Darling and ErdËos (1956) proved the following remarkable limit theorem ... below that a DarlingâErdËos-type theorem for self-normalized sums holds true if.
The Annals of Probability 2003, Vol. 31, No. 2, 676–692 © Institute of Mathematical Statistics, 2003
˝ THEOREM FOR SELF-NORMALIZED SUMS1 DARLING–ERDOS B Y M IKLÓS C SÖRG O˝ , BARBARA S ZYSZKOWICZ
AND
Q IYING WANG
Carleton University, Carleton University and Carleton University and Australian National University Let X, X1 , X2 , . . . be i.i.d. nondegenerate random variables, Sn = n 2 2 j =1 Xj and Vn = j =1 Xj . We investigate the asymptotic behavior in distribution of the maximum of self-normalized sums, max1≤k≤n Sk /Vk , and the law of the iterated logarithm for self-normalized sums, Sn /Vn , when n
X belongs to the domain of attraction of the normal law. In this context, we establish a Darling–Erd˝os-type theorem as well as an Erd˝os–Feller– Kolmogorov–Petrovski-type test for self-normalized sums.
1. Introduction and main results. Let X, X1 , X2 , . . . be a sequence of nondegenerate i.i.d. random variables and put Sn = nj=1 Xj for their partial sums, n ≥ 1. Darling and Erd˝os (1956) proved the following remarkable limit theorem for the maximum of the standardized sums. R ESULT A.
If EX = 0 and E|X|3 < ∞, then, for every t ∈ R,
(1)
√ lim P a(n) max Sk /(σ k) ≤ t + b(n) = exp(−e−t ),
n→∞
1≤k≤n
where σ 2 = EX2 , a(n) = (2 log log n)1/2 and b(n) = 2 log log n + 12 log log log n − 12 log(4π ). We assume throughout that log x = log(max{e, x}). Darling and Erd˝os (1956) actually established Result A for independent random variables that are not necessarily identically distributed. Several extensions have relaxed their third-moment condition in the i.i.d. case. Oodaira (1976) and Shorack (1979) independently showed that (1) holds when an absolute moment of order 2 + δ is finite for some δ > 0. Einmahl (1989) and Einmahl and Mason (1989) proved that the Darling–Erd˝os theorem holds for i.i.d. random variables whenever (2)
EX2 I(|X|≥x) = o (log log x)−1
as x → ∞.
Received July 2001; revised March 2002. 1 Supported by NSERC Canada grants of M. Csörg˝o and B. Szyszkowicz at Carleton University.
AMS 2000 subject classifications. Primary 60F05, 60F15; secondary 62E20. Key words and phrases. Darling–Erd˝os theorem, Erd˝os–Feller–Kolmogorov–Petrovski test, selfnormalized sums.
676
˝ THEOREM DARLING–ERDOS
677
Einmahl (1989) showed the condition (2) to also be necessary in the i.i.d. case. 2 Einmahl √ (1989) also concluded that if only EX = 0 and EX < ∞ are assumed, then σ k in Result A needs to be replaced by the less natural Bk as in his theorem that follows. If EX = 0 and EX2 < ∞, then, for every t ∈ R,
R ESULT B.
lim P a(n) max Sk /Bk ≤ t + b(n) = exp(−e−t ),
(3) where
n→∞
Bn2
=
n
j =1
1≤k≤n
EX2 I
√ (|X|≤ j /(log log j )2 ) .
For an extension of the Darling–Erd˝os theorem to stable law, we refer to Bertoin (1998). In this paper, we show that Results A and B can be merged into one result via the use of the natural random normalizer. Furthermore, in this context even the second-moment condition is not required anymore, for it is seen below that a Darling–Erd˝os-type theorem for self-normalized sums holds true if only X belongs to the domain of attraction of the normal law under some weak additional conditions. Write Vn2 = nj=1 Xj2 and l(x) = EX2 I(|X|≤x) . The following is our main theorem. T HEOREM 1. Suppose that EX = 0 and l(x) is a slowly varying function at ∞, satisfying l(x 2 ) ≤ Cl(x) for some C > 0. Then, for every t ∈ R, we have
(4)
lim P a(n) max Sk /Vk ≤ t + b(n) = exp(−e−t ).
n→∞
1≤k≤n
The proof of Theorem 1 is based on an extension of the truncation techniques of Feller (1946) and Theorem 1 of Einmahl and Mason (1989) for the maximum of normalized sums of bounded independent random variables. Utilizing this method, we also succeed in refining the self-normalized law of the iterated logarithm (LIL). In fact, we obtain the following Erd˝os–Feller–Kolmogorov–Petrovski (EFKP)type test for self-normalized sums [cf. Petrovski (1935), Erd˝os (1942) and Feller (1943, 1946)]. T HEOREM 2. Suppose that EX = 0 and l(x) is a slowly varying function at ∞, satisfying l(x 2 ) ≤ Cl(x) for some C > 0. Then (5)
P (Sn ≥ Vn φn , i.o.) = 0
or = 1
accordingly as (6)
J (φ) ≡
∞ φn −φn2 /2 e 0 and
l(s) (log log n)4 ηn = inf s : s ≥ b + 1, 2 ≤ . s n Furthermore, we let Zj = Xj I(|Xj |>ηj ) , Sn∗
=
Yj = Xj I(|Xj |≤ηj ) , n j =1
Yj∗ ,
Bn2
=
n j =1
Yj∗ = Yj − EYj , EYj∗2 .
˝ THEOREM DARLING–ERDOS
679
Since l(x) is a nondecreasing function, slowly varying at ∞, it follows that
P (|X| ≥ x) = o l(x)/x 2
(7) (8)
ηn → ∞
and E|X|I(|X|≥x) = o l(x)/x ,
and nl(ηn ) = ηn2 (log log n)4
for every large enough n
and, under the condition that l(x 2 ) ≤ Cl(x) for some C > 0, we also have (9)
l(ηk ) ≤ l(n2 ) ≤ Cl(n) ≤ Al(ηn )
for n ≤ k ≤ n3/2 and n large enough,
as well as Bn2 ∼
n j =1
EYj2 ∼ nl(ηn ) ∼ ηn2 (log log n)4 .
Consequently, under their respective conditions, we may assume without loss of generality that (8) and (9) hold for n ≥ 1, as well as that Bn2 = nl(ηn ) = ηn2 (log log n)4
(10)
for n ≥ 1.
It will be seen that these assumptions will not affect the proofs of the main results, which are based on the following six lemmas. L EMMA 1.
We have ∞
(11)
P |X| ≥ ηk (log log k)3 < ∞,
k=1 ∞
1 P (|X| ≥ ηk ) < ∞. (log log k)6 k=1
(12) P ROOF. for k ≥ 1,
Write τj = ηj (log log j )3 . It follows in terms of (8) and (9) that, ∞
∞ 1 1 = 2 j l(ηj )(log j )(log log j )2 j =k τj log j j =k 3/2
k A ≥ j −1 l(ηk )(log k)(log log k)2 j =k
(13)
≥
A Ak = 2. 2 l(ηk )(log log k) τk
By using (13) and noting that τj +1 ≤ j 2 for j large enough, we get ∞ j =1
P (|X| ≥ τj ) =
∞ k=1
kP (τk ≤ |X| < τk+1 )
˝ B. SZYSZKOWICZ AND Q. WANG M. CSÖRGO,
680
=
∞
τk2 P (τk ≤ |X| < τk+1 )
k=1
≤A
∞
τk2 P (τk ≤ |X| < τk+1 )
k=1
≤A ≤A
k τk2
∞
1
2 j =1 τj log j
∞
1
2 j =k τj log j
EX2 I(|X|≤τj +1 )
∞
1 < ∞. j log j (log log j )2 j =1
This proves (11). Similarly, we obtain that ∞
1 P (|X| ≥ ηj ) (log log j )6 j =1 ∞
≤
P (ηk ≤ |X| < ηk+1 )
∞
ηk2 P (ηk ≤ |X| < ηk+1 )
k=1
≤A ≤A
(log log j )−6
j =1
k=1
≤A
k
∞
1
2 j =1 τj log j
k τk2
EX2 I(|X|≤ηj +1 )
∞
1 < ∞. j log j (log log j )2 j =1
The proof of Lemma 1 is now complete. L EMMA 2.
We have n
(14)
(|Zj | + E|Zj |) ≤ 5Bn (log log n)2
a.s.
j =1
P ROOF. Let τj = Bj log log j and Zj∗ = Xj I(ηj 0, and, hence, for j large enough, k 2 (j ) k=k1 (j )
k −1/10 ≤ (log j )−1/20 .
˝ B. SZYSZKOWICZ AND Q. WANG M. CSÖRGO,
684
This, together with (12) and (24), implies that ∞
P (Gk ∩ Hk ) ≤
k=1
∞
k −1/10
∞
P (|X| ≥ ηj )
j =mk
k=1
≤
n k −1
P (|X| ≥ ηj )
j =1
≤A
k 2 (j )
k −1/10
k=k1 (j )
∞
1 P (|X| ≥ ηj ) < ∞. 6 (log log j ) j =1
This completes the proof of (18). We next prove (19). Let N1 = {k : nj=1 E|Zj | ≤ 12 Bn / log log n for all nk−1 ≤ n ≤ nk } and N2 = N − N1 . Using (12), we have
k −1/10 ≤ A
k∈N2
k −1 n
k∈N2
≤A ≤A
1 3 n=nk−1 n(log log n)
∞
n 1 E|Zj | nBn (log log n)2 j =1 n=1 ∞
E|Zj |
j =1
≤A ≤A ≤A ≤A
∞
1
n3/2 l 1/2 (ηn )(log log n)2 n=j
∞
∞ 1 E|X|I(ηk ≤|X| an + bn u) = ev−u < ∞. n→∞ P (Sn > an + bn v)
1 < lim
Therefore, via (33)–(35), Lemma 6 follows immediately from Theorem 2 of Feller (1970). We are now ready to prove the main results.
˝ B. SZYSZKOWICZ AND Q. WANG M. CSÖRGO,
688
P ROOF
OF
Write Kn = exp(log1/3 n),
T HEOREM 1.
1 = k :
k
|Zj | ≤ Bk / log log k ,
j =1
2 = k :
k
E|Zj | ≤ Bk / log log k
j =1
and = {k : Kn ≤ k ≤ n} ∩ 1 ∩ 2 . Recalling (27), we obtain from Theorem 3.1 in Shao (1995) that lim supn→∞ Sn∗ /(2Bn2 log log n)1/2 = 1 a.s. Thus,
(36)
max Sk∗ /Bk ≤ 2 log log Kn ≤ 2/3 log log n
1≤k≤Kn
a.s.
√ for n sufficiently large. Since b(n) > a(n) 2 log log n, using (20), (21) and (36), it is easy to see that a(n) max Sk∗ /Bk − b(n) → −∞ k ∈ /
a.s.
Similarly, using (18), (19) and lim sup Sn /(2Vn2 log log n)1/2 = 1
(37)
a.s.
n→∞
[cf. Griffin and Kuelbs (1989)], we get a(n) max Sk /Vk − b(n) → −∞ k ∈ /
a.s.
These facts combined with Lemma 7 of Einmahl (1989) and Lemma 5, together imply that Theorem 1 will follow if we can prove (38)
Sk Sk∗ Ln ≡ a(n) max − = o(1) k∈ Vk Bk
a.s.
Recalling (12), we get ∞ (log log j )2 j =1
Bj4 ≤ A
(39)
∞
k=1 ∞
EX4 I(|X|≤ηj )
EX4 I(ηk
