A note on the almost sure central limit theorems for the maxima and sums
Journal of Inequalities and Applications
A note on the almost sure central limit theorems for the maxima and sums
Qing-pei Zang 0
0 School of Mathematical Science, Huaiyin Normal University , Huaian, 223300 , China
In this note, we derive, under some natural assumptions, a general pattern of the almost sure central limit theorem in the joint version for the maxima and sums. 1 Introduction and main results Let {X, Xn; n ≥ } be a sequence of independent and identically distributed (i.i.d.) random variables and Sn = kn= Xk , Mn = max≤k≤n Xk for n ≥ . If E(X) = , E(X) = , then the classical almost sure central limit theorem (ASCLT) has the simplest form as follows:
almost sure central limit theorem; maxima and partial sums; Lipschitz function
-
≤ x = (x) a.s. for all x ∈ R,
Mk – bk ≤ x = G(x) a.s. for all x ∈ CG,
ak
Mk – bk ≤ x = G(x) a.s. for any x ∈ CG
ak
with G(x) being one of the extreme value distributions, i.e.,
∧(x) = exp – exp(–x) ,
α(x) = ⎨⎧ , x < ,
⎩ exp{–x–α}, x ≥ ,
α(x) = ⎨⎧ exp{–(–x)α}, x < ,
⎩ , x ≥ ,
Theorem A Let X, X, . . . be independent random variables with partial sums Sn. Assume
that for some numerical sequences an > and bn, we have
with some (possibly degenerate) distribution function H.
Suppose, moreover, that
E Sn – bn = O() for some v >
( ≤ k ≤ l)
for some positive constants C, β.
Assume finally that
dk = O
Then, if f is a bounded Lipschitz function on the real line or the indicator function of a Borel
set A ⊂ R with λ(∂A) = , we have
–∞
f (x) dH(x) a.s.
Now, we may state our main result as follows.
kn= dk .
Assume, in addition, that f (x, y) is a bounded Lipschitz function. Then
–∞ –∞
f (x, y) (dx)G(dy) a.s.
Remark . It can be seen by routine approximation arguments similar, e.g., to those in
Lacey and Philipp [] that, under the conditions of Theorem ., the result in (.) holds
for indicator functions, i.e.,
≤ x,
Remark . The result of Berkes and Csáki [] shows that the a.s. central limit theorem
remains valid even with the sequence of weights
dk =
which at least includes a ‘halfway’ from logarithmic to ordinary averaging. Moreover,
Hörmann [] shows that this sequence obeys the a.s. central limit theorem for all ≤ α < .
Due to the similar conditions on the sequence of weights, our result also holds for this
sequence provided ≤ α < .
2 Proof of our main result
The following notations will be used throughout this section: Sn = kn= Xk , Sk,n =
in=k+ Xi, Mn = max≤i≤n Xi, and Mk,n = maxk+≤i≤n Xi for n ≥ . Furthermore, a b and
a ∼ b stand for a = O(b) and ab → , respectively, and (x) is the standard normal
distribution function. The proof of our main result is based on the following lemmas.
Lemma Under the assumptions of Theorem ., we have
√Skk , Mka–k bk , f
, ≤ k ≤ l.
Proof It is easy to see that
≤ Cov f
=: L + L + L.
L = .
√Skk , Mka–k bk , f
√Skk , Mka–k bk , f
√Skk , Mka–k bk , f
√Skk , Mka–k bk , f
= E min
P(Ml = Mk,l)
= k
F(x) l–k F(x) k– dF(x)
I(Ml = Mk,l)
where we used the fact that
–∞
for any nondecreasing function ψ on [, ]. In order to verify the last relation, let F–(t) =
sup{x : F(x) ≤ t}, and let U be a random variable uniformly distributed on (, ). Then
F(F–(t)) ≤ t for all t ∈ (, ) and the random variable Y = F–(U) has distribution F . Thus,
the left-hand side of the above inequality equals
Eψ F(Y ) = Eψ F F–(U)
Thus, using (.)-(.), we get the desired result.
√Skk , Mka–k bk ,
We will also prove the following auxiliary result.
E|ξl – ξk,l|p ≤ p–E|ξl – ξk,l|.
Furthermore, we obtain that
≤ E f
Sk
P(Ml = Mk,l) + E √
The relations in (.), (.) imply the claim in Lemma .
The following lemma will also be used.
Proof We can write
l=m
lp=m
l=m
lp=m
l=m
lp=m
This and the relation
≤ p
imply the desired result.
We will also prove the following lemma.
Lemma For every p ∈ N, we have
≤k≤l≤n
Proof This lemma can be obtained from Lemmas and by making slight changes in the
proof of Lemma of Hörmann [].
The following result will be needed in the proof of our main result.
≤k≤l≤n
= O
where Dn =
Proof This result follows from Lemma in Hörmann [].
nl→im∞ P √Snn ≤ x,
Mn – bn ≤ y = (x)G(y) for x, y ∈ R.
an
Then, in view of the dominated convergence theorem, we have
–∞ –∞
f (x, y) (dx)G(dy).
Hence, in order to complete the proof, it is sufficient to show
= a.s.
verges almost surely. This completes the proof of Theorem ..
→ , the convergence of the subsequence implies that the whole sequence
conCompeting interests
The authors declare that they have no competing interests.
Acknowledgements
The author would like to thank the Editor and the two referees for careful reading of the paper and for their comments
which have led to the improvement of this work. This work was supported by the Natural Science Research Project of
Ordinary Universities in Jiangsu Province, P.R. China. Grant No. 12KJB110003.
1. Brosamler , GA: An almost everywhere central limit theorem (...truncated)