Self-normalized Cramér-Type Large Deviations for Independent Random Variables
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1 Self-normalized Cramér-Type Large Deviations for Independent Random Variables Qi-Man Shao National University of Singapore and University of Oregon
2 1. Introduction Let X, X 1, X 2,, X n be i.i.d. random variables and let S n = X i, Vn 2 = Xi 2. Chernoff s large deviation: If Ee t 0X < for some t 0 > 0, then x > EX, P ( ) 1/n Sn n x inf t 0 e tx Ee tx. Self-normalized large deviation (Shao, 1997): If EX = 0 or EX 2 =, then x > 0 ( ) 1/n P S n /V n xn 1/2 λ(x) where λ(x) = sup c 0 inf t 0 Eet(cX x( X 2 +c 2 )/2) 1
3 Cramér s moderate deviation: Assume EX = 0 and σ 2 = EX 2 <. If Ee t 0 X 1/2 < for t 0 > 0, then ( ) S P n σ n x 1 1 Φ(x) uniformly in 0 x o(n 1/6 ). If Ee t0 X < for t 0 > 0, then for x 0 and x = o(n 1/2 ) ( ) S P n σ n x { x 3 = exp λ( x } ( ) 1 + O( 1 + x ) ), 1 Φ(x) n n n where λ(t) is the Cramér s series. In particular, ( ) Sn ln P σ n x n 1 2 x2 n for x n and x n = o( n). 2
4 Self-normalized moderate deviations (Shao, 1997, 1999): If EX = 0 and EX 2 I{ X x} is slowly varying, then ln P (S n /V n > x n ) x 2 n/2 for x n and x n = o( n). If EX = 0 and E X 3 <, then P (S n /V n x) 1 Φ(x) uniformly in 0 x o(n 1/6 ). 1 3
5 What if {X n, n 1} are independent random variables? 4
6 2. Self-normalized Cramér type large deviation for independent random variables Let X 1, X 2,, X n be independent random variables with EX i = 0 and EXi 2 <. Put S n = X i, Vn 2 = Xi 2, B 2 n = EXi 2, L n,p = E X i p. Petrov (1968): Suppose there exist positive constants t 0, c 1, c 2, such that P (S n /B n x) 1 Φ(x) ln Ee tx i c i for t t 0, 1 lim sup c i 3/2 <, n n lim inf n B n/ n > 0. If x 0 and x = o( n), then { x 3 = exp λ n ( x } ( ) 1 + O( 1 + x ) ) n n n where λ n is a power series. 5
7 Wang and Jing (1999): If X i is symmetric with E X i 3 <, then { P (S n /V n x) Φ(x) A min (1+ x 3 ) L n,3 B 3 n }, 1 e x2 /2. If X 1,, X n i.i.d. with σ 2 = EX 2 1 and E X 1 10/3 <, then there exists an absolute constant 0 < η < 1 such that P (S n /V n x) Φ(x) AE X 1 10/3 σ 10/3 n e ηx2 /2 6
8 Chistyakov and Götze (1999): If X 1, X 2, are symmetric with finite third moments, then ( ) P (S n /V n x) = (1 Φ(x)) 1 + O(1)(1 + x) 3 Bn 3 L n,3 for 0 x B n /L 1/3 n,3, where O(1) is bounded by an absolute constant. 7
9 Can the assumption of symmetry be removed? 8
10 Let for x > 0. n,x = (1 + x)2 B 2 n (1 + x)3 + Bn 3 EXi 2 I { Xi >B n /(1+x)} E X i 3 I { Xi B n /(1+x)} Theorem 1 [Jing-Shao-Wang (2003)] There is an absolute constant A such that P (S n xv n ) = e O(1) n,x 1 Φ(x) for all x 0 satisfying (H1) x 2 max 1 i n EX 2 i B 2 n. (H2) n,x (1 + x) 2 /A, where O(1) A. 9
11 Theorem 1 provides a very general framework. The following results are direct consequences of the above general theorem. Theorem 2 Let {a n, n 1} be a sequence of positive numbers. Assume that a 2 n Bn/ 2 max 1 i n EX2 i and ε > 0, B 2 n EXi 2 I { Xi >εb n /(1+a n )} 0 as n. Then ln P (S n /V n x) ln(1 Φ(x)) holds uniformly for x (0, a n ). 1 The next corollary is a special case of Theorem 2 and may be of independent interest. Corollary 1 Suppose that B n c n for some c > 0 and that {Xi 2, i 1} is uniformly integrable. Then, for any sequence of real numbers x n satisfying x n and x n = o( n), ln P (S n /V n x n ) x 2 n/2. 10
12 When the X i s have a finite pth moment, 2 < p 3, we obtain Chistyakov and Götze result without assuming any symmetric condition. Theorem 3 Let 2 < p 3 and set L n,p = E X i p, d n,p = B n /L 1/p n,p. Then, P (S n /V n x) ( 1 + x ) p = 1 + O(1) 1 Φ(x) d n,p for 0 x d n,p, where O(1) is bounded by an absolute constant. In particular, if d n,p as n, we have uniformly in 0 x o(d n,p ). P (S n /V n x) 1 Φ(x) 1 By the fact that 1 Φ(x) 2e x2 /2 /(1 + x) for x 0, we have the following exponential non-uniform Berry-Esseen bound P (S n /V n x) (1 Φ(x)) A(1 + x) p 1 e x2 /2 /d p n,p holds for 0 x d n,p. 11
13 For i.i.d. random variables, Theorem 3 simply reduces to Corollary 2 Let X 1, X 2, be i.i.d. with EX i = 0, σ 2 = EX 2 1, E X 1 p < (2 < p 3). Then, there exists an absolute constant A such that P (S n /V n x) 1 Φ(x) ( (1 + x) p E X 1 p ) = 1 + O(1) n (p 2)/2 σ p for 0 x n 1/2 1/p σ/(e X 1 p ) 1/p, where O(1) A. 12
14 Question: Can condition (H1) be removed? Shao (2003): Theorem 2 remains valid under (H2). That is, if a n and ε > 0, EXi 2 I { Xi >εb n /(1+a n )} 0 as n, B 2 n then ln P (S n /V n x) ln(1 Φ(x)) holds uniformly for x (0, a n ). 1 13
15 3. Self-normalized law of the iterated logarithm Theorem 4 (Shao (2003)) Let X 1, X 2, be independent random variables with EX i = 0 and 0 < EXi 2 <. Assume that B n and that ε > 0, Bn 2 EXi 2 I { Xi >εb n /(log log B n ) 1/2 } 0 as n. Then, lim sup n S n = 1 a.s. V n (2 log log B n ) 1/2 14
16 4. Self-normalized central limit theorem Gine-Götze-Mason (1995): Let {X n, n 1} be i.i.d. Then EX = 0 and max 1 i n X i /V n 0 in probability S n /V n d. N(0, 1) Egorov (1996): If X i are independent and symmetric, then S n /V n d. N(0, 1) max 1 i n X i /V n 0 in probability 15
17 Shao and Zhou (2003): Let {X n, n 1} be independent. Suppose the following conditions are satisfied where a n satisfies Then max X i /V n 0 in probability, 1 i n {E(X i /V n )} 2 0, ( S ) n E 0 max(v n, a n ) E X2 i a 2 n + Xi 2 = 1. S n /V n d. N(0, 1) 16
18 5. An application to the Student-t statistic Let where s 2 = 1 n 1 T n = S n s n, (X i S n /n) 2. T n and S n /V n are closely related via the following identity: T n = S ( n n 1 ) 1/2. V n n (S n /V n ) 2 Hence { Sn ( n ) 1/2 } {T n x} = x V n n + x 2 1 and the results for S n /V n remain valid for T n. 17
19 6. The main idea of proof of Theorem 1 It suffices to show that and P (S n xv n ) (1 Φ(x))e A n,x P (S n xv n ) (1 Φ(x))e A n,x for all x > 0 satisfying (H1) and (H2). Let b := b x = x/b n. Observe that, by the Cauchy inequality Thus, we have xv n (x 2 + b 2 V 2 n )/(2b). P (S n xv n ) P (S n (x 2 + b 2 Vn 2 )/(2b)) = P (2bS n b 2 Vn 2 x 2 ). Therefore, the lower bound follows from the following proposition immediately. Proposition 1 There exists an absolute constant A > 1 such that P (2bS n b 2 V 2 n x 2 ) = (1 Φ(x))e O(1) n,x for all x > 0 satisfying (H1) and (H2), where O(1) A. 18
20 As for the upper bound: when 0 < x 2, this bound is a direct consequence of the Berry-Esseen bound. For x > 2, let and define X i = X i I { Xi τ}, Sn = S (i) n τ := τ n,x = B n /(1 + x) X i, V 2 n = = S n X i, V (i) n = (V 2 n X 2 i ) 1/2, B2 n = Noting that for any s, t R 1, c 0 and x 1, we have Hence, x c + t 2 t + (x 2 1)c, X 2 i, E X i 2. {s + t x c + t 2 } {s (x 2 1) 1/2 c}. P (S n xv n ) P ( S n x V n ) + P (S n xv n, max X i > τ) 1 i n P ( S n x V n ) + P (S n xv n, X i > τ) P ( S n x V n ) + P ( S n x V n ) + P (S (i) n P (S (i) n 19 (x 2 1) 1/2 V (i) n, X i > τ) (x 2 1) 1/2 V n (i) )P ( X i > τ).
21 By the inequality (1 + y) 1/2 1 + y/2 y 2 for any y 1, we have P ( S n x V n ) = P ( S n x( B 2 n + ( X i 2 E X i 2 )) 1/2 ) ( P Sn x B { n B ( X n 2 i 2 E X i 2 ) 1 B4 ( ( X i 2 E X }) i 2 )) 2 n := K n. Thus, the upper bound follows from the next two propositions: Proposition 2 There is an absolute constant A such that P (S (i) n xv n (i) ) (1 + x 1 1 ) exp( x 2 /2 + A n,x ) 2πx Proposition 3 There exists an absolute constant A such that K n (1 Φ(x))e A n,x for all x > 2 satisfying conditions (H1) and (H2). 20
22 References [1] Shao, Q.M. (1997). Self-normalized large deviations. Ann. Probab. 25, [2] Shao, Q.M. (1999). A Cramér type large deviation result for Student s t-statistic. J. Theoret. Probab. 12, [3] Jing, B.Y., Shao, Q.M. and Wang, Q.Y. (2003). Selfnormalized Cramér type large deviations for independent random variables. Ann. Probab. 31, [4] Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23, [5] Griffin, P. and Kuelbs, J. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab. 17, [6] Logan, B. F., Mallows, C. L., Rice, S. O. and Shepp, L. A. (1973). Limit distributions of self-normalized sums. Ann. Probab. 1, [7] Wang, Q. Y. and Jing, B.-Y. (1999). An exponential nonuniform Berry-Esseen bound for self-normalized sums. Ann. Probab. 27, [8] Chistyakov, G. P. and Götze, F. (1999). Moderate deviations for self-normalized sums. Preprint. 21
23 [9] Gine, E., Götze, F. and Mason, D. M. (1997). When is the Student t-statistic asymptotically standard normal? Ann. Probab. 25, [10] Hahn, M.G., Mason, D.M, Weiner, D.C. (eds) (1991). Sums, Trimmed Sumsand Extremes. Birkhauser, Boston. 22
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