The Annals of Probability 2003, Vol. 31, No. 3, 1228–1240 © Institute of Mathematical Statistics, 2003
DONSKER’S THEOREM FOR SELF-NORMALIZED PARTIAL SUMS PROCESSES B Y M IKLÓS C SÖRG O˝ ,1 BARBARA S ZYSZKOWICZ1
AND
Q IYING WANG
Carleton University, Carleton University and Australian National University Let X, X1 , X2 , . . . be a sequence of nondegenerate i.i.d. random variables with zero means. In this paper we show that a self-normalized version of Donsker’s theorem holds only under the assumption that X belongs to the domain of attraction of the normal law. A thus resulting extension of the arc sine law is also discussed. We also establish that a weak invariance principle holds true for self-normalized, self-randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero, if and only if max1≤j ≤n |Xj |/Vn →P 0, as n → ∞, where Vn2 = nj=1 Xj2 .
1. Introduction and main results. Let X, X1 , X2 , . . . be a sequence of nondegenerate i.i.d. random variables and let Sn =
n
Xj ,
j =1
Vn2 =
n j =1
Xj2 ,
n = 1, 2, . . . .
The classical weak invariance principle states that, on an appropriate probability space, as n → ∞, (1)
[nt] 1 1 sup √ (Xj − EXj ) − √ W (nt) = oP (1) nσ n 0≤t≤1 j =1
if and only if
Var(X) = σ 2 < ∞,
where {W (t), 0 ≤ t < ∞} is a standard Wiener process. This invariance principle in probability is a stronger version of Donsker’s classical functional central limit theorem. The normalizer (nσ 2 )−1/2 in (1) is that in the classical central limit theorem when Var(X) < ∞. In contrast to the well-known classical central limit theorem, Giné, Götze and Mason (1997) obtained the following self-normalized version of the central limit theorem. As n → ∞,
(2)
n 1 D (Xj − EXj ) → N (0, 1) Vn j =1
if and only if
x 2 P (|X| > x) = 0. x→∞ EX 2 I(|X|≤x) lim
Received October 2001; revised May 2002. 1 Supported by NSERC Canada grants.
AMS 2000 subject classifications. Primary 60F05, 60F17; secondary 62E20. Key words and phrases. Donsker’s theorem, self-normalized sums, arc sine law.
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SELF-NORMALIZED DONSKER THEOREM
The latter condition is well known to be equivalent to saying that X belongs to the domain of attraction of the normal law. This beautiful theorem was conjectured by Logan, Mallows, Rice and Shepp (1973). For a short summary of developments that have eventually led to Gine, Götze and Mason (1997), we refer to the Introduction of the latter paper. The result in (2) shows that when the normalizer in the classical central limit theorem is replaced by an appropriate sequence of random variables then the central limit theorem holds under a weaker moment condition than in the classical case. Thus, in the light of (2), it is natural to ask whether a self-normalized version of the weak invariance principle (1) could also hold under the same weaker assumption. As the following theorem shows, the answer to this paramount question is affirmative. T HEOREM 1.
As n → ∞, the following statements are equivalent:
(a) EX = 0 and X is in the domain of attraction of the normal law. (b) S[nt0 ] /Vn →D N (0, t0 ) for t0 ∈ (0, 1]. (c) S[nt] /Vn →D W (t) on (D[0, 1], ρ), where ρ is the sup-norm metric for functions in D[0, 1], and {W (t), 0 ≤ t ≤ 1} is a standard Wiener process. (d) On an appropriate probability space for X, X1 , X2 , . . . , we can construct a standard Wiener process {W (t), 0 ≤ t < ∞} such that √ sup S[nt] /Vn − W (nt)/ n = oP (1). 0≤t≤1
Assuming appropriate conditions, we mention two immediate analogs of Theorem 1 when {Xj , j ≥ 1} is a sequence of independent random variables with EXj = 0 and finite variances EXj2 . Write sn2 = nj=1 EXj2 . If the Lindeberg condition holds, namely, for all ε > 0,
sn−2
n j =1
EXj2 I(|Xj |>εsn ) → 0
as n → ∞,
then it is readily seen that Vn2 /sn2 →P 1. Hence it follows easily from classical results [e.g., Prohorov (1956)] that SKn (t) /Vn →D W (t) on (D[0, 1], ρ), where 2 ≤ ts 2 }. Kn (t) = sup{m : sm n By using a similar method as in the proof of Theorem 1 (cf. Section 2), we can also redefine {Xj , j ≥ 1} on a richer probability space together with a sequence of independent normal random variables {Yj , j ≥ 1} with mean zero and Var(Yj ) = Var(Xj ) such that [nt] sup S[nt] /Vn − Yj /sn = oP (1) 0≤t≤1 j =1
provided that the Lindeberg condition holds.
˝ B. SZYSZKOWICZ AND Q. WANG M. CSÖRGO,
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Furthermore, we prove also the following result for self-normalized, selfrandomized partial sums processes of independent random variables. T HEOREM 2. Let X1 , X2 , . . . be independent symmetric random variables around mean zero. Then P
max |Xj |/Vn → 0
(3)
1≤j ≤n
as n → ∞,
if and only if D
SK˜ n (t) /Vn → W (t)
(4)
on (D[0, 1], ρ),
where K˜ n (t) = sup{m : Vm2 ≤ tVn2 }. We mention that (3) is equivalent to the condition that X is in the domain of attraction of the normal law if {Xj , j ≥ 1} is a sequence of i.i.d. random variables [cf. O’Brien (1980)]. Also, it is readily seen that the Lindeberg condition implies (3). However, it is not clear at this moment whether or not Theorem 2 still holds for general independent random variables, that is, without assuming {Xj , j ≥ 1} to be symmetric. In the i.i.d. case, for X being symmetric, Griffin and Mason (1991) attribute to Roy Erikson the proof of (2). That Sn /Vn →D N (0, 1), as n → ∞, with X1 , X2 , . . . as in Theorem 2, is due to Egorov (1996). This result in turn inspired us to prove Theorem 2. The proofs of Theorems 1 and 2 will be given in the next section. We conclude this section with some immediate corollaries of Theorem 1, which are also of independent interest. With x ≥ 0, write
G1 (x) = P
sup W (t) ≤ x , 0≤t≤1
G3 (x) = P
1 0
G2 (x) = P
sup |W (t)| ≤ x , 0≤t≤1
W 2 (t) dt ≤ x ,
G4 (x) = P
1 0
|W (t)| dt ≤ x .
Our first corollary is an extension of the original Erd˝os and Kac (1946) invariance principle to the corresponding functionals of self-normalized sums. C OROLLARY 1. Let EX = 0 and X be in the domain of attraction of the normal law. Then, as n → ∞, we have (i) P (max1≤k≤n Sk /Vn ≤ x) → G1 (x) for x ≥ 0, and P (min1≤k≤n Sk /Vn ≤ x) → 1 − G1 (−x) for x < 0; (ii) P (max1≤k≤n |Sk |/Vn ≤ x) → G2 (x) for x ≥ 0; (iii) P (n−1 nk=1 (Sk /Vn )2 ≤ x) → G3 (x) for x ≥ 0; (iv) P (n−1 nk=1 |Sk /Vn | ≤ x) → G4 (x) for x ≥ 0.
SELF-NORMALIZED DONSKER THEOREM
1231
We note in passing that the same results also hold true for the corresponding functionals of SK˜ n (·) /Vn as in Theorem 2. Erd˝os and Kac (1947) gave a further demonstration of their (1946) invariance principle by deducing a general form of Lévy’s arc sine law (1939) via assuming a central limit theorem. Namely, let X1 , X2 , . . . be independent random variables with EXj = 0, EXj2 = 1 and assume that Lindeberg’s condition holds true, that is, as n → ∞, we have n−1/2 Sn →D N (0, 1). Then, √ lim P (n /n ≤ x) = (2/π ) arcsin x, (5) 0 ≤ x ≤ 1, n→∞
n
where n = j =1 I0x) = o(l(x)); E|X|α I(|X|≤x) = o(x α−2 l(x)) for α > 2.
P ROOF. It follows from Theorem 2 of Feller [(1966), page 275] that (a) holds if and only if (b) does. If (b) holds, then (c) follows from Lemma 6.2 of Griffin and Kuelbs (1989) with θ ↓ 0, and by noting that E|X| I(|X|≤x) = α
(6)
x 0 α
y α dP (|X| ≤ x)
= x P (|X| > x) + α
x
y α−1 P (|X| > y) dy,
0
we get (d). On the other hand, it can be easily shown that (c) implies (b) and (d) implies (b) via using (6) again. Therefore, the proof of Lemma 1 is now complete. The next result is due to Sakhanenko (1980, 1984, 1985). L EMMA 2. Let X1 , X2 , . . . be independent random variables with EXj = 0 and σj2 = EXj2 < ∞ for each j ≥ 1. Then we can redefine {Xj , j ≥ 1} on a richer probability space together with a sequence of independent N (0, 1) random variables, Yj , j ≥ 1, such that for every p > 2 and x > 0, i n i Xj − σj Yj ≥ x ≤ (Ap)p x −p E|Xj |p , P max i≤n
j =1
j =1
j =1
where A is an absolute positive constant. L EMMA 3. Let aj , j ≥ 1, be a sequence of nonnegative constants and put A(n) = nj=1 aj . If an+1 /A(n) → 0 as n → ∞, then we have (7)
A
−1/2
(n)
n−1
aj +1 A−1/2 (j ) = O(1).
j =1
If in addition A([tn])/A(n) → 1 for any t > 0 as n → ∞, then (8)
[nA(n)]−1/2
n−1
j 1/2 aj +1 A−1/2 (j ) = o(1)
j =1
and (9)
1 n−1 j aj +1 = o(1). nA(n) j =1
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SELF-NORMALIZED DONSKER THEOREM
P ROOF. To prove (7), we assume without loss of generality that aj +1 √/A(j ) ≤ 1/2 for all j ≥ 1. Noting that 1 ≤ A(j + 1)/A(j ) ≤ 3/2 for j ≥ 1 and 1 + y ≤ 1 + y/2 for y ≥ 0, we get I (n) := A−1/2 (n)
n−1
aj +1 A−1/2 (j )
j =1
= A−1/2 (n)
n−1
1/2
A1/2 (j + 1) 1 + aj +1 A−1 (j )
−1
j =1
+ A−1/2 (n)
n−1
1/2 A (j + 1) − A1/2 (j )
j =1
≤ ≤
1 −1/2 (n) 2A
1/2 1 3 2 2
n−1
aj +1 A1/2 (j + 1)A−1 (j ) + 1 − a1 A−1/2 (n)
j =1
I (n) + 1.
This implies (7), since 12 ( 32 )1/2 < 1. If A([tn])/A(n) → 1, then for any t > 0, A(n) − A([tn]) →0 A1/2 (n)A1/2 ([tn])
as n → ∞.
Hence, by using (7), letting n → ∞ and then t → 0, we have [nA(n)]−1/2
n−1
j 1/2 aj +1 A−1/2 (j )
j =1
≤ t 1/2 A−1/2 (n)
[tn]
aj +1 A−1/2 (j ) + A−1/2 (n)
j =1
≤ O(1)t 1/2 +
n−1 j =[tn]
aj +1 A−1/2 (j )
A(n) − A([tn]) = o(1). A1/2 (n)A1/2 ([tn])
This gives (8). The proof of (9) is similar, and hence omitted. This also completes the proof of Lemma 3.
We now are ready to prove that (a) implies (d). Put b = inf x ≥ 1 : l(x) > 0 and
l(s) 1 ηj = inf s : s ≥ b + 1, 2 ≤ , s j
j = 1, 2, . . . .
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˝ B. SZYSZKOWICZ AND Q. WANG M. CSÖRGO,
Furthermore, let Bn2 = nl(ηn ), Xj∗
= Xj I(|Xj |≤ηj )
Sn∗
and
=
n j =1
Xj∗ .
By Lemma 2, we can redefine {Xj , j ≥ 1} on a richer probability space together with a sequence of independent N (0, 1) random variables, Yj , j ≥ 1, such that for any x > 0, i i ∗ ∗ ∗ P max (Xj − EXj ) − σ j Yj ≥ x i≤n
j =1
(10)
≤ Ax −3
j =1
n
E|X|3 I(|X|≤ηj ) ,
j =1
where σj∗2 = Var(Xj∗ ). Let {W (t), 0 ≤ t < ∞} be a standard Wiener process such that W (n) =
n
Yj ,
n = 1, 2, 3, . . . .
j =1
We have √ sup S[nt] /Vn − W (nt)/ n
1/n≤t≤1
[nt] −1 ∗ −1/2 ≤ sup Bn σ j Yj − n W (nt) 1/n≤t≤1 j =1
(11)
[nt] −1 ∗ ∗ −1 ∗ + sup Bn S[nt] − ES[nt] − Bn σ j Yj 1/n≤t≤1 j =1
+
sup 1/n≤t≤1
1
S[nt] − 1 S ∗ − ES ∗ [nt] V B [nt] n
n
:= I1 (n) + I2 (n) + I3 (n). Therefore, (d) will follow from (a) if we can prove that, as n → ∞, (12)
Ij (n) = oP (1),
j = 1, 2, 3,
on assuming that EX = 0 and X is in the domain of attaction of the normal law. We now proceed to prove (12). Since X belongs to the domain of attraction of the normal law, that is, x 2 P (|X| > x) = o(l(x)), by Lemma 1 l(x) is a slowing
SELF-NORMALIZED DONSKER THEOREM
1235
varying function at ∞. Hence it can be easily shown that (13)
j l(ηj ) ≤ ηj2 ≤ (j + 1)l(ηj )
for j ≥ 1,
(14) l(ηj +1 )/ l(ηj ) → 1 (15) (16) (17)
as j → ∞,
|EXj∗ | ≤ E|X|I(|X|>ηj ) = o ηj−1 l(ηj ) = o(Bj /j )
∗ ∗2 ∗ 2
Var(Xj ) = EXj − (EXj ) = 1 + o(1) l(ηj )
E|Xj∗ |3 ≤ E|X|3 I(|X|≤ηn ) = o ηn l(ηn ) = o(Bn3 /n)
as j → ∞, as j → ∞, as j → ∞,
and, as n → ∞, n 1 X2 → 1 Bn2 j =1 j
(18)
in probability.
Let η0 = 0. Noting that l(ηn ) = nk=1 EX2 I(ηk−1 0, by (13) and Lemma 3, we get, as n → ∞,
(19)
n 1 EX2 I(ηj ηj ) + E|Xj |I(|Xj |>ηj ) ≥ ε
j =1
n 2n 2ε−1 E|X|I(|X|>ηn ) + 1/2 1/2 E|X|I(ηk ηj ) + E|Xj |I(|Xj |>ηj ) ) = oP (1). This, together with (18), implies that, as n → ∞, (1)
I3 (n) =
(22)
n
Bn −1 Bn |Xj |I(|Xj |>ηj ) + E|Xj |I(|Xj |>ηj ) = oP (1). Vn j =1
We continue to use the notations I1 (n) and I2 (n) introduced in (11). Noting that sup0 ε = 0.
|t−s|≤h
We first verify tightness, that is, (b). Let P and E denote conditional probability and conditional expectation respectively, given X1 , X2 , . . . . Recalling the definition of K˜ n (t), it is readily seen that for any ε > 0, P
|Xn (t) − Xn (kh)| > ε
sup kh εVn
4
Xj εj
j =K˜ n (kh)+1
≤ Aε
−4
Vn−4
τ (h)+1 n
2
Xj2
j =K˜ n (kh)+1 τn (h)+1
≤ Aε−4 Vn−2
1≤j ≤n
j =K˜ n (kh)+1
Xj2 h + max Xj2 /Vn2 ,
where τn (h) = min K˜ n {(k + 1)h}, n − 1 . Therefore, for any ε > 0,
P
sup Xn (t) − Xn (s) > ε
|t−s|≤h
≤
≤
1 + P h
1≤j ≤n
kh ε − max |Xj |/Vn
sup
P
k:kh≤1
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|Xn (t) − Xn (kh)| > ε/2
sup kh 0. Then λ is measurable with respect to (D, D), where D denotes the σ -field of subsets of D generated by the finite-dimensional subsets of D, and is continuous except on a set of Wiener measure 0. Now, if Sn (t) := S[nt] /Vn , then λ(Sn (. . .)) is exactly 1/n times the number of positive sums among S1 , . . . , Sn−1 . Hence, Theorem 1 and the continuous mapping theorem imply that (5) holds. This completes the proof of Corollary 2. Acknowledgments. The authors thank a referee and Editors for their valuable comments and suggestions. N OTE ADDED IN PROOF. While attending the 8th International Vilnius Conference on Probability Theory and Mathematical Statistics in June 2002, we learned that, using a different method than that of CsSzW (2001) and the present paper, the equivalence of (a), (b) and (c) as in Theorem 1 was also proved in Raˇckauskas, A. and Suquet, Ch. (2000, 2001).
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[email protected] [email protected]
Q. WANG C ENTRE FOR M ATHEMATICS AND I TS A PPLICATIONS M ATHEMATICAL S CIENCES I NSTITUTE AUSTRALIAN N ATIONAL U NIVERSITY C ANBERRA , ACT 0200 AUSTRALIA E- MAIL :
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