TY - JOUR
AU1 - Chen, Pingyan
AU2 - Sung, Soo
AB - Let 


r
≥
1


$r\geq1$
, 


1
≤
p
<
2


$1\leq p<2$
, and 


α
,
β
>
0


$\alpha, \beta>0$
 with 


1
/
α
+
1
/
β
=
1
/
p


$1/\alpha+1/\beta=1/p$
. Let 


{

a

n
k


,
1
≤
k
≤
n
,
n
≥
1
}


$\{a_{nk}, 1\leq k\leq n,n\geq1\}$
 be an array of constants satisfying 



sup

n
≥
1



n

−
1



∑

k
=
1

n



|

a

n
k


|

α

<
∞


$\sup_{n\geq1}n^{-1}\sum^{n}_{k=1}|a_{nk}|^{\alpha}<\infty$
, and let 


{

X
n

,
n
≥
1
}


$\{ X_{n},n\geq1\}$
 be a sequence of identically distributed 



ρ
∗



$\rho^{*}$
-mixing random variables. For each of the three cases 


α
<
r
p


$\alpha< rp$
, 


α
=
r
p


$\alpha=rp$
, and 


α
>
r
p


$\alpha>rp$
, we provide moment conditions under which 
			



∑

n
=
1

∞


n

r
−
2


P

{

max

1
≤
m
≤
n


|

∑

k
=
1

m


a

n
k



X
k

|
>
ε

n

1
/
p


}

<
∞
,

∀
ε
>
0
.


$$\sum^{\infty}_{n=1}n^{r-2}P \Biggl\{ \max_{1\leq m\leq n} \Biggl\vert \sum^{m}_{k=1}a_{nk}X_{k} \Biggr\vert >\varepsilon n^{1/p} \Biggr\} < \infty,\quad \forall \varepsilon>0. $$
 We also provide moment conditions under which 
			



∑

n
=
1

∞


n

r
−
2
−
q
/
p


E


(

max

1
≤
m
≤
n


|

∑

k
=
1

m


a

n
k



X
k

|
−
ε

n

1
/
p


)

+
q

<
∞
,

∀
ε
>
0
,


$$\sum^{\infty}_{n=1}n^{r-2-q/p} E \Biggl( \max_{1\leq m\leq n} \Biggl\vert \sum^{m}_{k=1}a_{nk}X_{k} \Biggr\vert -\varepsilon n^{1/p} \Biggr)_{+}^{q}< \infty,\quad \forall\varepsilon>0, $$
 where 


q
>
0


$q>0$
. Our results improve and generalize those of Sung (Discrete Dyn. Nat. Soc. 2010:630608, 2010) and Wu et al. (Stat. Probab. Lett. 127:55–66, 2017).
TI - On complete convergence and complete moment convergence for weighted sums of ρ ∗ $\rho^{*}$ -mixing random variables
JF - Journal of Inequalities and Applications
DO - 10.1186/s13660-018-1710-2
DA - 2018-06-01
UR - https://www.deepdyve.com/lp/springer-journals/on-complete-convergence-and-complete-moment-convergence-for-weighted-3N8tm29bSO
SP - 1
EP - 16
VL - 2018
IS - 1
DP - DeepDyve
ER -