A generalization of Cramér large deviations for martingales

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1 A generalization of Cramér large deviations for martingales Xiequan Fan, Ion Grama, Quansheng Liu To cite this version: Xiequan Fan, Ion Grama, Quansheng Liu. A generalization of Cramér large deviations for martingales. Comptes rendus hebdomadaires des séances de l Académie des sciences, Elsevier, 204, 352, pp <hal-00695> HAL Id: hal Submitted on 27 Sep 204 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 A generalization of Cramér large deviations for martingales Xiequan Fan a,b Ion Grama a Quansheng Liu a a Univ. Bretagne-Sud, UMR 6205, LMBA, F Vannes, France b Regularity Team, Inria and MAS Laboratory, Ecole Centrale Paris - Grande Voie des Vignes, Châtenay-Malabry, France Received *****; accepted after revision Presented by Abstract In this note, we give a generalization of Cramér s large deviations for martingales, which can be regarded as a supplement of Fan, Grama and Liu Stochastic Process. Appl., 203). Our method is based on the change of probability measure developed by Grama and Haeusler Stochastic Process. Appl., 2000). To cite this article: A. Name, A. Name2, C. R. Acad. Sci. Paris, Ser. I ). Résumé Une généralisation des grandes déviations de Cramér pour les martingales. Dans cette note, nous donnons une généralisation des grandes déviations de Cramér pour les martingales, qui peut être considéré comme un supplément de Fan, Grama et Liu Stochastic Process. Appl., 203). Notre méthode est basée sur le changement de mesure de probabilité développé par Grama et Haeusler Stochastic Process. Appl., 2000). Pour citer cet article : A. Name, A. Name2, C. R. Acad. Sci. Paris, Ser. I ).. Introduction Assume that η,..., η n is a sequence of independent and identically distributed i.i.d.) centered real valued random variables satisfying the following Cramér condition : E exp{c 0 η } < for some c 0 > 0. Denote by σ 2 = Eη, 2 ξ i = η i / nσ) and X n = n ξ i. Cramér [] has established the following asymptotic expansion of the tail probabilities of X n, for all 0 x = on /6 ) as n, ) ) PX n > x) = Φx) + o), ) addresses: fanxiequan@hotmail.com Xiequan Fan), ion.grama@univ-ubs.fr Ion Grama), quansheng.liu@univ-ubs.fr Quansheng Liu). Preprint submitted to the Académie des sciences 29 août 204

3 where Φx) = 2π x { } exp t2 dt 2 is the standard normal distribution function. More precise results can be found in Feller [4], Petrov [0,], Saulis and Statulevičius [5], Sakhanenko [4] and [2] among others. Let ξ i, F i ) i=0,...,n be a sequence of martingale differences defined on some probability space Ω, F, P), where ξ 0 = 0 and {, Ω} = F 0... F n F are increasing σ-fields. Set X 0 = 0, X k = k ξ i, k =,..., n. 2) Denote by X the quadratic characteristic of the martingale X = X k, F k ) k=0,...,n : X 0 = 0, X k = k Eξi 2 F i ), k =,..., n. 3) Consider the stationary case for simplicity. For the martingale differences having a 2 + p)th moment, i.e. ξ i 2+p < for some p 0, ], expansions of the type ) in the range 0 x = o log n) have been obtained by Haeusler and Joos [8], Grama [5] and Grama and Haeusler [7]. If the martingale differences are bounded ξ i C/ n and satisfy X n L 2 /n a.s. for two positive constants C and L, expansion ) has been firstly established by Račkauskas [2,3] in the range 0 x = on /6 ), and then this range has been extended to a larger one 0 x = on /4 ) by Grama and Haeusler [6] with a method based on change of probability measure. Recently, Fan et al. [3] have generalized the result of Grama and Haeusler [6] to a much larger range 0 x = on /2 ) for ξ i satisfying the following conditional Bernstein condition : for a positive constant C, Eξ k i F i ) 2 k! C n ) k 2Eξ 2 i F i ) for all k 2 and all i n. 4) It is worth noting that the conditional Bernstein condition does not imply that ξ i s are bounded. The aim of this note is to extend the expansion of Fan et al. [3] to the case of martingale differences satisfying the following conditional Cramér condition considered in Liu and Watbled [9] : sup Eexp{C 0 n ξi } F i ) C, 5) i where C 0 and C are two positive constants. It is worth noting that, in general, condition 5) does not imply the conditional Bernstein condition 4), unless neξ 2 i F i ) are all bounded from below by a positive constant. Thus our result is not a consequence of Fan et al. [3]. Throughout this paper, c and c α, probably supplied with some indices, denote respectively a generic positive absolute constant and a generic positive constant depending only on α. Moreover, θ stands for any value satisfying θ. 2. Main Results The following theorem is our main result, which can be regarded as a parallel result of Fan et al. [3] under the conditional Cramér condition : A) sup i n Eexp{c 0 n /2 ξ i } F i ) c ; 2

4 A2) X n δ 2 a.s., where δ is nonnegative and usually depends on n. Theorem 2. Assume conditions A) and A2). Then there exists a positive absolute constant α 0, such that for all 0 x α 0 n /2 and δ α 0, the following equalities hold PX n > x) {θc Φ x) = exp x 3 log n α0 + x 2 δ x) + δ) )} 6) n n and PX n < x) Φ x) x = exp {θc 3 log n α0 + x 2 δ x) + δ) )}, 7) n n where θ. In particular, for all 0 x = o min{n /6, δ } ) as min{n, δ }, ) ) PX n x) = Φ x) + o). 8) From 6), we find that there is an absolute constant α 0 > 0 such that for all 0 x α 0 n /2 and δ α 0, log PX n > x) Φ x) c x 3 log n α 0 + x 2 δ x) + δ) ). 9) n n Note that this result can be regarded as a refinement of the moderate deviation principle MDP) under conditions A) and A2). Let a n be any sequence of real numbers satisfying a n and a n n /2 0 as n. If δ 0 as n, then inequality 9) implies the MDP for X n with the speed a n and good rate function x 2 /2; for each Borel set B, x 2 inf x B o 2 lim inf n a 2 n log P ) X n B a n lim sup n a 2 n ) x 2 log P X n B inf a n x B 2, where B o and B denote the interior and the closure of B, respectively see Fan et al. [3] for details). 3. Sketch of the proof Let ξ i, F i ) i=0,...,n be a martingale differences satisfying the condition A). For any real number λ with λ c 0 n /2, define Z k λ) = k e λξ i Ee λξ i Fi ), k =,..., n, Z 0λ) =. Then Zλ) = Z k λ), F k ) k=0,...,n is a positive martingale and for each real number λ with λ c 0 n /2 and each k =,..., n, the random variable Z k λ) is a probability density on Ω, F, P). Thus we can define the conjugate probability measure P λ on Ω, F), where dp λ = Z n λ)dp. 0) Denote by E λ the expectation with respect to P λ. Setting b i λ) = E λ ξ i F i ) and η i λ) = ξ i b i λ) for i =,..., n, 3

5 we obtain the decomposition of X n similar to that of Grama and Haeusler [6] : X n = B n λ) + Y n λ), ) where B n λ) = b i λ) and Y n λ) = η i λ). Note that Y k λ), F k ) k=,...,n is also a sequence of martingale differences w.r.t. P λ. In the sequel, we establish some auxiliary lemmas which will be used in the proof of Theorem 2.. We first give upper bounds for the conditional moments. Lemma 3. Assume condition A). Then E ξ i k F i ) k! c 0 n /2 ) k c, k 3. Proof. Applying the elementary inequality x k /k! e x to x = c 0 n /2 ξ i, we have, for k 3, ξ i k k! c 0 n /2 ) k exp{c 0 n /2 ξ i }. 2) Taking conditional expectations on both sides of the last inequality, by condition A), we obtain the desired inequality. Remark It is worth noting that both condition A) and the conditional Bernstein condition 4) imply the following hypothesis. A ) There exists ε > 0, usually depends on n, such that E ξ i k F i ) c k! ε k for all k 2 and all i n. When ε = c 2 / n, condition A ), together A2), yields Theorem 2.. Using Lemma 3., we obtain the following two technical lemmas. Their proofs are similar to the arguments of Lemmas 4.2 and 4.3 of Fan et al. [3]. Lemma 3.2 Assume conditions A) and A2). Then, for all 0 λ 4 c 0n /2, B n λ) λ c λδ 2 + λ 2 n /2 ). 3) Lemma 3.3 Assume conditions A) and A2). Then, for all 0 λ 4 c 0n /2, Ψ nλ) λ2 2 c λ2 δ 2 + λ 3 n /2 ), where Ψ n λ) = log Ee λξi F i ). The following lemma gives the rate of convergence in the central limit theorem for the conjugate martingale Y i λ), F i ) under the probability measure P λ. Its proof is similar to that of Lemma 3. of Fan et al. [3] by noting the fact that Eξi 2 F i ) c/n. Lemma 3.4 Assume conditions A) and A2). Then, for all 0 λ 4 c 0n /2, sup x P λ Y n λ) x) Φx) c λ log n + + δ n n 4 ).

6 Proof of Theorem 2.. The proof of Theorem 2. is similar to the arguments of Theorems 2. and 2.2 in Fan et al. [3] with ε = c 0 4. However, instead of using Lemmas 4.2, 4.3 and 3. of [3], we shall make use n of Lemmas 3.2, 3.3 and 3.4 respectively. Acknowledgements We thank the reviewer for his/her thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the publication. The work has been partially supported by the National Natural Science Foundation of China Grants No and No. 0039), and by Hunan Provincial Natural Science Foundation of China Grant No. JJ200). References [] Cramér, H., 938. Sur un nouveau théorème-limite de la théorie des probabilités. Actualite s Sci. Indust., 736, [2] Fan, X., Grama, I. and Liu, Q., 203. Sharp large deviations under Bernstein s condition. C. R. Acad. Sci. Paris, Ser. I 35, [3] Fan, X., Grama, I. and Liu, Q., 203. Cramér large deviation expansions for martingales under Bernstein s condition. Stochastic Process. Appl. 23, [4] Feller, W., 943. Generalization of a probability limit theorem of Cramér. Trans. Amer. Math. Soc., [5] Grama, I., 997. On moderate deviations for martingales. Ann. Probab. 25, [6] Grama, I. and Haeusler, E., Large deviations for martingales via Cramér s method. Stochastic Process. Appl. 85, [7] Grama, I. and Haeusler, E., An asymptotic expansion for probabilities of moderate deviations for multivariate martingales. J. Theoret. Probab. 9, 44. [8] Haeusler, E. and Joos, K., 988. A nonuniform bound on the rate of convergence in the martingale central limit theorem. Ann. Probab. 6, No. 4, [9] Liu, Q. and Watbled, F., Exponential ineqalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment. Stochastic Process. Appl. 9, [0] Petrov, V. V., 954. A generalization of Cramér s limit theorem. Uspekhi Math. Nauk 9, [] Petrov, V. V., 975. Sums of Independent Random Variables. Springer-Verlag. Berlin. [2] Račkauskas, A., 990. On probabilities of large deviations for martingales. Liet. Mat. Rink. 30, [3] Račkauskas, A., 995. Large deviations for martingales with some applications. Acta Appl. Math. 38, [4] Sakhanenko, A. I., 99. Berry-Esseen type bounds for large deviation probabilities, Siberian Math. J. 32, [5] Saulis, L. and Statulevičius, V. A., 978. Limite theorems for large deviations. Kluwer Academic Publishers. 5

7 The proofs of Lemmas 3.2 and 3.3 are given below. Proof of Lemma 3.2. Recall that 0 λ 4 c 0n /2. By the relation between E and E λ on F i, we have b i λ) = Eξ ie λξ i F i ) Ee λξ i Fi ), i =,..., n. Jensen s inequality and Eξ i F i ) = 0 imply that Ee λξ i F i ). Since Eξ i e λξ i F i ) = E ξ i e λξ i ) F i ) 0, by Taylor s expansion for e x, we find that B n λ) Eξ i e λξ i F i ) = λ X n + Using Lemma 3., we obtain ) E ξ i λξ i ) k k! F i k=2 n k=2 k=2 k=2 ) ξ i λξ i ) k E k! F i. 4) E ξ k+ ) λ k i F i k! c k + )λ k c 0 n /2 ) k c 2 λ 2 n /2. 5) Condition A2) together with 4) and 5) imply the upper bound of B n λ): B n λ) λ + λδ 2 + c 2 λ 2 n /2. Using Lemma 3., we have E ) e λξ i ) + F i + E λξ i ) k k! F i + k=2 k=2 c λ k c 0 n /2) k + c 3 λ 2 n. 6) This inequality together with 5) and condition A2) imply the lower bound of B n λ): B n λ) n ) ) Eξ i e λξ i F i ) + c 3 λ 2 n λ X n k=2 ) ) ) E ξ i λξ i ) k k! F i + c 3 λ 2 n λ λδ 2 c 2 λ 2 n /2 ) + c 3 λ 2 n ) λ λδ 2 c 4 λ 2 n /2. The proof of Lemma 3.2 is finished. 6

8 Proof of Lemma 3.3. Recall that 0 λ 4 c 0n /2. Since Eξ i F i ) = 0, it is easy to see that Ψ n λ) = ) log Ee λξ i F i ) λeξ i F i ) λ2 2 Eξ2 i F i ) + λ2 2 X n. Using a two-term Taylor s expansion of log + x), x 0, we obtain Ψ n λ) λ2 n 2 X n = ) Ee λξ i F i ) λeξ i F i ) λ2 2 Eξ2 i F i ) Since Ee λξ i F i ), we find that 2 + θ Ee λξi F i ) )) 2 Ee λξi F i ) ) 2. Ψ nλ) λ2 2 X n n F Eeλξi i ) λeξ i F i ) λ2 2 Eξ2 i F i ) + 2 k=3 ) 2 Ee λξ i F i ) λ k k! Eξk i F i ) In the same way as in the proof of Lemma 3.2, by Lemma 3., we have Ψ nλ) λ2 2 X n c 3λ 3 n /2. k=2 ) 2 λ k k! Eξk i F i ). Combining this inequality with condition A2), we obtain the desired inequality. 7