measure lower than 4(A`). This implies that the set ff 6= f k g \ 6n k+? [ is contained in a subtower A 0 k of A k such that `>k A` j=2n k+ T j A k (A 0 k) 4 `>k (A`) We have (A 0 k) 4 `>k 0 k?`(a k ) < 2 (A k) Let n be a positive integer and k be such that n k n < n k+. We have On the set S n (f k ) = n on the set 3n k+? [ 4n k+?n? [ j=0 j=2n k+ T j (A k n A 0 k); which is included in the preceding one, we have S n (f) = S n (f k ) = n 20 T j A k

22 Thus (S n (f) = n) n k+ (A k n A 0 k) 2 n k+(a k ) = 6 p b nk 6 p b n which proves the theorem. Proof of Lemma 4.9. Dene A 0 = 3n? [ k=n? A \ T?k B In the tower A; T A; ; T 4n? A, above each point of A 0 there is at least n points of B. Indeed, if! 2 A 0, there exists k between n and 3n? such that T k! 2 B. Then, either T k+j! 2 A \ B for each 0 j n or T k?j! 2 A \ B for each 0 j n. (Here we used two facts if! 0 2 B, then T j! 0 2 B either for any 0 j n or for any 0 j?n; and if! 0 2 T k A for one k between n and 3n?, then T j! 0 2 A for any j between n and?n.) We claim that this implies n(a 0 ) (B) Here is a proof of this claim for each subset K of E = f0; ; ; 4n? g, dene A 0 K = f! 2 A 0 j T k (!) 2 B for k 2 K and T k (!) =2 B for k 2 E n Kg The sets A 0 K form a partition of A0. We have A 0 K = ; if card(k)< n. Sets T k A 0 K, for K E and k 2 K, are two by two disjoint and included in B. Thus (B) (T k A 0 K) n KE k2k KE References (A 0 K) = n(a) [A] Azuma, K., Weighted sums of certain dependent random variables, T^ohoku Math. J. 9 (967), [Ba-Ka] [Br] Baum, L.E. and Katz, M., Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 20 (965), Bryc, On large deviations for uniformly strong mixing sequences, Stochastic Processes and their Applications 4 (992), [Br-Dem] Bryc, W. and Dembo, A., Large deviations and strong mixing, Ann. Inst. Henri Poincare 32 (996), [Bu-Den] [C] [dj-ro] Burton, R. and Denker, M., On the central limit theorem for dynamical systems, Trans. Amer. Math. Soc. 302 (987,), Cramer, H., Sur un nouveau theoreme-limite de la theorie des probabilites, Actualites Scientiques et Industrielles (Hermann, Paris) 736 (938), del Junco, A. and Rosenblatt, J., Counterexamples in ergodic theory and in number theory, Math. Ann. 245 (979),

23 [Dem-Z] [Du] [F] [G] [Ha-He] [I-L] [K] Dembo, A. and Zeitouni, O., Large Deviations Techniques and Applications, Jones and Bartlett, Boston, 993 (Or Applications of Mathematics 38, Springer, 998). Durrett, R., Probability Theory and Examples, Wadsworth and Brooks/Cole, Pacic Grove, California, 99. Feller, W., An Introduction to Probability Theory and its Applications, Vol. II, second edition, John Wiley, New York, 97. Gordin, M.I., The central limit theorem for stationary processes, Soviet Math. Doklady 0 (969), Hall, P. and Heyde, C.C., Martingale Limit Theory and its Applications, Academic Press, New York, 980. Ibragimov, I.A. and Linnik Y.V., Independant and Stationary Sequences of Random Variables, Wolters-Noordho, Groningen, 97. Katok A., Constructions in Ergodic Theory, An unpublished manuscript. [Kr] Krengel, U., On the speed of convergence in the ergodic theorem, Monatsh. Math. 86 (978), 3-6. [N] Nagaev, S.V., Some limit theorems for large deviations, Theory of Probab. and its Appl. 0 (965), [P] Petrov, V.V., Limit Theorems of Probability Theory, Oxford Science Public., 995. [Ra] Rackauskas, A., On probabilities of large deviations for martingales, Lietuvos Matematikos Rinkinys 30 (990), [Re] Revesz, P., The Laws of Large Numbers, Academic Press, New York and London, 968. [Sc] Schonmann, R.H., Exponential convergence under mixing, Prob. Th. Rel. Fields 8 (989), [Sh] Shields, P., The Theory of Bernoulli Shifts, University of Chicago Press, 973. [V] [V2] [V3] [V4] [V-W] [Y] Volny, D., On limit theorems and category for dynamical systems, Yokohama Math. J. 38 (990), Volny, D., Approximating martingales and the central limit theorem for strictly stationary processes, Stochastic Processes and their Applications 44 (993), Volny, D., Invariance principles and Gaussian approximation for strictly stationary processes (to appear(trans. Amer. Math. Soc.)). Volny, D., Martingale approximation of stochastic processes and limit theorems, In preparation. Volny, D. and Weiss, B., Coboundaries in L 0, In preparation. Yoshihara, K., Summation Theory for Weakly Dependent Sequences, Sanseido, Tokyo, 992. Lesigne Laboratoire de Mathematiques et Physique Theorique UPRES-A 6083 CNRS, Universite Francois Rabelais, Parc de Grandmont F Tours, France address lesigne@univ-tours.fr Volny Laboratoire d'analyse et Modeles Stochastiques, UPRES-A 6085 CNRS, Universite de Rouen, F-7682 Mont-Saint-Aignan Cedex, France address volny@univ-rouen.fr 22