PROBABILITY AND MOMENT INEQUALITIES UNDER DEPENDENCE

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1 Statistica Sinica , doi: PROBABILITY AND MOMENT INEQUALITIES UNDER DEPENDENCE Weidong Liu, Han Xiao and Wei Biao Wu Shanghai Jiao Tong University, Rutgers University and The University of Chicago Abstract: We establish Nagaev and Rosenthal-type inequalities for dependent random variables. The imposed dependence conditions, which are expressed in terms of functional dependence measures, are directly related to the physical mechanisms of the underlying processes and are easy to work with. Our results are applied to nonlinear time series and kernel density estimates of linear processes. Key words and phrases: m-dependence approximation, Nagaev inequality, nonlinear process, non-stationary process, physical dependence measure, Rosenthal inequality. 1. Introduction Probability and moment inequalities play an important role in the study of properties of sums of random variables. A number of inequalities have been derived for independent random variables; see the recent collection by Lin and Bai The celebrated Nagaev and Rosenthal inequalities are two useful ones. We first start with the Nagaev inequality. Let X 1,..., X n be mean 0 independent random variables and S n = n X i. Further assume that for all i, X i p := E X i p 1/p <, p > 2. By Corollary 1.7 in Nagaev 1979, for a positive number x, one has { a p x 2 } PS n x PX i y i + exp n EX2 i 1 X i y i n + EXp i 1 βx/y 0 X i y i, 1.1 βxy p 1 where y 1,..., y n > 0, y = max i {y i, 1 i n}, β = p/p + 2 and a p = 2e p p With µ n,p = E X i p

2 1258 WEIDONG LIU, HAN XIAO AND WEI BIAO WU and y i = xβ for all 1 i n, one obtains from 1.1 that P S n x p µn,p p x p + 2 exp a px 2 ; 1.2 see Corollary 1.8 in Nagaev If the random variables X i, i, are independent and identically distributed i.i.d., then 1.1 implies P S n x p n X0 p p p x p + 2 exp a px 2 n X Inequalities of this type have applications in insurance and risk management. For example, for a small level α 0, 1, if x = x α is such that the right hand side of 1.2 is α, then the α-quantile or value-at-risk of S n is bounded by x α since PS n x α α. Inequality 1.2 suggests two types of bounds for the tail probability PS n x: if x 2 is around the variance µ n,2 = vars n, then one can use the Gaussian-type tail exp a p x 2 /µ n,2. If x is larger, the algebraic decay tail µ n,p /x p is needed. In dealing with temporal or time series data, the X i are often dependent. Then the problem naturally arises on how to generalize the Nagaev inequality to dependent random variables. The latter problem is quite challenging and very few results have been obtained. Under some boundedness conditions on conditional expectations, Basu 1985 derived a similar result. However, the imposed conditions there appear too restrictive and they exclude many commonly used time series models. Nagaev 2001 considered uniformly mixing processes, a very strong type of dependence condition. In Nagaev 2007 he considered martingales. Bertail and Clémençon 2010 dealt with functionals of positive recurrent geometrically ergodic Markov chains; see also Rio The Rosenthal inequality provides a bound for the moment E S n p. Rosenthal 1970 proved that if X i are i.i.d., then there exists a constant B p such that µ n,2 E S n p B p maxµ n,p, µ p/2 n, The calculation of the constant B p has been extensively discussed in the literature; see Pinelis and Utev 1984, Johnson, Schechtman, and Zinn 1985, Ibragimov and Sharakhmetov 2002, among others. Hitczenko 1990 obtained the best constant for a martingale version of the Rosenthal inequality. Under various types of strong mixing conditions, the Rosenthal type inequalities have been obtained for dependent random variables; see Shao 1988, 1995, 2000, Peligrad 1985, 1989, Utev and Peligrad 2003, and Rio Rio 2009 and Merlevède and Peligrad 2011 used projections and conditional expetations. Pinelis 2006 applied domination technique to obtain moment inequalities for supermartingales.

3 PROBABILITY AND MOMENT INEQUALITIES UNDER DEPENDENCE 1259 In this paper we establish Nagaev and Rosenthal-type inequalities for dependent random variables under easily verifiable dependence conditions. We assume that X n is a stationary causal process of the form X i = g, ε i 1, ε i, 1.5 where ε i, i Z, are i.i.d. random variables. We adopt the functional dependence measure introduced by Wu Let ε i, ε j, i, j Z, be i.i.d. random variables; let F n =, ε n 1, ε n and X n = gf 1, ε 0, ε 1,..., ε n. Define the dependence measure θ n,p = X n X n p and the tail sum Θ m,p = θ i,p. Assume throughout the paper that the short-range dependence condition Θ 0,p < holds. Let S n = n k=1 X k and Sn = max 1 i n S i. Our Rosenthal and Nagaev-type inequalities are expressed in terms of θ n,p and Θ m,p, and are presented in Sections 2 and 3, respectively. Those inequalities are applied to nonlinear time series and kernel density estimates of linear processes. Section 4 provides an extension to non-stationary processes. 2. A Rosenthal-type Inequality Throughout the paper we let c p denote a constant that only depends on p, and whose values may change from place to place. Theorem 1 provides a Rosenthal-type inequality for the maximum partial sum S n. Peligrad, Utev, and Wu 2007 proved that, for p 2, S n p c p n 1/2[ X 1 p + i=m j 3/2 ES j F 0 p ]. This inequality can be viewed as a generalization of the Burkholder 1973, 1988 inequality to stationary processes. The Rosenthal inequality has a different flavor in that it relates higher moments of S n to its variance. Rio 2009 showed a Rosenthal-type inequality for stationary processes: for 2 < p 3, one has S n p c p n 1/2 σ N + c p n 1/p X 0 p + 1/2 N + D N, 2.1 where N = min{i : 2 i n} and σ N = X N 1 l=0 2 l/2 ES 2 l F 0 2,

4 1260 WEIDONG LIU, HAN XIAO AND WEI BIAO WU N = N 1 l=0 N 1 2 2l/p ES 2 2 l F 0 ES 2 2 l p/2, D N = 2 l/p ES 2 l F 0 p. l=0 Merlevède and Peligrad 2011 obtained the following: for all p > 2, S n p c p n 1/p [ X 0 p + k=1 ES k F 0 p k 1+1/p + k=1 ES 2 k F 0 δ p/2 k 1+2δ/p 1/2δ ], 2.2 where δ = min1, p 2 1. For 2 < p 3, 2.2 is a maximal version of Rio s inequality 2.1. If the X i are independent, then ESk 2 F 0 = ESk 2 = k X and 2.2 reduces to 1.4. A key step in applying 2.2 is to deal with the quantity ESk 2 F 0 δ p/2 k=1 k 1+2δ/p k=1 ES 2 k F 0 ES 2 k δ p/2 k 1+2δ/p + k=1 [ES 2 k ]δ k 1+2δ/p. In doing so, one needs to control ES k F 0 p and ES 2 k F 0 ES 2 k p/2. The computation of the latter can be quite involved. Merlevède and Peligrad 2011 provided an inequality in terms of individual summands that involve terms such as EX i X j F 0 and EX j F 0. The latter quantities can be controlled by using various mixing coefficients in Bradley 2007, Rio 2000, and Dedecker et al Our Theorem 1 provides an upper bound for S n p using the functional dependence measure θ n,p which is easily computable in many applications; see Wu We do not need to deal with the quantity ES 2 k F 0 ES 2 k p/2. Our inequality is powerful enough so that the behavior of S n p for p near boundary can also be depicted; see Example 1. In order to provide explicit constants in our Rosenthal-type inequality, we need the following version of the Rosenthal inequality for independent variables taken from Johnson, Schechtman, and Zinn 1985 X i p 14.5p log p µ 1/2 n,2 + µ1/p n,p. 2.3 We also need a version of the Burkholder inequality due to Rio 2009: X 1,..., X n are martingale differences and p 2, then 2 X i p 1 X i 2 p. 2.4 p if

5 PROBABILITY AND MOMENT INEQUALITIES UNDER DEPENDENCE 1261 Theorem 1. Assume EX 1 = 0 and E X 1 p <, p > 2. Then Sn p n 1/2[ 87p θ j,2 + 3p 1 1/2 θ j,p + 29p ] log p log p X 1 2 +n 1/p[ 87pp 1 1/2 log p j=n+1 j 1/2 1/p θ j,p + 29p log p X 1 p ]. 2.5 Note that, since p > 2, we have θ j,2 θ j,p. Then the term n θ j,2 + j=n+1 θ j,p in 2.5 can be equivalently replaced by Θ 1,2 +Θ n+1,p. Hence we can rewrite 2.5 as Sn p c p n 1/2 Θ 1,2 + X c p n 1/p[ minj, n 1/2 1/p θ j,p + X 1 p ]. If the X i are independent, then θ j,2 = 0 and θ j,p = 0 for all j 1, and 2.5 reduces to the traditional Rosenthal inequality 1.4. The presence of Θ 1,2 and minj, n1/2 1/p θ j,p is due to dependence. It is generally convenient to apply Theorem 1 since the functional dependence measure θ j,p is directly related to the data-generating mechanism of the underlying processes, and in many cases it can be easily computed; see Wu 2011 for examples of linear and nonlinear processes. Proof of Theorem 1. For i 0 and j 0 let S i,j = i X k,j, where X k,j = EX k ε k j,..., ε k. k=1 Note that X k,j, k Z, is j-dependent. Namely X k,j and X k,j are independent if k k > j. We write X k as X k = X k X k,n + X k,j X k,j 1 + X k, Then S n p max S i S i,n + 1 i n p max S i,j S i,j 1 + max 1 i n p For the second term in 2.7, max S i,j S i,j 1 1 i n p S n,j S n,j 1 p + max 0 i n 1 k=n i 1 i n S i, p X k,j X k,j p

6 1262 WEIDONG LIU, HAN XIAO AND WEI BIAO WU Note that {X k,j X k,j 1, 1 k n} are also j-dependent. Moreover, X n k,j X n k,j 1, 0 k n 1, are martingale differences with respect to σε n k j, ε n k j+1,.... Thus, { n k=n i X k,j X k,j 1, 0 i n 1} is a nonnegative submartingale with respect to σε n i j, ε n i j+1,.... By the Doob inequality, we have max 0 i n 1 k=n i X k,j X k,j 1 p p/p 1 S n,j S n,j 1 p. 2.9 Write Y i,j = ij n k=1+i 1j X k,j X k,j 1, where a b := mina, b for two real numbers a and b. With l = n/j + 1, we have S n,j S n,j 1 = l. Y i,j 2.10 Observe that Y 1,j, Y 3,j,... are independent and Y 2,j, Y 4,j,... are also independent. By 2.3, S n,j S n,j 1 p 14.5p [ 2 2 Y i,j + Y i,j log p i is odd i is even 1/p ] 1/p + Y i,j p p + Y i,j p p By 2.4 we have, for 1 i l, i is odd i is even Y i,j p p 1 1/2 [ij n i 1j] 1/2 X 1,j X 1,j 1 p p 1 1/2 [ij n i 1j] 1/2 θ j,p ; Y i,j 2 [ij n i 1j] 1/2 θ j, Thus 2.11 implies that for 1 j n S n,j S n,j 1 p 29p nθj,2 + p 1 1/2 n 1/p j 1/2 1/p θ j,p log p By and noting that p/p 1 2 when p 2, we obtain max S i,j S i,j 1 1 i n p 87p n θ j,2 + p 1 1/2 n 1/p log p j 1/2 1/p θ j,p. 2.14

7 PROBABILITY AND MOMENT INEQUALITIES UNDER DEPENDENCE 1263 For the first term in 2.7, using the Burkholder inequality 2.4 and a similar argument as 2.8 and 2.9, we have max S i S i,n 3p 1 1/2 n 1/2 1 i n p j=n+1 θ j,p For the third term in 2.7, noting that X k,0, k Z, are independent, again by 2.3 and the Doob inequality, we have max 1 i n S i,0 29p p log p n 1/2 X n 1/p X 1 p. This, together with 2.7, 2.15, and 2.14, implies Theorem 1. Example 1. Consider the nonlinear time series that is expressed in the form of iterated random functions see for example Diaconis and Freedman 1999: X i = F X i 1, ε i = F εi X i 1, 2.16 where F is a bivariate measurable function and ε i, i Z, are i.i.d. innovations. Assume that there exist x 0 and p > 2 such that and the Lipschitz constant κ p := x 0 F ε0 x 0 p < 2.17 F ε0 x F ε0 x p L p := sup x x x x < By Theorem 2 in Wu and Shao 2004, conditions 2.17 and 2.18 imply that 2.16 has a stationary ergodic solution with X 0 p x 0 + κ p /1 L p =: K p. Also the functional dependence measure θ i,p L i p X 0 X 0 p 2K p L i p. We now apply Theorem 1 to the process X i. Assume EX i = 0 and let A = 1/2 1/p. Then j=n+1 θ j,p + n A n ja θ j,p j AL L j j p + 2K p n p j=n+1 min L j p, n A j A L j p Elementary manipulations show that there exists a constant C A such that, if L p 1 1/n, the right hand side of 2.19 is less than C A /1 L p, while if L p 1 1/n, it is less than C A /n A 1 L p 1+A. Combining these two cases,

8 1264 WEIDONG LIU, HAN XIAO AND WEI BIAO WU we obtain an upper bound for 2.19 as C A min1/1 L p, 1/n A 1 L p 1+A. Hence by Theorem 1, we obtain S n p c p n 1/2 θ j,2 + c p n 1/2 K 2 + c p n 1/p K p +n 1/2 K p C A min 1 1 L p, 1 n A 1 L p 1+A 2K 2c p n 1/2 1 L 2 + c pn 1/2 K p 1 L p min1, n A 1 L p A If there exists a positive constant λ 0 such that L p < 1 λ 0, then the second term in 2.20 has magnitude On 1/p, which together with the first term mimic the classical Rosenthal inequality 1.4. If this well-separateness condition is violated, then we can have a quite interesting behavior. Let L p = 1 r n with r n 0. If r n 1/n, then the second term in 2.20 has order On 1/p rn 1/p 5/2. If r n 1/n, then the order becomes On 1/2 rn 2. As a specific example, consider the ARCH model with F εi x = ε i a 2 +b 2 x 2 1/2, where the ε i are i.i.d. standard normal and a, b > 0 are parameters. Then L p = bε 0 p. Choose p 0 such that L p0 = 1. Note that E X 0 p 0 = since, for some C > 0, PX 0 x Cx p 0 as x see Goldie Since L p = L p0 + O p p 0, if p p 0 = Or n, L p 1 = Or n. Overall, as r n 0, the second term in 2.20 has bound n 1/2 rn 2 min1, nr n A. 3. A Nagaev-type Inequality Nagaev-type inequalities under dependence have been much less studied than Rosenthal-type inequalities for dependent random variables. If we just apply the Markov inequality and 2.5, we only obtain that PS n x S n p p x p n p/2 = O x p. In comparison, the bound On/x p in 1.3 is much sharper. We also observe that, according to Borovkov 1972, the Nagaev inequality 1.2 also holds for Sn, the maximum of absolute partial sums, when X 1,..., X n are mean zero independent variables: PS n x p µn,p p x p + 2 exp 2x 2 e p p µ n,2 We will need the Gaussian-like tail function G q y = e jq y 2, y > 0, q >

9 PROBABILITY AND MOMENT INEQUALITIES UNDER DEPENDENCE 1265 Note that sup y 1 G q ye y2 = G q 1e. Hence if y 1, G q y G q 1ee y2. In this section C, C 1,... denote constants that do not depend on x and n. Theorem 2. i Assume that ν := Then for all x > 0, µ j <, where µ j = j p/2 1 θ p j,p 1/p PSn n x c p ν p+1 x p + X 1 p p +4 exp c pµ 2 j x2 nν 2 θj,2 2 ii Assume that Θ m,p = Om α, α > 1/2 1/p. constants C 1, C 2 such that for all x > 0, PS n x C 1Θ p 0,p n x p + 4G 1 2/p + 2 exp c px 2 n X Then there exist positive C2 x. 3.4 nθ0,p iii If Θ m,p = Om α, α < 1/2 1/p, then a variant of 3.4 holds: PS n x C 1Θ p 0,p np1/2 α x p C 2 x + 4G p 2/p+1 n 2p 1 2αp/2+2p. Θ 0,p 3.5 If the X i are independent, then θ j,2 = θ j,p = 0 for all j 1, and hence 3.3 reduces to the traditional Nagaev inequality 3.1. We remark that those inequalities are non-asymptotic and they hold for any n and x. The exponential term in 3.3 decays to zero very quickly as j. If x = nν 1+1/p y with y > 0, then µ 2 j x2 /nν 2 θ 2 j,2 j1 2/p y 2 and exp c pµ 2 j x2 nν 2 θj,2 2 exp c p j 1 2/p y 2 is an upper bound for the second term in 3.3. Consider the two cases y 1 and y < 1 separately, we conclude that there exists constants c p and c p such that the second term in 3.3 is bounded by c p exp[ c p x 2 /nν 2+2/p ]. We now compare conditions on dependence in i and ii. Consider the special case θ j,p = j β. Then 3.2 requires β > 3/2, and ii only requires β > 3/2 1/p. On the other hand, 3.2 implies Θ m,p = om 1/p 1/2 since 2m 1 j=m θ j,p 2m 1 j=m θ1/q j,p q, where q = 1+1/p, and m p/2 1/1+p 2m 1 j=m θ1/q j,p

10 1266 WEIDONG LIU, HAN XIAO AND WEI BIAO WU 2m 1 j=m µ j = o1. In case iii the dependence is stronger; as compensation, we need a larger numerator n p1/2 α than n, and the term n 2p 1 2αp/2+2p in 3.5 is also larger than n. Proof of Theorem 2. i We use the decomposition 2.6. Let λ j, j = 1,..., n be a positive sequence such that n λ j 1. For i Z and l N, write i l := i/l l. Define M i,j = i X k,j X k,j 1 and Mn,j = max M i,j i n k=1 Then S n,n S n,0 = n k=1 X k,n X k,0 = n M n,j. For each 1 j n, as in 2.10, let Y i,j = ij n k=1+i 1j X k,j X k,j 1, 1 i l, where l = n/j + 1. Define Ws,j e = s 1 + 1i /2 Y i,j and Ws,j o = s 1 1i /2 Y i,j. Then PMn,j 3λ j x P max M i i n j,j 2λ j x + P max M i i n j,j M i,j λ j x P max Wl,j e λ jx s l + n j P max i j M i,j λ j x + P max Wl,j o λ jx s l. 3.7 Since Y 2,j, Y 4,j,..., are independent, from 3.1 and 2.12 we obtain Pmax Wl,j e λ n/je Y 2,j p jx c p s l λ j x p + 2 exp n j p/2 1 θ c p x p λ p j p j,p λ j x 2 c p nθ 2 j,2 λ j x exp c p nθ 2 j, A similar inequality holds for Ws,j o. For the last term in 3.7, noting that, by 2.4 and the Doob inequality, E max M i,j p 2 p 1 E M j,j p + max 1 i j 1 i j we have PMn,j n j p/2 1 p θ 3λ j x c p x p λ p j j X k,j X k,j 1 p c p j p/2 θ p j,p, k=i j,p λ j x exp c p. 3.9 nθ 2 j,2 Since X 1,0 X 1 2 and X 1,0 p X 1 p, by 3.1, we have P max S n X 1 p p i,0 x c p 1 i n x p + 2 exp c px 2 n X

11 PROBABILITY AND MOMENT INEQUALITIES UNDER DEPENDENCE 1267 Choose λ j = µ j /ν. By 2.6 and 2.15, we obtain 3.3 in view of Θ p/p+1 n+1,p θ p/p+1 l p/2 1/p+1θ p/p+1 l,p n l,p, PS n 5x l=n+1 l=n+1 PMn,j 3λ j x +P max S i S i,n x + P max S i,0 x. 1 i n 1 i n ii Let 0 = τ 0 < τ 1 <... < τ L = n be a sequence of integers. As in 3.6, write S i,n S i,0 = L l=1 M i,l, where M i,l = Then there exists a constant c p > 0 such that i X k,τl X k,τl 1. M i,l p τ l c p θ i,p =: c p θl,p and Mi,l 2 θ i,2 := θ l,2. i +τ i l 1 +τ l 1 k=1 τ l Let M n,l = max i n M i,l. Again let λ 1,... λ L be a positive sequence with L l=1 λ l 1. With the argument in 3.7, 3.8, and 3.9, we similarly obtain P M n,l 3 λ n τ l x c p x p p/2 1 l θ p l,p λ p l λ l x exp c p n θ 2 l,2 Let µ l = τ p/2 1 l θ p l,p Θ p 0,p 1/p+1, ν L = L l=1 µ l, and λ l = µ l / ν L. Using 3.10, we have L PSn 5x P M n,l 3 λ l x l=1 +P max S i S i,n x + P max S i,0 x 1 i n 1 i n n L λ l x 2 c p x p νp+1 L + 4 exp c p n θ l=1 l,2 2 n p/2 Θ p n+1,p n X 0 p p +c p x p + c p x p + 2 exp c px 2 n X We now show that the above relation implies 3.4. Let A = 1/2 1/pp/1 + p and B = αp/1 + p. Since θ l,p Θ τl 1 +1,p, we have ν L = L l=1 τ p/2 1 l θ p l,p 1/p+1.

12 1268 WEIDONG LIU, HAN XIAO AND WEI BIAO WU = O1 = O1 L l=1 L τ 1/2 1/p l τ A l τ B l=1 l 1. τ α l 1 p/1+p If A < B, choose ρ A/B, 1, L = 1 + log log n/log ρ 1, and τ l = n ρl l, 1 l L. Since A < ρb and τl A /τl 1 B nρl l A ρb, elementary calculation shows that L l=1 τ l A /τl 1 B = O1. Let x = nθ 0,p ν 1+1/p y, then λ l x 2 n θ 2 l,2 = µ2 l ν2/p L Θ2 0,p y2 θ l,2 2 µ2+2/p l Θ 2 0,p y2 θ l,2 2 L = τ 1 2/p l y 2, and the second term on the right hand side of 3.10 is bounded by l=1 4 exp c p l 1 2/p y 2, which implies 3.4. If A > B, let r = A/B 1/A B, τ l = n/r L l, and L = 1 + log n 1/log r. Then ν L = On A B. Since A B = 1/2 1/p αp/1 + p, we have, by 3.10, P S n 5x = Onp1/2 α Θ p 0,p x p + 4 Let x = nθ 0,p ν L y, then λ l x 2 n θ 2 l,2 and 3.5 follows. = µ2 l Θ2 0,p y2 θ 2 l,2 L λ l x 2 exp c p + 2 exp c px 2 n X 0 2. l=1 n θ 2 l,2 = θ l,p /Θ 0,p 2p/p+1 τ p 2/p+1 l y 2 θ l,2 /Θ 0,p 2 τ p 2/p+1 l y 2, Remark 1. We consider the boundary case of Theorem 2 with α = 1/2 1/p. Recall the proof of the Theorem 2 for the definitions of A, B, and ν L. Now we have A = B. Let τ l = 2 l for 1 l < L, where L = log n/log 2, we have ν L = Olog n. Then the argument there implies the following upper bound: there exist positive constants C 1 and C 2 such that for all x > 0, P Sn x C 1Θ p 0,p nlog np+1 [ C 2 x ] x p + 4G p 2/p+1. Θ 0,p n log n Example 2. Kernel Density Estimation Consider estimating the marginal density of the linear process Y i = j=0 a jε i j, where a j j=0 are real coefficients and the ε j are i.i.d. innovations with density f ε satisfying f := sup u [f ε u+ f εu ] <. Based on the data Y 1,..., Y n, we estimate the marginal density f of Y i by ˆf n u = 1 u Yi K, nb n b n

13 PROBABILITY AND MOMENT INEQUALITIES UNDER DEPENDENCE 1269 where b n is the bandwidth sequence with b n 0 and nb n, and K is a bounded kernel function with support [ 1, 1]. We want an upper bound for the tail probability P ˆf n u E ˆf n u x. The latter problem has been studied for i.i.d. random variables; see Louani 1998, Gao 2003, and Joutard 2006, among others. However, the case of dependent random variables has been largely untouched. Assume a 0 = 1. Let W i 1 = a jε i j = Y i ε i, F i 1 =..., ϵ i 2, ϵ i 1, and where ˆf nu = 1 nb n X i = 1 1 [ u Yi ] Fi 1 E K = 1 b n n Kvf ε u W i 1 vb n dv. X i, We compute the functional dependence measure of X i. Let W i 1 = W i 1 a i ε 0 + a i ε 0, where ε 0, ε l, l Z, are i.i.d. Then X i X i a i ε 0 ε 0 κ, where 1 κ = f 1 Kv dv. Assume E ε 0 q <, q > 0. Since X i κ and X i κ, we obtain E X i X i p E[min2κ, a i ε 0 ε 0 κ p ] 2κ p E[min1, a i ε 0 ε 0 minp,q ] 2κ p a i minp,q E[ ε 0 ε 0 minp,q ]. Hence the the functional dependence measure of X i, θ i,p = O a i min1,q/p. Assume that i p/2 1 a i minq,p 1/p+1 <. By Theorem 2i, there exists constants C 1, C 2, C 3 > 0 such that, for all y > 0, P[ ˆf nu E ˆf nu y] = P X X n nex 1 ny C 1n ny p + C 3 exp C 2ny n Since D i := Ku Y i /b n E[Ku Y i /b n F i 1 ], i = 1,..., n, are martingale differences bounded by K 2 = 2 sup u Ku and EDi 2 F i 1 b n K 3, where 1 K 3 = f 1 K2 vdv, by Freedman 1975 s martingale exponential inequality, we obtain P[ f n u ˆf [ nu y] 2 exp [ = 2 exp nb n y 2 ] 2nb n yk 2 + 2nb n K 3 nb n y 2 2yK 2 + 2K 3 ]. 3.12

14 1270 WEIDONG LIU, HAN XIAO AND WEI BIAO WU For sufficiently large n, b n K 3. So by 3.11 and 3.12, we have the upper bound [ nb n y 2 ] P[ f n u Ef n u 2y] 3 exp + C 1n 2yK 2 + 2K 3 ny p. Hence, if y log n/ nb n, the tail probability P[ f n u Ef n u 2y] has an upper bound with order n/ny p = n 1 p y p. 4. Extension to Non-stationary Processes The inequalities for the stationary case can be generalized to causal nonstationary processes without essential difficulties. Consider the non-stationary process X i = g i, ε i 1, ε i, 4.1 where ε i, i Z, are i.i.d. and the g i are measurable functions. If g i does not depend on i, then 4.1 reduces to the stationary process 1.5. For any random vector X 1,..., X n, one can always find g 1,..., g n and independent random variables ε i uniformly distributed over [0, 1] such that X i n and g iε 1,..., ε i n have the same distribution see for example Rosenblatt 1952 and Wu and Mielniczuk We define a uniform functional dependence measure. Again let ε i, ε j, i, j Z, be i.i.d. and assume for all i that E X i p <, p > 2, and EX i = 0. For m 0 let θm,p = sup X i g i, ε i m 1, ε i m, ε i m+1,..., ε i p, i and define the tail sum Θ m,p = j=m θ j,p. A careful check of the proofs of Theorems 1 and 2 suggest that they remain valid if we instead use the uniform functional dependence measure θm,p. The details are omitted. Acknowledgements We are grateful to Dr. Xiaohui Chen and two referees for their helpful comments and suggestions that resulted in a much improved version. Weidong Liu s research was supported by NSFC, Grant No , the Program for Professor of Special Appointment Eastern Scholar at Shanghai Institutions of Higher Learning, Foundation for the Author of National Excellent Doctoral Dissertation of PR China and the startup fund from SJTU. Han Xiao s research was supported in part by the US National Science Foundation DMS Wei Biao Wu s research was supported in part by the US National Science Foundation DMS and DMS

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