R E S E A R C H
Open Access
Strong law for linear processes
Xiangdong Liu
1*, Zongping Guan
1and Tien Chung Hu
2*Correspondence:
1Department of Statistics, Jinan
University, Guangzhou, 510630, China
Full list of author information is available at the end of the article
Abstract
In this paper, the authors obtain the strong approximations, the laws of the single logarithm, the functional laws of the single logarithm, and the limit law of the single logarithm for linear processes generated by arrays of independent and identically distributed random variables, and they show that they are all equivalent under the same moment conditions.
MSC: 60F05; 60F25
Keywords: linear process; strong law; array of independent and identically distributed random variables; law of the single logarithm; limit law of the single logarithm
1 Introduction and main result
Throughout the paper, we assume that{εi, –∞<i<∞}(or{εn,i, –∞<i<∞,n≥}) is a sequence (or an array) of independent and identically distributed (i.i.d.) random variables with the same distribution as the random variableε,{ai, –∞<i<∞}is a sequence of real constants with
∞
i=–∞
|ai|<∞, a=
∞
i=–∞
ai= .
Define
Xn=
∞
i=–∞
aiεn–i, n≥
orXnk=
∞
i=–∞
aiεn,k–i,k≥,n≥
. (.)
We call{Xn,n≥} (or {Xnk,k ≥,n≥}) the linear process of {εi, –∞<i<∞} (or
{εni, –∞<i<∞,n≥}). In particular ifai= fori= –, –, . . . , we call{Xn,n≥}(or
{Xnk,k≥,n≥}) the one-side linear processes. Denote byK the set of all absolutely continuousf on [, ] withf() = and(f(x))dx≤; it is well known thatKis a
com-pact, convex, and symmetric subset of the Banach spaceC[, ], which is the set of real continuous functions on [, ] with the uniform norm.
For the one-side linear processes, Philipps and Solo [] established the strong law of large numbers and the law of the iterated logarithm, Wanget al.[] established the strong approximation to a Gaussian process. Lu and Qiu [] obtained the functional law of the iterated logarithm for the one-side linear processes as follows.
Theorem A Suppose that Eε= ,Eε= ,and∞
i=i|ai|<∞.Then the stochastic pro-cess sequence{√ S(nt)
anlog logn, ≤t≤,n≥}converges to and clusters throughoutKin the
uniform norm with probability,that is, ⎧
⎪ ⎨ ⎪ ⎩
limn→∞inff∈Ksup≤t≤|√aSn(ntlog log) n–f(t)|= a.s. and
lim infn→∞sup≤t≤|√ S(nt)
anlog logn–f(t)|= a.s. for every f ∈K,
where
S(nt) =
[nt]
j=
Xj+
nt– [nt]X[nt]+, ≤t≤,n≥,
[·]is the greatest integer function,andlogx=ln(max{e,x})for x> .
By the same argument as Strassen [], Theorem A implies the classical Hartman-Wintner law of the iterated logarithm (cf.Hartman and Wintner [])
lim sup n→∞
n
j=Xj
anlog logn= a.s. and lim infn→∞
n
j=Xj
anlog logn= – a.s.
Recently, Tanet al.[] proved the above-mentioned conclusions without the condition
∞
i=i|ai|<∞.
There is a substantial difference between the a.s. limiting behavior for sequences of i.i.d. random variables and arrays of i.i.d. random variables. For example, if Eε= , Eε= , Eε<∞, Hu and Weber [] proved the following law of the single logarithm:
lim sup n→∞
n
j=εn,j
√
nlogn= a.s. and lim infn→∞
n
j=εn,j
√
nlogn= – a.s. (.)
Hence the classical Hartman-Wintner law of the iterated logarithm (cf. Hartman and Wintner []) does not hold for arrays. The above result was improved by Liet al.[] and by Qi [] who simultaneously and independently proved that
Eε= , Eε= , and E |ε|
log|ε|<∞
are necessary and sufficient for (.) to hold.
Afterwards, using the strong approximations for partial sums of i.i.d. random variables, Liet al.[] obtained more general results as follows.
Theorem B Letα> .On a suitable probability space one can redefine{ε,εn,i, ≤i≤
[nα],n≥}without changing the distribution,and there exist a sequence of independent
real standard Wiener processes{Wn(t), ≤t≤,n≥}and a sequence of independent standard normal random variables{Zn,n≥},such that the following are equivalent:
Eε= , Eε= , and E |ε|
p
lim sup n→∞
[nα]
j= εn,j
√
nαlogn= a.s. and lim infn→∞
[nα]
j= εn,j
√
nαlogn= – a.s., (.)
⎧ ⎨ ⎩
limn→∞inff∈Ksup≤t≤|(nαUnlog(tn))/ –f(t)|= a.s. and lim infn→∞sup≤t≤|
Un(t)
(nαlogn)/ –f(t)|= a.s. for every f∈K,
(.)
[nα]
j= εn,j nα/ –Zn
=o(logn)/ a.s., (.)
sup
≤t≤
Un(t)
nα/ –Wn(t)
=o(logn)/ a.s., (.)
where p= + /αand
Un(t) =
[[nα]t]
j=
εn,j+
nαt–nαtεn,[[nα]t]+, ≤t≤,n≥. (.)
Remark . Theorem B also holds if (.) is replaced by
lim sup n→∞
max≤m≤[nα]|m
j=εn,j|
√
nαlogn = a.s. (.)
In fact, by the equivalence of (.) and (.),Eε= ,Eε= . By Markov’s inequality, for fixedδ> , whennis large enough,
max
≤m≤[nα]–P
[nα]
j=m+
εn,j > (δ/)
nαlogn
≤δ–E|ε|(logn)–< /.
Hence, by Ottaviani’s inequality (see Lemma . in Ledoux and Talagrand []), whennis large enough,
P
max
≤m≤[nα]
m
j=
εn,j
> ( +δ)
nαlogn
=P
max
≤m≤[nα]
m
j=
εn,j
> ( +δ/)
nαlogn+ (δ/)nαlogn
≤ P{|
[nα]
j= εn,j|> ( +δ/)
√
nαlogn}
–max≤m≤[nα]–P{|[n α]
j=m+εn,j|> (δ/)
√
nαlogn}
≤P
[nα]
j=
εn,j
> ( +δ/)
nαlogn
.
By the Borel-Cantelli lemma, (.) implies that
∞
n=
P
[nα]
j=
εn,j
> ( +δ/)
nαlogn
Hence
∞
n=
P
max
≤m≤[nα]
m
j=
εn,j
> ( +δ)
nαlogn
<∞.
By the Borel-Cantelli lemma again
lim sup n→∞
max≤m≤[nα]|m
j=εn,j|
√
nαlogn ≤ +δ a.s.
Note thatδ> is arbitrary and
lim sup n→∞
max≤m≤[nα]|m
j=εn,j|
√
nαlogn ≥lim sup
n→∞
|[nα]
j= εn,j|
√
nαlogn= a.s.,
(.) holds.
Very few results for a moving average process of an array of i.i.d. random variables are known. In this paper, we will extend Theorem B to the linear processes as follows.
Theorem . Letα> .On a suitable probability space one can redefine{ε,εn,i, –∞<i<
∞,n≥} without changing the distribution,and there exist a sequence of independent real standard Wiener processes{Wn(t), ≤t≤,n≥}and a sequence of independent standard normal random variables{Zn,n≥},such that the following are equivalent:
Eε= , Eε= , and E |ε|
p
logp|ε|<∞, (.)
lim sup n→∞
[nα]
j= Xn,j
anαlogn= a.s. and lim infn→∞ [nα]
j= Xn,j
anαlogn= – a.s., (.)
⎧ ⎨ ⎩
limn→∞inff∈Ksup≤t≤| Sn(t)
(anαlogn)/ –f(t)|= a.s. and lim infn→∞sup≤t≤|(anSnαlog(t)n)/ –f(t)|= a.s. for every f ∈K,
(.)
[nα]
j= Xn,j nα/ –aZn
=o(logn)/ a.s., (.)
sup
≤t≤
Snn(α/t)–aWn(t)
=o(logn)/ a.s., (.)
lim n→∞
alognmax≤m≤n
|[mα]
j= Xm,j|
mα/ = a.s., (.)
where p= + /αand
Sn(t) =
[[nα]t]
j=
Xn,j+
nαt–nαtX
n,[[nα]t]+, ≤t≤,n≥.
2 Proof of main result
To prove our main result, the following lemmas are needed.
Lemma . Letα> and{Y,Yn,j, ≤j≤[nα],n≥}be an array of identically distributed
random variables(without independence assumption)with EY= and Elog|Yp||pY|<∞,p= + /α.Then
√
nαlogn≤maxj≤[nα]|Yn,j| → a.s. (.)
Proof Note thatElog|Yp||pY|<∞is equivalent to
∞
n=
nαP|Y|>δnαlogn<∞, ∀δ> .
Hence∀δ>
∞
n=
P
max
≤j≤[nα]|Ynj|>δ
nαlogn≤ ∞
n=
nαP|Y|>δnαlogn<∞,
which implies that (.) holds by the Borel-Cantelli lemma.
Lemma . Letα> and{Y,Yn,j, ≤j≤[nα],n≥}be an array of i.i.d.random variables
with EY = and Elog|Yp||pY|<∞,p= + /α.Then
Esup n≥
max≤m≤[nα]|mj=Yn,j|
√
nαlogn <∞. (.)
Proof Let{Yn,n≥}be a sequence of independent random variables with common dis-tribution asY. By the same argument as Theorem in Li and Spˇataru [] or Theorem . in Chen and Wang [],
∞
M
∞
n=
P
[nα]
j=
Yj >y
nαlogn
dy<∞
holds for some large enoughM> . For allx≥M, by Markov ’s inequality, whennis large enough,
max
≤m≤[nα]–P
[nα]
j=m+
Yn,j >x
nαlogn
≤(M)–E|Y|(logn)–≤/.
Hence by Ottaviani’s inequality (see Lemma . in Ledoux and Talagrand []), for all x≥M, whennis large enough,
P
max
≤m≤[nα]
m
j=
Yn,j >x
nαlogn
≤P
[nα]
j=
Ynj >x
nαlogn/
Then
Esup n≥
max≤m≤[nα]|mj=Yn,j|
√
nαlogn
= ∞ P sup n≥
max≤m≤[nα]|mj=Yn,j|
√
nαlogn >x
dx
≤M+
∞ M ∞ n= P max
≤m≤[nα]
m j=
Yn,j >x
nαlogn
dx
≤M+
∞ M ∞ n= P
[nα]
j= Ynj >x
nαlogn/
dx
= M+
∞ M ∞ n= P
[nα]
j= Yj >y
nαlogn
dy (x= y)
<∞.
Proof of Theorem. We only prove that (.)⇒(.), (.)⇒(.) and (.)⇒(.)
⇒(.), the proofs of (.)⇒(.)⇒(.), (.)⇒(.)⇒(.) are similar to those in Liet al.[].
(.)⇒(.). Note that
sup
≤t≤
Un(t) –
[[nα]t]
j=
εn,j
≤≤maxj≤[nα]|εn,j|
and
sup
≤t≤
Sn(t) –
[[nα]t]
j=
Xn,j
≤≤maxj≤[nα]|Xn,j|,
whereUn(t) is as Theorem B, and
sup
≤t≤
Snn(α/t)–aWn(t)
= sup
≤t≤
nα/ Sn(t) –
[[nα]t]
j=
Xn,j
+ [[nα]t]
j=
Xn,j–a
[[nα]t]
j=
εn,j
+a [[nα]t]
j=
εn,j–Un(t)
+a
Un(t)
nα/ –Wn(t)
≤
nα/≤maxj≤[nα]|Xn,j|+
nα/ sup
≤t≤
[[nα]t]
j=
Xn,j–a
[[nα]t]
j=
εn,j
+ |a|
nα/≤maxj≤[nα]|εn,j|+|a| sup
≤t≤
Unn(α/t)–Wn(t)
Therefore, to prove (.), by Lemma . and Theorem B it is enough to prove that
√
nαlogn sup ≤t≤
[[nα]t]
j=
Xn,j–a
[[nα]t]
j=
εn,j
→ a.s. (.)
Givenm> , set
Ynm(t) =
[[nα]t]
j=
m
i=–m aiεn,j–i,
am= , ai= m
k=i+
ak, i= , . . . ,m– ,
a–m= , ai= i–
k=–m
ak, i= –m+ , –m+ , . . . , ,
εnj= m
i=
aiεn,j–i, εnj=
i=–m aiεn,j–i.
Then
Ynm(t) =
m
i=–m ai
[[nα]t]
j=
εn,j+ (εn–εn,[[nα]t]+εn,[[nα]t]+–εn) (.)
and
[[nα]t]
j=
Xn,j=Ynm(t) +
[[nα]t]
j=
|i|>m
aiεn,j–i. (.)
For everyi, by Lemma .,
√
nαlogn sup ≤t≤|
εn,[[nα]t]–i| ≤√
nαlogn≤maxm≤[nα]|εn,m–i| → a.s.
So
√
nαlogn sup ≤t≤|
εn,[[nα]t]| → a.s.
and
√
nαlogn sup ≤t≤|
εn,[[nα]t]+| → a.s.
Furthermore
lim n→∞
|εn|
√
nαlogn=nlim→∞ |εn|
√
Hence
√
nαlogn sup ≤t≤|
εn–εn,[[nα]t]+εn,[[nα]t]+–εn| → a.s. (.)
By (.)-(.) and Remark .
lim sup n→∞
√
nαlogn sup ≤t≤
[[nα]t]
j=
Xn,j–a
[[nα]t]
j=
εn,j
=lim sup n→∞
√
nαlogn sup ≤t≤
|i|>m ai
[[nα]t]
j=
εn,j–i–
|i|>m ai
[[nα]t]
j=
εn,j
≤lim sup n→∞
√
nαlogn sup ≤t≤
|i|>m ai
[[nα]t]
j=
εn,j–i
+lim sup n→∞
√
nαlogn sup ≤t≤
|i|>m ai
[[nα]t]
j=
εn,j
≤
|i|>m
|ai|sup n≥
√
nαlogn≤maxm≤[nα]
m j=
εn,j–i
+
|i|>m ai
. (.)
By the stationarity of{ε,εn,i, –∞<i<∞,n≥}and Lemma .
E
∞
i=–∞ |ai|sup
n≥
√
nαlogn≤maxm≤[nα]
m j=
εn,j–i ≤ ∞
i=–∞
|ai|Esup n≥
√
nαlogn≤maxm≤[nα]
m j=
εn,j–i = ∞
i=–∞ |ai|
Esup
n≥
√
nαlogn≤maxm≤[nα]
m j=
εn,j
<∞.
Hence
∞
i=–∞ |ai|sup
n≥
√
nαlogn≤maxm≤[nα]
m j=
εn,j–i
<∞ a.s.
Therefore lettingm→ ∞in (.)
lim sup n→∞
√
nαlogn sup ≤t≤
[[nα]t]
j=
Xn,j–a
[[nα]t]
j=
εn,j
= a.s.,
i.e.(.) holds.
(.)⇒(.). Setan,i= [nα]
j= aj–i. Note that [nα]
j= Xn,j=
∞
i=–∞an,iεn,i, and{ [nα]
(.) implies that
∞
n=
P
∞
i=–∞
an,iεn,i >M
anαlogn
<∞ (.)
holds for anyM> . Let{ε,εn,i, –∞<i<∞,n≥}be an independent copy of{ε,εn,i, –∞<
i<∞,n≥}, Hence by (.), for{ε,εn,i, –∞<i<∞,n≥}
∞
n=
P
∞
i=–∞
an,iεn,i >M
anαlogn
<∞ (.)
also holds. Setεs=ε–εandεns,i=εn,i–εn,i, it is clear that{εs,εns,i, –∞<i<∞,n≥}is an array of independent symmetrical, identically distributed random variables. Then by (.) and (.)
∞
n=
P
∞
i=–∞
an,iεns,i > M
anαlogn
<∞. (.)
By the comparison principle (see Lemma . of Ledoux and Talagrand []), (.) implies that
∞
n=
P
[nα/]
i=[nα/]
an,iεsn,i > M
anαlogn
<∞. (.)
Note that∞i=–∞ai=a= , then for nlarge enough,|an,i| ≥ |a|/ holds uniformly for [nα/]≤i≤[nα/]. By (.) and the comparison principle (see Lemma . of Ledoux
[]) again
∞
n=
P
[nα/]
i=[nα/] εsn,i
> M
nαlogn
<∞. (.)
Then by the same argument as Liet al.(p. in []), (.) implies thatElog|εsp||εps|<∞, and
consequentlyElog|εp||pε|<∞holds. By (.)⇒(.), we have
lim sup n→∞
|∞i=–∞an,i(εn,i–Eεn,i)|
aE(ε–Eε)nαlogn = a.s. (.)
Consequently
lim n→∞
∞
i=–∞an,i(εn,i–Eεn,i)
nα = a.s.
By (.), we also have
lim n→∞
∞
i=–∞an,iεn,i
nα = a.s.,
(.)⇒(.). By Embrechtset al.([], p., line ),
lim n→∞
max√≤m≤nZm
logn = a.s.,
which with (.) implies (.) immediately. (.)⇒(.). By (.),
lim sup n→∞
|[nα]
j= Xn,j|
anαlogn≤ a.s.
The rest proof is similar to that of (.)⇒(.). The proof of the theorem is completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have equal contributions. All authors read and approved the final manuscript.
Author details
1Department of Statistics, Jinan University, Guangzhou, 510630, China.2Department of Mathematics, National Tsing Hua
University, Hsinchu, 300, Taiwan, Republic of China.
Acknowledgements
The authors would like to thank the referees and the editors for the helpful comments and suggestions. The work of Liu was supported by the National Natural Science Foundation of China (Grant No. 71471075), the work of Guan was supported by the National Natural Science Foundation of China (Grant No. 11271161), the work of Hu was partially supported by National Science Council Taiwan under the grant number NSC 101-2118-M-007-001-MY2.
Received: 4 December 2014 Accepted: 14 April 2015
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