The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT
We prove a new bound for the Rosenblatt coefficient of the normalized partial sums of a sequence of m-dependent random variables; this bound is used to prove a general result, from which the Almost Sure Central Limit Theorem can be deduced.
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irk-123456789-45332009-11-26T12:00:40Z The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT Giuliano, R. We prove a new bound for the Rosenblatt coefficient of the normalized partial sums of a sequence of m-dependent random variables; this bound is used to prove a general result, from which the Almost Sure Central Limit Theorem can be deduced. 2008 Article The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT / R. Giuliano // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 30–38. — Бібліогр.: 9 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4533 519.21 en Інститут математики НАН України |
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We prove a new bound for the Rosenblatt coefficient of the normalized partial sums of a sequence of m-dependent random variables; this bound is used to prove a general result, from which the Almost Sure Central Limit Theorem can be deduced. |
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Giuliano, R. The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT |
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Giuliano, R. |
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Giuliano, R. |
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The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT |
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The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT |
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The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT |
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The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT |
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The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT |
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rosenblatt coefficient of dependence for m–dependent random sequences with applications to the asclt |
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Інститут математики НАН України |
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The Rosenblatt coefficient of dependence for m–dependent random sequences with applications to the ASCLT / R. Giuliano // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 30–38. — Бібліогр.: 9 назв.— англ. |
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2025-07-02T07:45:18Z |
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2025-07-02T07:45:18Z |
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1836520381331013632 |
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Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 30–38
UDC 519.21
RITA GIULIANO
THE ROSENBLATT COEFFICIENT OF
DEPENDENCE FOR m–DEPENDENT RANDOM
SEQUENCES WITH APPLICATIONS TO THE ASCLT
We prove a new bound for the Rosenblatt coefficient of the normalized partial sums
of a sequence of m-dependent random variables; this bound is used to prove a general
result, from which the Almost Sure Central Limit Theorem can be deduced.
Introduction
Let (Xn)n∈N be a sequence of normalized centered i. i. d random variables. Put
Sn = X1 + · · ·+Xn, Un =
Sn√
n
.
In paper [4], it was proved that
(1.1) sup
A,x
∣∣P (Up ∈ A, Uq ≤ x)− P (Up ∈ A)P (Uq ≤ x)
∣∣ ≤ H 4
√
p
q
,
where H is a suitable constant depending on the sequence (Xn)n∈N only and where the
sup is taken over A ∈ B(R) and x ∈ R.
It is well known that covariance inequalities of the Rosenblatt type such as (1.1) are
a crucial tool in the proof of Almost Sure Limit Theorems, see papers [2], [5], and [9] for
some literature on this topic.
Here, we deal with a more general case than the one, considered in [4], of a sequence
of i.i.d random variables. More precisely, the aim of the present paper is twofold: first,
in Theorem (2.3), we prove an inequality similar to (1.1) for the case of a sequence
of m–dependent random variables (Xn)n∈N. Note that we do not assume the identical
distribution of (Xn)n∈N; note, moreover, that the constant H in the second member of
our inequality (see the statement of Theorem (2.3)) is absolute.
Using the inequality of Theorem (2.3), we prove a general result [Theorem (2.5) of
this paper] which is, in some sense, a generalization of the ASCLT to some kind of Borel
sets A such that ∂A is not necessarily of Lebesgue measure 0. We deduce the ASCLT as
a corollary of Theorem (2.5) (Corollary (2.6)).
The paper is organized as follows: Section 2 contains the statements of the main
results [i.e. Theorem (2.3), Theorem (2.5), and Corollary (2.6)]. In Section 3, we prove
Theorem (2.3). In Section 4, we prove Theorem (2.5) and Corollary (2.6).
Throughout the whole paper, the symbol H denotes a constant which may not have
the same value in all cases.
2000 AMS Mathematics Subject Classification. Primary 60F05, Secondary 60G10.
Key words and phrases. Rosenblatt coefficient; m-dependent sequences; Almost Sure Central Limit
Theorem.
This article was partially supported by the M.I.U.R. Italy.
30
THE ROSENBLATT COEFFICIENT 31
1. The main results
Let (Xn)n∈N be a sequence of m-dependent real centered random variables with
(2.1) sup
n
E[X2+δ
n ] < +∞
for a suitable δ ∈ (0, 1].
In the sequel, we put α = δ(6δ + 8)−1. Moreover, we set Sn = X1 + X2 + · · · + Xn,
vn = V arSn,
Un =
Sn√
vn
and assume that
(2.2) lim inf
n→∞
vn
n
> 0.
The first result proved in this paper is
(2.3) Theorem. There exists an absolute constant H such that, for every pair of inte-
gers p, q with p ≤ q, the following bound holds:
sup
A,x
∣∣P (Up ∈ A, Uq ≤ x)− P (Up ∈ A)P (Uq ≤ x)
∣∣ ≤ H( 4
√
vp
vq
+
1
qα
)
,
where the sup is taken over A ∈ B(R) and x ∈ R.
Theorem (2.3) will be used to prove the second main result of this paper [Theorem
(2.5) below].
For a fixed Borel set A ⊆ R, consider the two sequences (Tn) and (Wn) defined,
respectively, as
Tn =
∑n
i=1 1A(U2i)
n
; Wn =
∑n
i=1
1
i 1A(Ui)
logn
, n ≥ 1.
Put
(2.4) φ(n) =
vn
n
.
(2.5) Theorem. In addition to the hypotheses of Theorem (2.3), assume that the se-
quence (φ(n)) defined in (2.4) is not decreasing, and let A ⊆ R be a finite union of
intervals. Then, P–a.s. the two sequences (Tn)n≥1 and (Wn)n≥1 have the same limit
points as n→∞.
Denote, by λ, the Lebesgue measure on R and, by μ, the Gaussian measure on R, i.e.
μ(A) =
∫
A
1√
2π
e−x2/2 λ(dx), A ∈ B(R).
Theorem (2.5) has the following consequence:
(2.6) Corollary (ASCLT). There exists a P–null set Γ such that, for every ω ∈ Γc,
we have
lim
n→∞
∑n
i=1
1
i 1A(Ui)
logn
= μ(A)
for every Borel set A ⊆ R such that λ(∂A) = 0.
32 RITA GIULIANO
2. The proof of Theorem (2.3)
We start with some preparatory results.
For every integer n ≥ 1, we put
Πn = sup
x∈R
∣∣∣P (Un ≤ x
)− Φ(x)
∣∣∣,
where Φ is the distribution function of the standard normal law. In [6], the following
Berry-Esseen-type result is proved:
(3.1) Theorem. Let (Xn)n∈N be a sequence of m-dependent random variables verifying
(2.1) and (2.2). Then, for every integer n,
Πn ≤ H
nα
,
where H is an absolute constant.
(3.2) Definition. The concentration function of a r.v. S is defined as
Q(ε) = sup
x∈R
P (x < S ≤ x+ ε), ε ∈ R
+.
In the sequel, we denote, by Qn, the concentration function of Un.
The following result gives an estimate of Qn. It is similar to the one given in [8] for a
sequence of i.i.d. random variables, but here the constant H is absolute (i.e. it doesn’t
depend on the sequence (Xn)n∈N).
(3.3) Lemma. There is an absolute constant H such that, for every ε ∈ R+,
Qn(ε) ≤ H
(
ε+
1
nα
)
.
Proof. Denoting the distribution function of Un by Fn, Theorem (3.1) yields
max
{∣∣Fn(x+ ε)− Φ(x+ ε)
∣∣, ∣∣Fn(x)− Φ(x)
∣∣} ≤ Πn ≤ H
nα
.
Hence,
P (x < Un ≤ x+ ε) = Fn(x + ε)− Fn(x)
≤ ∣∣Fn(x+ ε)− Φ(x+ ε)
∣∣+ ∣∣Fn(x) − Φ(x)
∣∣+ Φ(x+ ε)− Φ(x)
≤ H
nα
+
1√
2π
ε ≤ H
(
ε+
1
nα
)
.
The following lemma is stated in [1] without proof:
(3.4) Lemma. If S and T are random variables, then, for every pair of real numbers
a, b with b ≥ 0, we have
P
(
S + T ≤ a− b)− P (|T | > b
) ≤ P (S ≤ a)
≤ P (S + T ≤ a+ b
)
+ P
(|T | > b
)
.
Proof. The first inequality follows from the inclusion
{S + T ≤ a− b} ⊆ {S ≤ a} ∪ {|T | > b}.
The second inequality follows from the first one applied to the pair of random variables
S + T,−T and to the pair of numbers a+ b, b.
We now begin the proof of Theorem (2.3).
THE ROSENBLATT COEFFICIENT 33
Let p, q be two integers with p ≤ q; let (Yn)n∈N be an independent copy of (Xn)n∈N,
and put
Vq =
Y1 + · · ·Yp +Xp+1 + · · ·Xq√
vq
.
Put, moreover,
Z = Vq − Uq =
(Y1 −X1) + · · ·+ (Yp −Xp)√
vq
=
Rp√
vq
.
If we set
H = {Up ∈ A}, K = {Uq ≤ x},
our aim is to give a bound for |P (H ∩K)− P (H)P (K)|.
Let ε > 0 be any positive real number, and put
K1 = {Vq ≤ x− ε}, K2 = {Vq ≤ x+ ε}, F = {|Z| > ε}.
By Lemma (3.4) (applied to S = Uq, T = Z, a = x, b = ε), we can write
P (K1)− P (F ) ≤ P (K) ≤ P (K2) + P (F ).
Hence,
(3.5)
|P (H ∩K)−P (H)P (K)| ≤ max
{|P (H ∩K)−P (K1)P (H)+P (F )P (H)|,
|P (H ∩K)− P (K2)P (H)− P (F )P (H)|}
≤ max
{|P (H ∩K)−P (K1)P (H)|, |P (H ∩K)−P (K2)P (H)|}+ P (F ).
In what follows, we estimate the three quantities in the last member, i.e. |P (H ∩K)−
P (K1)P (H)|, |P (H ∩K)− P (K2)P (H)| and P (F ).
We start with P (F ). We have
(3.6) P (F ) = P (|Rp| > ε
√
vq) ≤
E
[|Rp|
]
ε
√
vq
≤ V ar1/2(Rp)
ε
√
vq
.
Now, since (Xn)n∈N and (Yn)n∈N are independent and have the same law,
(3.7) V ar(Rp) = 2V ar(Sp) = 2vp
.
From (3.6) and the (3.7), we conclude that
(3.8) P (F ) ≤ H
ε
√
vp
vq
.
We now pass to the terms |P (H ∩K)− P (K1)P (H)| and |P (H ∩K)− P (K2)P (H)|.
We give the details only for |P (H ∩ K) − P (K2)P (H)|, since the proof is identical for
the other quantity.
We need some more lemmas.
(3.9) Lemma. Let g be a Lipschitzian function defined on R, with Lipschitz constant
β. Then ∣∣E[g(Uq)]− E[g(Vq)]
∣∣ ≤ H β
√
vp
vq
.
Proof. Arguing as for relation (3.6) and using (3.7), we get∣∣E[g(Uq)]− E[g(Vq)]
∣∣ ≤ E[|g(Uq)− g(Vq)|
] ≤ β E[|Uq − Vq|]
= β
E[|Rp|]√
vq
≤ β V ar1/2(Rp)√
vq
≤ H β
√
vp
vq
.
34 RITA GIULIANO
In the sequel, we denote, by Q̃q, the concentration function of Vq.
(3.10) Lemma. Let z ∈ R and g = 1(−∞,z]. Then, for every η > 0, we have∣∣E[g(Uq)]− E[g(Vq)]
∣∣ ≤ H
η
√
vp
vq
+Qq(η) + Q̃q(η).
Proof. Put
h(t) =
(
1 +
z − t
η
)
1(z,z+η](t), g̃(t) = g(t) + h(t).
Then g̃ is Lipschitzian with the Lipschitz constant 1/η. So, by Lemma (3.9),
(3.11)
∣∣E[g̃(Uq)]− E[g̃(Vq)]
∣∣ ≤ H
η
√
vp
vq
.
On the other hand, h has support contained in (z, z+ η] and is bounded by 1. Hence,
we have trivially
(3.12)
∣∣E[h(Uq)− h(Vq)]
∣∣ ≤ Qq(η) + Q̃q(η).
Now, recalling that g = g̃ − h, we can write∣∣E[g(Uq)]− E[g(Vq)]
∣∣ = ∣∣E[(g̃ − h)(Uq)]− E[(g̃ − h)(Vq)]
∣∣
≤ ∣∣E[g̃(Uq)]− E[g̃(Vq)]
∣∣+ ∣∣E[h(Uq)− h(Vq)]
∣∣ ,
and the conclusion follows from relations (3.11) and (3.12).
The next lemma concerns the concentration function Q̃n of Vn. Its proof is iden-
tical to the proof of Lemma (3.3), since it is immediate to see that also the sequence
(Y1, Y2, . . . , Yp, Xp+1, . . . ) is m-dependent.
(3.13) Lemma. There is an absolute constant H such that, for every ε ∈ R
+,
Q̃n(ε) ≤ H
(
ε+
1
nα
)
.
We go back to the proof of the main result (2.3). Since H and K2 are independent,
we can write
|P (H ∩K)− P (K2)P (H)| = P (H)
∣∣P (K|H)− P (K2|H)
∣∣
= P (H)
∣∣EH [f(Uq)]− EH [g(Vq)]
∣∣,
where f = 1(−∞,x] and g = 1(−∞,x+ε]. We denote, by EH , the expectation with respect
to the probability law P (·|H). By summing and subtracting EH [g(Uq)], we see that the
above quantity is not greater than
(3.14)
P (H)
∣∣EH [g(Uq)]− EH [g(Vq)]
∣∣+ P (H)EH [|f − g|(Uq)]
= |E[g(Uq)]− E[g(Vq)]
∣∣+ E[|f − g|(Uq)]
≤ H
ε
√
vp
vq
+ 2Qq(ε) + Q̃q(ε),
using Lemma (3.10) and observing that the function f − g is bounded by 1 and has the
interval (x, x + ε] as its support.
Estimate (3.14) holds not only for |P (H ∩ K) − P (K2)P (H)|, but also for |P (H ∩
K)− P (K1)P (H)|.
THE ROSENBLATT COEFFICIENT 35
We now insert relations (3.8) and (3.14) into (3.5) and obtain
|P (H ∩K)− P (H)P (K)| ≤ H
ε
√
vp
vq
+ 2Qq(ε) + Q̃q(ε)
≤ H
(
1
ε
√
vp
vq
+ ε+
1
qα
)
by Lemmas (3.3) and (3.13). The above inequality holds for every ε > 0; by passing to
the infimum in ε, we get
|P (H ∩K)− P (H)P (K)| ≤ H
(
4
√
vp
vq
+
1
qα
)
.
4. The proof of Theorem (2.5) and the ASCLT
Let’s start with the proof of Theorem (2.5). It is sufficient to consider the case where
A is of the form A = (−∞, x]. The proof is split in two steps: (i) and (ii).
Put
(4.1) an = log2
(
1 +
1
n
)
.
(i) Here, we prove that (Sn) and (Hn) have the same limit points, where
Hn =
∑2n
i=1 ai1A(Ui)
n
;
This is equivalent to proving that the sequence
Tn −Hn +
a2n1A(U2n)
n
=
∑n
i=1 1A(U2i)−∑2n−1
i=1 ai1A(Ui)
n
tends to 0 as n→ ∞, P–a.s. Now, the numerator of the fraction in the second member
above can be written as
n∑
i=1
1A(U2i)−
n∑
i=1
2i−1∑
j=2i−1
aj1A(Uj) =
n∑
i=1
(
1A(U2i)−
2i−1∑
j=2i−1
aj1A(Uj)
)
=
n∑
i=1
2i−1∑
j=2i−1
aj
(
1A(U2i)− 1A(Uj)
)
(note that
∑2i−1
j=2i−1 aj = log2(2i)− log2(2i−1) = 1). Put now
(4.2) Ri =
2i−1∑
j=2i−1
aj
(
1A(U2i)− 1A(Uj)
)
.
Then we must prove that, P -a.s.
lim
n→∞
∑n
i=1Ri
n
= 0.
We write∑n
i=1Ri
n
=
∑n
i=1
(
Ri − E[Ri]
)
n
+
∑n
i=1E[Ri]
n
=
∑n
i=1 R̃i
n
+
∑n
i=1E[Ri]
n
and consider separately the two summands above.
For the first one, we apply the Gaal–Koksma law (see [8], p. 134) to the sequence
(R̃n)n:
36 RITA GIULIANO
(4.3) Theorem (Gaal–Koksma Strong Law of Large Numbers). Let (Xn)n be a
sequence of centered random variables with finite variance. Suppose that there exists a
constant β > 0 such that, for all integers m ≥ 0, n ≥ 0,
(4.4) E
[( m+n∑
i=m+1
Xi
)2
]
≤ H((m+ n)β −mβ
)
,
for a suitable constant H independent of m and n. Then, for each ρ > 0,
n∑
i=1
Xi = O
(
nβ/2(logn)2+ρ
)
, P − a.s.
We need a bound for Cov(R̃i, R̃j). It is easily seen that, for i ≤ j,
Cov(R̃i, R̃j) =
2i−1∑
h=2i−1
2j−1∑
k=2j−1
ahak
(
C(2i, 2j)− C(h, 2j)− C(2i, k) + C(h, k)
)
,
where
C(p, q)=Cov(1A(Up), 1A(Uq))=P (Up∈ A,Uq ∈ A)− P (Up∈ A)P (Uq ∈ A).
By Theorem (2.3), there exists a constantH such that, for every p, q with 2i−1 ≤ p ≤ 2i
and 2j−1 ≤ q ≤ 2j ,
C(p, q) ≤ H
(
4
√
vp
vq
+
1
qα
)
= H
(
4
√
p φ(p)
q φ(q)
+
1
qα
)
≤ H
(p
q
)α
≤ H 2−α|i−j|,
so that we obtain
Cov(R̃i, R̃j) ≤ H 2−α|i−j|
2i−1∑
h=2i−1
ah
2j−1∑
k=2j−1
ak = H 2−α|i−j|.
In particular, E[R̃2
i ] ≤ H . In order to use the Gaal–Koksma law, we evaluate
E
[( m+n∑
i=m+1
R̃i
)2
]
= E
[ m+n∑
i=m+1
R̃2
i + 2
∑
m+1≤i<j≤m+n
R̃iR̃j
]
≤ Hn+ 2H
∑
m+1≤i<j≤m+n
2−α|i−j| = H n+ 2H
n−1∑
r=1
(n− r)(2α)−r
≤ H n+ 2H n
n−1∑
r=0
(2α)−r ≤ H n = H
[
(m+ n)−m].
Hence, the condition in the Gaal–Koksma law holds with β = 1, and we obtain
n∑
i=1
R̃i = O
(√
n(log n)2+ρ
)
, P − a.s.,
which implies
lim
n→∞
∑n
i=1 R̃i
n
= 0, P − a.s.
We now prove that
lim
n→∞
∑n
i=1E[Ri]
n
= 0.
THE ROSENBLATT COEFFICIENT 37
By Cesaro’s theorem, it will be sufficient to prove that
lim
n→∞E[Rn] = lim
n→∞
2n−1∑
j=2n−1
aj
(
P (U2n ∈ A)− P (Uj ∈ A)
)
= 0
[recall formula (4.2)]. This is immediate by the relation
2i−1∑
j=2i−1
aj = log2(2
i)− log2(2
i−1) = 1
and by Theorem (3.1), which implies
lim
n→∞P (Un ∈ A) = μ(A).
(ii) We now prove that (Hn) and (Wn) have the same limit points. First, observe that
Wn =
∑n
i=1
1
i log 21A(Ui)
log2 n
.
Since the sequences (an) [see definition (4.1)] and (bn), where bn = 1
n log 2 , are equiv-
alent as n→∞, this amounts to show that (Hn) has the same limit points as
Vn =
∑n
i=1 ai1A(Ui)
log2 n
.
This is easy since, for 2r ≤ n < 2r+1, we can write∑2r
i=1 ai1A(Ui)
r + 1
≤ Vn ≤
∑2r+1
i=1 ai1A(Ui)
r
.
We pass to the proof of the ASCLT [Corollary (2.6)]. Consider first a Borel set A of
the form A = (−∞, x]. The Gaal–Koksma law applied to the sequence
1A(U2i)− P (U2i ∈ A)
gives, P–a.s,
lim
n→∞
(
Tn −
∑n
i=1 P (U2i ∈ A)
n
)
= lim
n→∞
∑n
i=1
(
1A(U2i)− P (U2i ∈ A)
)
n
= 0.
by an argument similar to that used above for the sequence (R̃n) [see below for the
definition of (R̃n)]. On the other hand, again by Cesaro’s theorem and Theorem (3.1),
we have
lim
n→∞
∑n
i=1 P (U2i ∈ A)
n
= lim
n→∞P (U2n ∈ A) = μ(A).
Hence, we get
(4.5) lim
n→∞Tn = μ(A), P − a.s.
Now, the classical techniques (similar to those used in the Glivenko–Cantelli theorem;
see, e.g., [3], p. 59) yield that the P -null set Γ such that (4.5) holds for ω ∈ Γc is
independent of A, and it is henceforth immediate that, on Γc, (4.5) holds also for Borel
sets A that are finite unions of disjoint intervals.
For a general set A with λ(∂A) = μ(∂A) = 0, fix ε > 0 and let Aε and Bε be finite
unions of disjoint intervals such that
Aε ⊆ A ⊆ Bε and μ(Bε \Aε) < ε.
38 RITA GIULIANO
Then ∑n
i=1 1Aε(U2i)
n
≤ Tn ≤
∑n
i=1 1Bε(U2i)
n
;
hence, by passing to the limit as n→∞, we get, for ω ∈ Γc,
(4.6) μ(Aε) ≤ lim inf
n→∞ Tn ≤ lim sup
n→∞
Tn ≤ μ(Bε);
since
(4.7) μ(Aε) ≤ μ(A) ≤ μ(Bε) ≤ μ(Aε) + ε
by passing to the limit as ε → 0 in (4.7) and then in (4.6), we deduce that limn→∞ Tn
exists for ω ∈ Γc and, moreover,
lim
n→∞Tn = μ(A), ω ∈ Γc.
Bibliography
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Letters 9 (1990), 201-205.
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Theory Prob. and Math. Statist. (Erevan, 1958) (1960), 38–44.
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