Cramér large deviation expansions for martingales under Bernstein s condition

Transcription

1 Cramér large deviation expansions for martingales under Bernstein s condition Xiequan Fan, Ion Grama, Quansheng Liu To cite this version: Xiequan Fan, Ion Grama, Quansheng Liu. Cramér large deviation expansions for martingales under Bernstein s condition. Stochastic Processes and their Applications, Elsevier, 23, 23, pp <hal-97843> HAL Id: hal Submitted on 2 Nov 23 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Cramér large deviation expansions for martingales under Bernstein s condition Xiequan Fan, Ion Grama and Quansheng Liu Université de Bretagne-Sud, LMBA, UMR CNRS 625, Campus de Tohannic, 567 Vannes, France Abstract An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales. Key words: expansions of large deviations; Cramér type large deviations; large deviations; moderate deviations; exponential inequality; Bernstein s condition; central limit theorem 2 MSC: Primary 6G42; 6F; 6E5; Secondary 6F5. Introduction Consider a sequence of independent and identically distributed i.i.d. centered real random variables,..., satisfying Cramér s condition exp{ } <, for some constant >. Denote 2 = 2 and = =. In 938, Cramér [5] established an asymptotic expansion of the probabilities of large deviations of, based on the powerful technique of conjugate distributions see also Esscher [8]. The results of Cramér imply that, uniformly in = /2, log P > Φ 3 = as, where Φ = 2 exp{ 2 /2} is the standard normal distribution. Various large deviation expansions for sums of independent random variables have been obtained by many authors, see for instance Feller [], Petrov [22], Rubin and Sethuraman [27], Statulevičius [29], Saulis and Statulevičius [28] and Bentkus and Račkauskas []. We refer to the book of Petrov Corresponding author. fanxiequan@hotmail.com X. Fan, quansheng.liu@univ-ubs.fr Q. Liu. ion.grama@univ-ubs.fr I. Grama, Preprint submitted to Elsevier June 7, 23

3 [23] and the references therein for a detailed account. Despite the fact that the case of sums of independent random variables is well studied, there are only a few results on expansions of type for martingales: see Bose [3, 4], Račkauskas [24, 25, 26], Grama [3, 4] and Grama and Haeusler [5, 6]. It is also worth noting that limit theorems for large and moderate deviation principle for martingales have been proved by several authors, see e.g. Liptser and Pukhalskii [2], Gulinsky and Veretennikov [7], Gulinsky, Liptser and Lototskii [8], Gao [2], Dembo [6], Worms [3] and Djellout [7]. However, these theorems are less precise than large deviation expansions of type. Let,F =,..., be a sequence of square integrable martingale differences defined on a probability space Ω,F,P, where = and {,Ω} = F... F F. Denote = =. Assume that there exist absolute constants > and such that max and = 2 F 2. Here and hereafter, the equalities and inequalities between random variables are understood in the P-almost sure sense. From the results in Grama and Haeusler [5], it follows that, for any constant > and log = /6, P > Φ = and, for any = log, P > Φ = + ++ log 3 as se also [4, 6] for more results in the last range. In this paper we extend the expansions 2 and 3 to the case of martingale differences,f =,..., satisfying the conditional Bernstein condition, F 2! 2 2 F, for 3 and, 4 where is a positive absolute constant. Note that in the i.i.d. case Bernstein s condition 4 is equivalent to Cramér s condition see Section 8 and therefore 2 implies Cramér s expansion. It is worth stressing that the remainder in expansion 2 is of the same order as that in in the stated range and therefore cannot be improved. As to the remainder in 3, from the rate of convergence result in Bolthausen [2] we conclude that it is also optimal. Another objective of the paper is to find an asymptotic expansion of large deviation for martingales in a wider range than that of 2. From Theorems 2. and 2.2 of the paper it follows that, for any constant > and log = /2, log P > Φ 3 = as. 5 This improves the corresponding result in [5] where 5 has been established only in the range [ log, /4 ] for some absolute constant >. The upper bound of the range and 2

4 the remainder in expansion 5 cannot be improved since they are of the same order as in the Cramér s expansion. The idea behind our approach is similar to that of Cramér for independent random variables with corresponding adaptations to the martingale case. We make use of the conjugate multiplicative martingale for changing the probability measure as proposed in Grama and Haeusler [5] see also [9]. However, we refine [5] in two aspects. First, we relax the boundedness condition, replacing it by Bernstein s condition 4. Secondly, we establish upper and lower bounds for the large deviation probabilities in the range [, /2 thus enlarging the range [, /4 ] established in [5]. In the proof we make use of a rate of convergence result for martingales under the conjugate measure. It is established under the Bernstein condition 4, unlike [5] where it is established only for bounded martingale differences. As a consequence, we improve the result on the rate of convergence in the central limit theorem CLT due to Bolthausen [2] see Theorem 3. below. The paper is organized as follows. Our main results are stated and discussedin in Section 2. A rate of convergence in the CLT for martingales is given in Section 3. Section 4 contains auxiliary assertions used in the proofs of the main results. Proofs are deferred to Sections 5, 6 and 7. We clarify the relations among the conditions of Bernstein, Cramér and Sakhanenko in Section 8. Throughout the paper, and, probably supplied with some indices, denote respectively a generic positive absolute constant and a generic positive constant depending only on. Moreover, s stand for values satisfying. 2. Main results 2.. Main theorems Assume that we are given a sequence of martingale differences,f =,...,, defined on some probability space Ω,F,P, where =, {,Ω} = F... F F are increasing -fields and =,..., are allowed to depend on. Set =, =, =,...,. 6 Let be the quadratic characteristic of the martingale =,F =,..., : =, = = F 2, =,...,. 7 = In the sequel we shall use the following conditions: A There exists a number, ] such that 2 F 2! 2 2 F, for 3 and ; 3

5 A2 There exists a number [, 2 ] such that 2. Note that in the case of normalized sums of i.i.d. random variables conditions A and A2 are satisfied with = and = see conditions A and A2 below. In the case of martingales and usually depend on such that = and =. The following two theorems give upper and lower bounds for large deviation probabilities. Theorem 2.. Assume conditions A and A2. Then, for any constant, and all, we have P > { } Φ exp log + 8 and P < Φ exp { } log +, 9 where the constant does not depend on,f =,...,, and. Theorem 2.2. Assume conditions A and A2. Then there is an absolute constant > such that, for all and, P > { Φ exp log + } 2 and P < Φ exp { log + } 2, where the constants and do not depend on,f =,...,, and. Using the inequality valid for, from Theorems 2. and 2.2, we obtain the following improvement of the main result of [5]. Corollary 2.. Assume conditions A and A2. Then there is an absolute constant > such that, for all min{ log, }, P > Φ = exp{ 3 } log + 2 and P < Φ = exp{ 3 3 } log +, 3 where does not depend on, but possibly depend on,f =,...,, and. For bounded martingale differences under condition A2, Grama and Haeusler [5] proved the asymptotic expansions 2 and 3 for [, min{ /2, }] and some small absolute constant, ]. Thus Corollary 2. extends the asymptotic expansions of [5] to 8 a larger range [, min{ log, } and non bounded martingale differences. 4

6 2.2. Remarks on the main theorems Combining the inequalities 8 and, we conclude that under A and A2 there is an absolute constant > such that, for all and, log P > Φ log +. 4 We show that this result can be regarded as a refinement of the moderate deviation principle MDP in the framework where A and A2 hold. Assume that A and A2 are satisfied with = and = as. Let be any sequence of real numbers satisfying and as. Then inequality 4 implies the MDP for with the speed and rate function 2 /2. Indeed, using the inequalities we deduce that, for any, 2+ 2 /2 Φ lim 2 logp > = 2 2. By a similar argument, we also have, for any, lim 2 logp < = /2,, The last two equalities are equivalent to the statement that: for each Borel set, 2 inf liminf logp limsup logp inf 2, 2 where and denote the interior and the closure of respectively, see Lemma 4.4 of [2]. Similar results can be found in Gao [2] for the martingale differences satisfying the conditional Cramér condition exp{ } F <. To show that our results are sharp, assume that = /, where,f =,..., is a sequence of martingale differences satisfying the following conditions: A Bernstein s condition There exists a positive absolute constant such that F 2! 2 2 F, for 3 and ; A2 There exists an absolute constant such that = 2 F 2. 5

7 These conditions are satisfied with some > and = if, for instance,, 2,..., are i.i.d. random variables with finite exponential moments see Section 8 for an explicit expression of the positive absolute constant. Corollary 2.2. Assume conditions A and A2. Then there is an absolute constant 2 > such that for any absolute constant > and all log 2 /2, we have log P = > = Φ and as. log P = < Φ = + 3 It is worth noting that the remainders of the expansions 5 and 6 are of the same order as in and therefore are optimal. Corollary 2.3. Assume conditions A and A2. Then, for all = log, P = > = + ++ log Φ and as. P = < Φ = + ++ log Notice that 7 extends expansion 3 proved in Grama and Haeusler [5] to the case of martingale differences satisfying the conditional Bernstein condition A. The Remark 2. of [5]andthesharprateofconvergenceintheCLTduetoBolthausen[2]hintthattheremainders of the expansions 7 and 8 are sharp. Corollary 2.4. Assume conditions A and A2. Then, for any absolute constant > and log = /6, P = > = Φ and as. P = < Φ = The remainders of the expansions 9 and 2 are of the same order as in in the stated range and therefore cannot be improved. Remark 2.. The results formulated above are proved under Bernstein s condition A. But they are also valid under some equivalent conditions which are stated in Section

8 3. Rates of convergence in the CLT Let,F =,..., be a sequence of martingale differences satisfying condition A and =,F =,..., be the corresponding martingale defined by 6. For any real satisfying <, consider the exponential multiplicative martingale =,F =,...,, where = = F, =,...,, =. For each =,...,, the random variable defines a probability density on Ω,F,P. This allows us to introduce, for <, the conjugate probability measure P on Ω,F defined by P = P. 2 Denote by the expectation with respect to P. For all =,...,, let = and = F. We thus obtain the well-known semimartingale decomposition: = +, =,...,, 22 where =,F =,..., is the conjugate martingale defined as = and =,F =,..., is the drift process defined as =, =,...,, 23 =, = =,...,. In the proofs of Theorems 2. and 2.2, we make use of the following assertion, which gives us a rate of convergence in the central limit theorem for the conjugate martingale under the probability measure P. Lemma 3.. Assume conditions A and A2. Then, for all <, sup P Φ + log +. If =, then = and P = P. So Lemma 3. implies the following theorem. Theorem 3.. Assume conditions A and A2. Then sup P Φ log

9 Remark 3.. By inspecting the proof of Lemma 3., we can see that Theorem 3. holds true when condition A is replaced by the following weaker one: C There exists a number, ] depending on such that 2 F 2 2 F, for = 3,5 and. Remark 3.2. Bolthausen see Theorem 2 of [2] showed that if and condition A2 holds, then sup P Φ 3 log+. 25 We note that Theorem 3. implies Bolthausen s inequality 25 under the less restrictive condition A. Indeed, by condition A2, we have 3/4 2 and then 3/4. For /2, it is easy to see that 3 log 3 log /4. Thus, inequality 24 implies 25 with = 4/3. 4. Auxiliary results In this section, we establish some auxiliary lemmas which will be used in the proofs of Theorems 2. and 2.2. We first prove upper bounds for the conditional moments. Lemma 4.. Assume condition A. Then and F 6!, for 2, F! 2 2 F, for 2. Proof. By Jensen s inequality and condition A, from which we get 2 F 2 4 F F, 2 F 2 2. We obtain the first assertion. Again by condition A, for 3, F 2! 2 2 F 6!. If is even, the second assertion holds obviously. If = 2 +,, is odd, by Hölder s inequality and condition A, it follows that 2+ F + F 2 F 2+ F 2 2!2+2! 2 2 F 2+! 2 2 F. 8

10 This completes the proof of Lemma 4.. The following lemma establishes a two sided bound for the drift process. Lemma 4.2. Assume conditions A and A2. Then for any constant, and all, Proof. By the relation between and on F, we have = F F, =,...,. Jensen s inequality and F = imply that F. Since F = F, for, by Taylor s expansion for, we find that F = = F = = + = + =2! F. 27 Using condition A, we obtain, for any constant, and all, + +! F + F! = =2 = = = Using condition A2, we get 2 and, for any constant, and all, +! F = =2 Condition A2 together with 27 and 29 imply the upper bound of : for any constant, and all,

11 Using Lemma 4., we have, for any constant, and all, + F +! F +6 =2 + =2 +, 2. 3 This inequality together with condition A2 and 29 imply the lower bound of : for any constant, and all, +, 2 = F +! F +, 2 = = , , 2, where the last line follows from the following inequality, for any constant, and all, , 2 +, , 2 +, 2. The proof of Lemma 4.2 is finished. Now, consider the predictable cumulant process Ψ = Ψ,F =,..., related with the martingale as follows: Ψ = log F. 3 = We establish a two sided bound for the process Ψ. Lemma 4.3. Assume conditions A and A2. Then, for any constant, and all, Ψ

12 Proof. Since F =, it is easy to see that Ψ = log F F F = Using a two-term Taylor s expansion of log+,, we obtain Ψ 2 2 = F F F = F = =3 = F 2. Since F, we find that Ψ 2 2 F F F = + 2 F 2 = +! F ! F. In the same way as in the proof of 28, using condition A and the inequality F cf. Lemma 4., we have, for any constant, and all, Ψ Combining this inequality with condition A2, we get, for any constant, and all, Ψ , which completes the proof of Lemma Proof of Theorem 2. For <, the assertion follows from Theorem 3.. It remains to prove Theorem 2. for any, and all. Changing the probability measure according to 2, we have, for all <, P > = { >} = exp{ +Ψ } { >} = exp{ +Ψ } { + >}. 32 = =2

13 Let = be the largest solution of the equation =, 33 where is given by inequality 26. The definition of implies that there exist,,, > such that, for all,, = and =, [,, ]. 35 From 32, using Lemmas 4.2, 4.3 and equality 33, we obtain, for all, P >, /2 { >}. 36 It is easy to see that {>} = Similarly, for a standard gaussian random variable, we have {>} = From 37 and 38, it follows {>} {>} 2sup P <. 37 P <. 38 P Φ. Using Lemma 3., we obtain the following bound: for all, { >} {>} + log From 36 and 39 we find that, for all, P >, /2 {>} + + log +. Since 2 /2 {>} = +2 /2 = Φ 4 2 2

14 and, for all,, Φ 2+ 2 /2, 2+, 2 /2 4 seefeller[], weobtainthefollowingupperboundontailprobabilities: forall, P > Φ 3 2, ,3 2 + log Next, we would like to compare Φ with Φ. By 34, 35 and 4, we get exp{ 2 /2} exp{ 2 /2} = + exp{ 2 /2} exp{ 2 /2} +,4 exp{ 2 2 /2} exp{, }. 43 So, we find that Φ = Φ exp {, }. 44 Implementing 44 in 42 and using 34, we obtain, for all, P > Φ exp{, } +,7 2 + log + exp{, } exp{, } +,7 2 +,7 log + +,7 log +. Taking = max{,7,,8 }, we prove the first assertion of Theorem 2.. The second assertion follows from the first one applied to the martingale =,...,. 6. Proof of Theorem 2.2 For <, the assertion follows from Theorem 3.. It remains to prove Theorem 2.2 for, where > is an absolute constant. Let = be the smallest solution of the equation 2 /2 2 =, 45 where isgivenbyinequality26. Thedefinitionofimpliesthat, forall. /2, it holds 2 = /2 3

15 and = [,.2 /2 ]. 47 From 32, using Lemmas 4.2, 4.3 and equality 45, we obtain, for all. /2, P > /2 { >}. 48 In the subsequent we distinguish two cases. First, let min{ /2, }, where > is a small absolute constant whose value will be given later. Note that inequality 39 can be established with replaced by, which, in turn, implies P > /2 {>} 2 + log +. By 4 and 4, we obtain the following lower bound on tail probabilities: P > Φ log Taking = 8 2, we deduce that, for all min{ /2, }, log + exp { log + }. 5 Implementing 5 in 49, we obtain P > { exp log } 5 Φ which is valid for all min{ /2, }. Next, we consider the case of min{ /2, } and. Let be an absolute constant, whose exact value will be chosen later. It is easy to see that { >} {< } P < where = + log + 4 /2, if. From Lemma 3., we have P < P < 5 Taking = /6 2, we find that P < 2 2 / , 52

16 Letting 8 5, it follows that Choosing = max we deduce that P < max{ 2, } } {8 5, and taking into account that min{ /2, }, P <.. Since the inequality 2 /2 Φ is valid for see Feller [], it follows that, for all min{ /2, }, P < Φ 2 /2. 53 From 48, 52 and 53, we obtain P > Φ { exp,6 3 + log } 54 which is valid for all min{ /2, }. Putting 5 and 54 together, we obtain, for all and, P > Φ { exp,7 3 + log }. 55 As in the proof of Theorem 2., we now compare Φ with Φ. By a similar argument as in 43, we have Φ = Φ exp { }. 56 Combining 46, 55 and 56, we obtain, for all and, P > Φ exp {,8 3 + log } 57 which gives the first conclusion of Theorem 2.2. The second conclusion follows from the first one applied to the martingale =,...,. 5

17 7. Proof of Lemma 3. The proof of Lemma 3. is a refinement of Lemma 3.3 of Grama and Haeusler [5] where it is assumed that 2, which is a particular case of condition A. Compared to the case where are bounded, the main challenge of our proof comes from the control of defined in 64 below. In this section, denotes a positive absolute number satisfying,, denotes a real number satisfying, which is different from, and denotes the density function of the standard normal distribution. For the sake of simplicity, we also denote, and by, and, respectively. We want to obtain a rate of convergence in the central limit theorem for the conjugate martingale =,F =,...,, where = =. Denote the quadratic characteristic of the conjugate martingale by = F 2, and set Δ = 2 F. It is easy to see that, for =,...,, Δ = 2 F = 2 F F F 2 F Since F and 2 + F, using condition A and Lemma 4., we obtain, for all 3 and all 4, F 2 + F F 2 F 2 exp{ } F 2! 2 2 F. Using Taylor s expansion for and Lemma, we have, for all 4, Δ Δ 2 F F 2 F + F 2 F 2 2 F 2 F F + F 2 +2 F! +Δ = F! = = F! Δ. 59 Therefore,

18 Thus the martingale satisfies the following conditions analogous to conditions A and A2: for all 4, B F 2 2 F, 5 3; B We first prove Lemma 3. for <. Without loss of generality, we can assume that 4, otherwise we take 4 in the assertion of the lemma. Set = + 2 and introduce a modification of the quadratic characteristic as follows: = {<} + {=}. 6 Notethat =, = andthat,f =,..., isapredictableprocess. Set = +,where [,. Let 4 be a free absolute constant, whose exact value will be chosen later. Consider the non-increasing discrete time predictable process = 2 2 +, =,...,. For any fixed R and any R and >, set for brevity, Φ, = Φ /. 6 In the proof we make use of the following two assertions, which can be found in Bolthausen s paper [2]. Lemma 7.. [2] Let and be random variables. Then sup P Φ sup P + Φ /2. Lemma 7.2. [2] Let be an integrable function of bounded variation, be a random variable and, > are real numbers. Then + sup P Φ + 2. Let 2 2 =, be a normal random variable independent of. Using a well-known smoothing procedure which employs Lemma 7., we get sup P Φ sup Φ, Φ + 2 Φ, Φ, sup + sup = sup + sup Φ, Φ + 2 Φ, Φ, Φ Φ sup Φ, Φ, + 3, 62 7

19 where Φ, = P and Φ, = P By simple telescoping, we find that Φ, Φ, = = Φ, Φ,. From this, taking into account that,f =,..., is a P -martingale and that we obtain where = 2 = 2 3 = = 2 2Φ, = 2 Φ,, Φ, Φ, = + 2 3, 63 = Φ, Φ, Φ, 2 2, 2, Φ, Δ Δ, 65 = Φ, Φ, Φ, Δ. 66 We now give estimates of, 2 and 3. To shorten notations, set = /. a Control of. Using a three-term Taylor s expansion, we have = = 6 3/ In order to bound we distinguish two cases as follows. Case : / /2. In this case, by the inequality + 2, it follows + 2 sup + 2 /2 /

20 Define = sup 3, where = / It is easy to see that is a symmetric integrable function of bounded variation, non-increasing in. Therefore, { / /2}. 68 Case 2: / > /2. Since 2, it follows that { / > /2} 2 { <2} { 2} Now we bound the conditional expectation of. Using condition B, we have 3 F Δ and 5 F 3 Δ,. 69 where Δ =. From the definition of the process see 6, it follows that Δ Δ =, 3 F Δ and 5 F Δ 3. 7 Thus, from 68, we obtain 3 { / /2} F 4 Δ. 7 From 69, by 7 and the inequality 2 2, we find 3 { / > /2} F 2 Δ, 72 where 2 = 2 { <2} +4 2 { 2}. Set = Then is a symmetric integrable function of bounded variation, non-increasing in. Returning to 67, by 7 and 72, we get where [ = 6 3/2 = = 3/2 Let us introduce the time change as follows: for any real [,], 3 ] F, 73 Δ. 74 = min{ : > }, where min =. 75 9

21 It is clear that, for any [,], the stopping time is predictable. Let =,...,+ be the increasing sequence of moments when the increasing stepwise function, [,], has jumps. It is clear that Δ = [, + and that =, for [, +. Since =, we have = 3/2 Δ = = = [, + 3/2 3/2. Set, for brevity, = Since Δ 2 2, we see that Taking into account that 4, we have +Δ +2 2, [,] = 2 2 +, [,]. 77 Since is symmetric and is non-increasing in, the last bound implies that 3/2. 78 /2 By Lemma 7.2, it is easy to see that /2 sup P Φ Since = Δ, cf. 76 and Δ 2 2, we get +Δ Thus 2 F = = F 2 F = Δ F = F F. 2

22 Then, by Lemma 7., we find that, for any [,], sup From 78, 79 and 8, we obtain P Φ sup P Φ /2 sup P Φ + 2 By elementary computations, we see that since Then 3/2 and. 82 log. 83 sup P Φ + 2 log. 84 b Control of 2. Set = sup 2 +, where = + 2 3/2. Then is a symmetric integrable function of bounded variation, non-increasing in. Since Δ = Δ, we have 2 2, + 2,2, where 2, = = 2,2 = = 2 Δ Δ, 2 Δ Δ. We first deal with 2,. Since, for any real, we have. 85 Note that Δ Δ 2 2 {=}, = 2 2 and 4. Then, using 85, we get the estimations 2, 2, and, by 79 with = and 8 with =, 2, sup P Φ + 2. We next consider 2,2. By 59, we easily obtain the bound Δ Δ Δ Δ. 2

23 With this bound, we get 2,2 = 2 Δ. Since, the right-hand side can be bounded exactly in the same way as in 74, with replacing 3/2. What we get is cf. 82 2,2 sup P Φ + 2. By elementary computations, we see that /2 /2 2, and, taking into account that 2 2, Then Collecting the bounds for 2, and 2,2, we get c Control of 3. By Taylor s expansion, 3 = 8 /2 2. 2,2 sup P Φ sup P Φ = Since Δ = Δ 2 2 and 4, we have Δ 2 Δ. 2 Δ Δ Using 87 and the inequalities, we obtain = 2 Δ. 22

24 Proceeding in the same way as for estimating in 74, we get 3 sup P Φ We are now in a position to end the proof of Lemma 3.. From 63, using 84, 86 and 88, we find Φ, Φ, sup P Φ log +. Implementing the last bound in 62, we come to sup P Φ sup from which, choosing = max{2,4}, we get P Φ log +, sup P Φ log +, 89 which proves Lemma 3. for <. For <, we can prove Lemma 3. similarly by taking = log +. We only need to note that in this case, instead of 83, 3/2 ln and log Equivalent conditions In the following we give several equivalent conditions to the Bernstein condition A. In the independent case equivalent conditions can be found in Saulis and Statulevičius [28]. For the convenience of the readers and motivated by the fact that in [28] the conditions are rather different from those used here, we decided to include independent proofs. Proposition 8.. The following three conditions are equivalent: I Bernstein s condition A. II Sakhanenko s condition There exists some positive absolute constant such that 3 exp{ } F 2 F, for. III There exists some positive absolute constant such that F 2! 2 2 F, for 3 and. 23

25 Proof. First we prove that I implies II. Let,. By condition I and Lemma 4., we find that 3 F = +3 F! = +3! +! F 2 = F 2 +3!! = =: 2 F. 9 Since = is a continuous function in [, ] and satisfies = and 2 3, there 2 exists, such that 2 =. Taking =, we obtain condition II from 9. Next we show that II implies III. By the elementary inequality!, for and, it follows that, for 3, Using condition II, for 3, F = F 3! 3 3 exp{ } F. F 3! 2 2 F 2! 2 2 F, where =, which proves condition III. It is obvious that III implies I with =. Proposition 8.2. If,..., are i.i.d., then Bernstein s condition, Cramér s condition and Sakhanenko s condition are all equivalent. Proof. According to Theorem 8., we only need to prove that Cramér s condition and Bernstein s condition are equivalent. We can assume that, a.s., =. First, from 3, we find that Bernstein s condition A implies Cramér s condition: 2 <. Second, we show that Cramér s condition, i.e. := <, implies Bernstein s condition A. By the inequality!, for and, it follows that Then, it is easy to see that, for 3,! =! ! ! 2, 2 { where 2 = 2 and = max, 23 }, which proves that condition A is satisfied. 2 24

26 Acknowledgements We would like to thank the two referees for their helpful remarks and suggestions. The work has been partially supported by the National Natural Science Foundation of China Grant no. 39 and Grant no. 744, and Hunan Provincial Natural Science Foundation of China Grant No.JJ2. Fan was partially supported by the Post-graduate Study Abroad Program sponsored by China Scholarship Council. References [] Bentkus, V. and Račkauskas, A., 99. On probabilities of large deviations in Banach spaces. Probab. Theory Relat. Fields, 86, [2] Bolthausen, E., 982. Exact convergence rates in some martingale central limit theorems. Ann. Probab.,, [3] Bose, A., 986a. Certain non-uniform rates of convergence to normality for martingale differences. J. Statist. Plann. Inference, 4, [4] Bose, A., 986b. Certain non-uniform rates of convergence to normality for a restricted class of martingales. Stochastics, 6, [5] Cramér, H., 938. Sur un nouveau théorème-limite de la théorie des probabilités. Actualite s Sci. Indust., 736, [6] Dembo, A., 996. Moderate deviations for martingales with bounded jumps. Electron. Comm. Proba.,, 7. [7] Djellout, H., 22. Moderate deviations for martingale differences and applications to mixing sequences. Stochastic Stochastic Rep., 73, [8] Esscher, F., 932. On the probability function in the collective theory of risk. Skand. Aktuarie Tidskr., 5, [9] Fan, X., Grama, I. and Liu, Q., 22. Hoeffding s inequality for supermartingales. Stochastic Process. Appl., 22, [] Feller, W., 943. Generalization of a probability limit theorem of Cramér. Trans. Amer. Math. Soc., [] Feller, W., 97. An introduction to probability theory and its applications, J. Wiley and Sons. [2] Gao, F. Q., 996. Moderate deviations for martingales and mixing random processes. Stochastic Process. Appl., 6, [3] Grama, I., 995. The probabilities of large deviations for semimartingales. Stochastic Stochastic Rep., 54, 9. [4] Grama, I., 997. On moderate deviations for martingales. Ann. Probab., 25,

27 [5] Grama, I. and Haeusler, E., 2. Large deviations for martingales via Cramér s method. Stochastic Process. Appl., 85, [6] Grama, I. and Haeusler, E., 26. An asymptotic expansion for probabilities of moderate deviations for multivariate martingales. J. Theoret. Probab., 9, 44. [7] Gulinsky, O. V. and Veretennikov, A. Yu., 993. Large deviations for discrete-time processes with averaging. VSP, Utrecht. [8] Gulinsky, O. V., Liptser, R. S. and Lototskii, S. V., 994. Large deviations for unbounded additive functionals of Markov processes with discrete time. J. Appl. Math. Stochastic Anal., 7. [9] Hall, P. and Heyde, C. C., 98. Martingale Limit Theory and Its Application. Academic Press. [2] Huang, C., 22. Moments, moderate and large deviations for a branching process in a random environment. Stochastic Process. Appl., 22, [2] Liptser, R.S. and Pukhalskii, A. A., 992. Limit theorems on large deviations for semimartingales. Stochastic Stochastic Rep., 38, [22] Petrov, V. V., 954. A generalization of Cramér s limit theorem. Uspekhi Math. Nauk, 9, [23] Petrov, V. V., 975. Sums of Independent Random Variables. Springer-Verlag. Berlin. [24] Račkauskas, A., 99. On probabilities of large deviations for martingales. Liet. Mat. Rink., 3, [25] Račkauskas, A., 995. Large deviations for martingales with some applications. Acta Appl. Math., 38, [26] Račkauskas, A., 997. Limit theorems for large deviations probabilites of certain quadratic forms. Lithuanian Math. J., 37, [27] Rubin, H. and Sethuraman, J., 965. Probabilities of moderate deviations. Sankhyā Ser. A, 37, [28] Saulis, L. and Statulevičius, V. A., 978. Limit theorems for large deviations. Kluwer Academic Publishers. [29] Statulevičius, V. A., 966. On large deviations. Probab. Theory Relat. Fields, 6, [3] Worms, J., 2. Moderate deviations for some dependent variables, part I: Martingales. Math. Methods Statist.,,