# (

Authors: Tunay Bilgin
Journal: International Journal of Analysis
 (2014)

---

## Abstract

We introduce the strong(
σ
,
f
)-asymptotically equivalent and strong (
σ
,
f
)-asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, σ-convergence, and lacunary sequences. Then we use these definitions to prove strong (
σ
,
f
)-asymptotically equivalent and strong (
σ
,
f
)-asymptotically lacunary equivalent analogues of Connor’s results in Connor, 1988, Fridy and Orhan’s results in Fridy and Orhan, 1993, and Das and Patel’s results in Das and Patel, 1989.


---

## Body

## 1. Introduction

Lets, l
∞, and c denote the spaces of all real sequences, bounded and convergent sequences, respectively. Any subspace of s is called a sequence space.Letσ be a mapping of the set of positive integers into itself. A continuous linear functional ϕ on l
∞, the space of real bounded sequences, is said to be an invariant mean or σ-mean if and only if(i)
ϕ
(
x
)
≥
0 when the sequence x
=
(
x
k
) has x
k
≥
0 for all k;
(ii)
ϕ
(
e
)
=
1, where e
=
(
1,1
,
1
,
…
);
(iii)
ϕ
(
x
)
=
ϕ
(
σ
x
) for all x
∈
l
∞, where σ
x
=
(
x
σ
(
n
)
). The mapping σ is one to one with σ
k
(
n
)
≠
n for all positive integers n and k, where σ
k denotes the kth iterate of the mapping σ at n. Thus ϕ extends the limit functional on c in the sense that ϕ
(
x
)
=
lim
⁡
x for all x
∈
c. If x
=
(
x
n
), write T
x
=
T
x
n
=
(
x
σ
(
n
)
).Several authors including Bilgin [1], Mursaleen [2], Savas [3], Schaefer [4], and others have studied invariant convergent sequences.The notion of modulus function was introduced by Nakano [5]. We recall that a modulus f is a function from [
0
,
∞
) to [
0
,
∞
) such that (i) f
(
x
)
=
0 if and only if x
=
0, (ii) f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
) for x
,
y
≥
0, (iii) f is increasing, and (iv) f is continuous from the right at 0. Hence f must be continuous everywhere on [
0
,
∞
). Kolk [6], Maddox [7], Öztürk and Bilgin [8], Pehlivan and Fisher [9], Ruckle [10], and others used a modulus function to construct sequence spaces.Following Freedman et al. [11], we call the sequence θ
=
(
k
r
) lacunary if it is an increasing sequence of integers such that k
0
=
0
,
h
r
=
k
r
-
k
r
-
1
→
∞ as r
→
∞. The intervals determined by θ will be denoted by I
r
=
(
k
r
-
1
,
k
r
] and q
r
=
k
r
/
k
r
-
1. These notations will be used throughout the paper. The sequence space of lacunary strongly convergent sequences N
θ was defined by Freedman et al. [11] as follows:
(1)
N
θ
=
x
=
x
i
∈
s
:
h
r
-
1
∑
i
∈
I
r
x
i
-
s
=
0
for
some
s
.Lacunary convergent sequences have been studied by Bilgin [12], Das and Mishra [13], Das and Patel [14], Savas and Patterson [15], and others. Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices in [16]. Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices in [17].Recently, the concept of asymptotically equivalent was generalized by Bilgin [12], Kumar and Sharma [18], Patterson and Savas [19], Savas and Basarr [20], and Patterson and Savas [15]. In this paper we introduce the strong (
σ
,
f
)-asymptotically equivalent and strong (
σ
,
f
)-asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, σ-convergence, and lacunary sequences.In addition to these definitions, natural inclusion theorems will also be presented.


## 2. Definitions and Notations

Now we recall some definitions of sequence spaces (see [7, 12, 15–17, 21–25]).Definition 1.
A sequence[
x
] is statistically convergent to L if
(2)
lim
⁡
n
⁡
1
n
the
number
of
k
≤
n
:
x
k
-
L
≥
ɛ
=
0
for
every
ɛ
>
0
,
denoted
by
s
t
-
lim
⁡
⁡
x
=
L
.Definition 2.
A sequence[
x
] is strongly (Cesàro) summable to L if
(3)
lim
⁡
n
⁡
1
n
∑
k
=
1
n
x
k
-
L
=
0
,
denoted
by
w
-
lim
⁡
⁡
x
=
L
.Definition 3.
Letf be any modulus; the sequence [
x
] is strongly (Cesàro) summable to L with respect to a modulus if
(4)
lim
⁡
n
⁡
1
n
∑
k
=
1
n
f
x
k
-
L
=
0
,
(
denoted
by
w
f
-
lim
⁡
⁡
x
=
L
)
.Definition 4.
Two nonnegative sequences[
x
] and [
y
] are said to be asymptotically equivalent if lim
⁡
k
(
x
k
/
y
k
)
=
1, (denoted by x
∽
y).Definition 5.
Two nonnegative sequences[
x
] and [
y
] are said to be asymptotically statistical equivalent of multiple L provided that for every ɛ
>
0
(5)
lim
⁡
n
⁡
1
n
the
number
of
k
≤
n
:
x
k
y
k
-
L
≥
ɛ
=
0
,
hhhhhhhhhhhhhhhhhhhh
denoted
by
x
∽
S
y
and simply asymptotically statistical equivalent, if L
=
1.Definition 6.
Two nonnegative sequences[
x
] and [
y
] are said to be strong asymptotically equivalent of multiple L provided that
(6)
lim
⁡
n
⁡
1
n
∑
k
=
1
n
x
k
y
k
-
L
=
0
(
denoted
by
x
∽
w
y
)
and simply strong asymptotically equivalent, if L
=
1.Definition 7.
Letθ be a lacunary sequence; the two nonnegative sequences [
x
] and [
y
] are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ
>
0
(7)
lim
⁡
r
⁡
1
h
r
the
number
of
k
∈
I
r
:
x
k
y
k
-
L
≥
ɛ
=
0
,
hhhhhhhhhhhhhhhhhhhh
denoted
by
x
∽
S
θ
y
and simply asymptotically lacunary statistical equivalent, if L
=
1.Definition 8.
Letθ be a lacunary sequence; the two nonnegative sequences [
x
] and [
y
] are said to be strong asymptotically lacunary equivalent of multiple L provided that
(8)
lim
⁡
r
⁡
1
h
r
∑
k
∈
I
r
‍
x
k
y
k
-
L
=
0
denoted
by
x
∽
N
θ
y
and simply strong asymptotically lacunary equivalent, if L
=
1.Definition 9.
Letf be any modulus; the two nonnegative sequences [
x
] and [
y
] are said to be f-asymptotically equivalent of multiple L provided that
(9)
lim
⁡
k
⁡
f
x
k
y
k
-
L
=
0
denoted
by
x
∽
f
y
and simply strong f-asymptotically equivalent, if L
=
1.Definition 10.
Letf be any modulus; the two nonnegative sequences [
x
] and [
y
] are said to be strong f-asymptotically equivalent of multiple L provided that
(10)
lim
⁡
n
⁡
1
n
∑
k
=
1
n
f
x
k
y
k
-
L
=
0
denoted
by
x
∽
w
f
y
and simply strong f-asymptotically equivalent, if L
=
1.Definition 11.
Two nonnegative sequences[
x
] and [
y
] are said to be S
σ-asymptotically statistical equivalent of multiple L provided that for every ɛ
>
0
(11)
lim
⁡
n
⁡
1
n
the
number
of
k
≤
n
:
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≥
ɛ
=
0
,
hh
uniformly
in
i
=
1,2
,
3
,
…
,
(
denoted
by
x
∽
S
σ
y
)
and simply σ-asymptotically statistical equivalent, if L
=
1.Definition 12.
Letθ be a lacunary sequence; two nonnegative sequences [
x
] and [
y
] are said to be S
σ
,
θ-asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ
>
0
(12)
lim
⁡
r
⁡
1
h
r
the
number
of
k
∈
I
r
:
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≥
ɛ
=
0
,
hh
uniformly
in
i
=
1,2
,
3
,
…
,
(
denoted
by
x
∽
S
σ
,
θ
y
)
and simply S
σ
,
θ-asymptotically lacunary statistical equivalent, if L
=
1.Definition 13.
Letθ be a lacunary sequence; two nonnegative sequences [
x
] and [
y
] are said to be strong σ-asymptotically lacunary equivalent of multiple L provided that
(13)
lim
⁡
r
⁡
1
h
r
∑
k
∈
I
r
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
0
,
uniformly
in
i
=
1,2
,
3
,
…
,
denoted
by
x
∽
N
σ
,
θ
y
and simply σ-asymptotically lacunary statistical equivalent, if L
=
1.Definition 14.
Letf be any modulus and let θ be a lacunary sequence; two nonnegative sequences [
x
] and [
y
] are said to be strong f-asymptotically lacunary equivalent of multiple L provided that
(14)
lim
⁡
r
⁡
1
h
r
∑
k
∈
I
r
f
x
k
y
k
-
L
=
0
,
denoted
by
x
∽
N
θ
,
f
y
and simply f-asymptotically lacunary equivalent, if L
=
1.Definition 15.
Letf be any modulus and let θ be a lacunary sequence; two nonnegative sequences [
x
] and [
y
] are said to be strong (
σ
,
f
)-asymptotically equivalent of multiple L provided that
(15)
lim
⁡
n
⁡
1
n
∑
k
=
1
n
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
0
,
uniformly
in
i
=
1,2
,
3
,
…
,
denoted
by
x
∽
w
σ
,
f
y
and simply (
σ
,
f
)-asymptotically equivalent, if L
=
1.
For(
i
)
=
i
+
1 we write [
w
f
] for w
σ
,
f. Hence the two nonnegative sequences [
x
] and y are said to be strong almost f-asymptotically equivalent of multiple L provided that
(16)
lim
⁡
n
⁡
1
n
∑
k
=
1
n
‍
f
x
k
+
i
y
k
+
i
-
L
=
0
,
uniformly
in
i
=
1,2
,
3
,
…
,
denoted
by
x
∽
[
w
f
]
y
and simply f-asymptotically equivalent, if L
=
1.
Forf
(
x
)
=
x for all x
∈
[
0
,
∞
) we write [
w
] for [
w
f
]. Hence the two nonnegative sequences [
x
] and [
y
] are said to be strong almost asymptotically equivalent of multiple L provided that
(17)
lim
⁡
n
⁡
1
n
∑
k
=
1
n
x
k
+
i
y
k
+
i
-
L
=
0
uniformly
in
i
=
1,2
,
3
,
…
,
denoted
by
x
∽
[
w
f
]
y
and simply strong almost asymptotically equivalent, if L
=
1.Definition 16.
Letf be any modulus and let θ be a lacunary sequence; two nonnegative sequences [
x
] and [
y
] are said to be strong (
σ
,
f
)-asymptotically lacunary equivalent of multiple L provided that
(18)
lim
⁡
r
⁡
1
h
r
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
0
,
uniformly
in
i
=
1,2
,
3
,
…
,
denoted
by
x
∽
N
σ
,
θ
,
f
y
and simply (
σ
,
f
)-asymptotically lacunary equivalent, if L
=
1.
For(
i
)
=
i
+
1 we write M
θ
,
f for N
σ
,
θ
,
f. Hence the two nonnegative sequences [
x
] and [
y
] are said to be strong almost f-asymptotically lacunary equivalent of multiple L provided that
(19)
lim
⁡
r
⁡
1
h
r
∑
k
∈
I
r
f
x
k
+
i
y
k
+
i
-
L
=
0
,
uniformly
in
i
=
1,2
,
3
,
…
,
denoted
by
x
∽
M
θ
,
f
y
and simply strong almost f-asymptotically lacunary equivalent, if L
=
1.
Forf
(
x
)
=
x for all x
∈
[
0
,
∞
) we write [
w
θ
] for w
θ
,
f. Hence the two nonnegative sequences [
x
] and [
y
] are said to be strong almost asymptotically lacunary equivalent of multiple L provided that
(20)
lim
⁡
r
⁡
1
h
r
∑
k
∈
I
r
‍
x
k
+
i
y
k
+
i
-
L
=
0
,
uniformly
in
i
=
1,2
,
3
,
…
,
denoted
by
x
∽
M
θ
y
and simply strong almost asymptotically lacunary equivalent, if L
=
1.


## 3. Main Theorems

We now give lemma to be used later.Lemma 17.
Letf be any modulus. Suppose for given ɛ
>
0, there exist n
o and i
o such that
(21)
1
n
∑
k
=
0
n
-
1
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
<
ɛ
∀
n
≥
n
o
,
i
≥
i
o
.
Then x
∽
w
σ
,
f
y.Proof.
Letɛ
>
0 be given. Chose n
1 and i
o such that
(22)
1
n
∑
k
=
0
n
-
1
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
<
ɛ
2
∀
n
≥
n
1
,
i
≥
i
o
.
It is sufficient to prove that there exists n
2 such that for n
≥
n
2, i
o
≥
i
≥
0
(23)
1
n
∑
k
=
0
n
-
1
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
<
ɛ
since, taking n
o
=
max
⁡
⁡
(
n
1
,
n
2
), (23) holds for n
>
n
o and for all i, which gives the result. Once i
o has been chosen, i
o is fixed, so
(24)
∑
k
=
0
i
o
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
R
say
.
Now, taking i
o
≥
i
≥
0 and n
>
i
o, we have
(25)
1
n
∑
k
=
0
n
-
1
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
1
n
∑
k
=
0
i
o
-
1
∑
k
=
i
o
n
-
1
‍
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≤
R
n
+
ɛ
2
.
Taking n sufficiently large, we can make R
/
n
+
ɛ
/
2
<
ɛ which gives (23) and hence the result.The next theorems show the relationship between the strong(
σ
,
f
)-asymptotically equivalence and the strong (
σ
,
f
)-asymptotically lacunary equivalence.Theorem 18.
Letf be any modulus. Then x
∽
N
σ
,
θ
,
f
y
⇔
x
∽
w
σ
,
f
y for every lacunary sequence θ.Proof.
Letx
∽
N
σ
,
θ
,
f
y. Then, given ɛ
>
0, there exist r
o and L such that
(26)
1
h
r
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
<
ɛ
for r
≥
r
o and i
=
k
r
-
1
+
1
+
v
,
v
≥
0. Let n
≥
h
r; write n
=
m
h
r
+
p, where 0
≤
p
≤
h
r; m is an integer. Since n
≥
h
r, m
≥
0. We have
(27)
1
n
∑
k
=
0
n
-
1
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≤
1
n
∑
k
=
0
(
m
+
1
)
h
r
-
1
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
1
n
∑
j
=
0
m
∑
k
=
j
h
r
(
j
+
1
)
h
r
-
1
f
x
σ
k
i
y
σ
k
i
-
L
≤
m
+
1
h
r
ɛ
n
≤
2
m
h
r
ɛ
n
for
m
≥
0
.
For h
r
/
n
≤
1, since m
h
r
/
n
≤
1, therefore,
(28)
1
n
∑
k
=
0
n
-
1
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≤
2
ɛ
.
Then by lemmax
∽
N
σ
,
θ
,
f
y implies x
∽
w
σ
,
f
y. It is easy to see that x
∽
w
σ
,
f
y implies x
∽
N
σ
,
θ
,
f
y for every θ.Proposition 19.
Letf be any modulus. Then x
∽
M
θ
,
f
y
⇔
x
∽
[
w
f
]
y for every lacunary sequence θ.Proof.
It follows from Theorem18 for σ
(
i
)
=
i
+
1 for all i
=
1,2
,
3
,
….Proposition 20.
x
∽
M
θ
y
⇔
x
∽
[
w
]
y for every lacunary sequence θ.Proof.
It follows from Proposition19 for f
(
x
)
=
x for all x
∈
[
0
,
∞
).Theorem 21.
Letf be any modulus. Then(1)
ifliminf
⁡
r
q
r
>
1, then x
∽
w
σ
,
f
y implies x
∽
N
σ
,
θ
,
f
y;
(2)
iflimsup
⁡
r
q
r
<
∞, then x
∽
N
σ
,
θ
,
f
y implies x
∽
w
σ
,
f
y;
(3)
if1
<
liminf
⁡
r
q
r
≤
limsup
⁡
r
q
r
<
∞, then x
∽
w
σ
,
f
y
⇔
x
∽
N
σ
,
θ
,
f
y.Proof.
Part (1): letx
∽
w
σ
,
f
y and liminf
⁡
r
q
r
>
1. There exists δ
>
0 such that q
r
=
(
k
r
/
k
r
-
1
)
≥
1
+
δ for sufficiently large r. We have, for sufficiently large r, that (
h
r
/
k
r
)
≥
δ
/
(
1
+
δ
). Then
(29)
1
k
r
-
1
∑
k
=
1
k
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≥
1
k
r
-
1
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
h
r
k
r
1
h
r
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≥
δ
1
+
δ
1
h
r
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
which yields that x
∽
N
σ
,
θ
,
f
y.
Part (2): iflimsup
⁡
r
q
r
<
∞, then there exists K
>
0 such that q
r
<
K for every r.
Now suppose thatx
∽
N
σ
,
θ
,
f
y and ɛ
>
0. There exists m
0 such that for every m
≥
m
0,
(30)
H
m
=
1
h
m
∑
k
∈
I
m
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
<
ɛ
,
∀
i
.
We can also findR
>
0 such that H
m
≤
R for all m. Let n be any integer with k
r
≥
n
>
k
r
-
1 where r
>
m
0. Now write
(31)
1
n
∑
k
=
0
n
-
1
f
x
σ
k
i
y
σ
k
i
-
L
≤
1
k
r
-
1
∑
k
=
1
k
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
1
k
r
-
1
∑
k
∈
I
1
f
x
σ
k
i
y
σ
k
i
-
L
+
∑
k
∈
I
2
f
x
σ
k
i
y
σ
k
i
-
L
+
⋯
+
∑
k
∈
I
r
f
x
σ
k
i
y
σ
k
i
-
L
=
1
k
r
-
1
k
1
k
1
∑
k
∈
I
1
f
x
σ
k
i
y
σ
k
i
-
L
+
k
2
-
k
1
k
2
-
k
1
×
∑
k
∈
I
2
f
x
σ
k
i
y
σ
k
i
-
L
+
⋯
+
k
m
0
-
k
m
0
-
1
k
m
0
-
k
m
0
-
1
∑
k
∈
I
m
0
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
+
⋯
+
k
r
-
k
r
-
1
k
r
-
k
r
-
1
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
k
1
k
r
-
1
H
1
+
k
2
-
k
1
k
r
-
1
H
2
+
⋯
+
k
m
0
-
k
m
0
-
1
k
r
-
1
H
m
0
+
⋯
+
k
r
-
k
r
-
1
k
r
-
1
H
r
≤
k
m
0
k
r
-
1
sup
⁡
1
<
k
<
k
m
0
⁡
H
k
+
sup
⁡
k
>
k
m
0
⁡
H
k
(
k
r
-
k
m
0
)
k
r
-
1
<
R
k
m
0
k
r
-
1
+
ɛ
K
from which we deduce that x
∽
w
σ
,
f
y.
Part (3): this immediately follows from(
1
) and (
2
).Proposition 22.
Letf be any modulus. Then(1)
ifliminf
⁡
r
q
r
>
1, then x
∽
[
w
f
]
y implies x
∽
M
θ
,
f
y;
(2)
iflimsup
⁡
r
q
r
<
∞, then x
∽
M
θ
,
f
y implies x
∽
[
w
f
]
y;
(3)
if1
<
liminf
⁡
r
q
r
≤
limsup
⁡
r
q
r
<
∞, then x
∽
[
w
f
]
y
⇔
x
∽
M
θ
,
f
y.Proof.
It follows from Theorem21 for σ
(
i
)
=
i
+
1 for all i
=
1,2
,
3
,
….Proposition 23.
Consider the following:(
1
)
ifliminf
⁡
r
q
r
>
1, then x
∽
[
w
]
y implies x
∽
M
θ
y;
(
2
)
iflimsup
⁡
r
q
r
<
∞, then x
∽
M
θ
y implies x
∽
[
w
]
y;
(
3
)
if1
<
liminf
⁡
r
q
r
≤
limsup
⁡
r
q
r
<
∞, then x
∽
[
w
]
y
⇔
x
∽
M
θ
y.Proof.
It follows from Proposition22 for f
(
x
)
=
x for all x
∈
[
0
,
∞
).In the following theorem we study the relationship between the strong(
σ
,
f
)-asymptotically lacunary equivalence and the strong σ-asymptotically lacunary equivalence.Theorem 24.
Letf be any modulus. Then(1)
ifx
∽
N
σ
,
θ
y, then x
∽
N
σ
,
θ
,
f
y;
(2)
iflim
⁡
t
→
∞
(
f
t
/
t
)
=
β
>
0, then x
∽
N
σ
,
θ
y
⇔
x
∽
N
σ
,
θ
,
f
y.Proof.
Part(
1
): let x
∽
N
σ
,
θ
y and ɛ
>
0. We choose 0
<
δ
<
1 such that f
(
u
)
<
ɛ for every u with 0
≤
u
≤
δ. We can write
(32)
1
h
r
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
1
h
r
∑
1
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
+
1
h
r
∑
2
f
x
σ
k
i
y
σ
k
i
-
L
,
where the first summation is over x
σ
k
(
i
)
/
y
σ
k
(
i
)
-
L
≤
δ and the second summation over x
σ
k
(
i
)
/
y
σ
k
(
i
)
-
L
>
δ with k
∈
I
r. By definition of f, we have
(33)
1
h
r
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≤
ɛ
+
2
f
1
δ
-
1
1
h
r
∑
k
∈
I
r
f
x
σ
k
i
y
σ
k
i
-
L
.
Therefore,x
∽
N
σ
,
θ
,
f
y.
Part(
2
): if lim
⁡
t
→
∞
(
f
(
t
)
/
t
)
=
β
>
0, then f
(
t
)
≥
β
t for all t
>
0. Let x
∽
N
σ
,
θ
,
f
y; clearly
(34)
1
h
r
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≥
1
h
r
∑
k
∈
I
r
β
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
β
1
h
r
∑
k
∈
I
r
x
σ
k
i
y
σ
k
i
-
L
.
Therefore, x
∽
N
σ
,
θ
y. By using (23) the proof is complete.Proposition 25.
Letf be any modulus. Then(1)
ifx
∽
M
θ
y, then x
∽
M
θ
,
f
y;
(2)
iflim
⁡
t
→
∞
(
f
(
t
)
/
t
)
=
β
>
0, then x
∽
M
θ
y
⇔
x
∽
M
θ
,
f
y.Proof.
It follows from Theorem24 for σ
(
i
)
=
i
+
1 for all i
=
1,2
,
3
,
….Finally we give the relation betweenS
σ
,
θ-asymptotically lacunary statistical equivalence and strong (
σ
,
f
)-asymptotically lacunary equivalence. Also we give relation between S
σ
,
θ-asymptotically lacunary statistical equivalence and strong (
σ
,
f
)-asymptotically equivalence.Theorem 26.
Letf be any modulus. Then(1)
ifx
∽
N
σ
,
θ
,
f
y, then x
∽
S
σ
,
θ
y;
(2)
iff is bounded and x
∽
S
σ
,
θ
y, then x
∽
N
σ
,
θ
,
f
y;
(3)
iff is bounded, then x
∽
N
σ
,
θ
,
f
y
⇔
x
∽
S
σ
,
θ
y.Proof.
Part(
1
): take ɛ
>
0 and x
∽
N
σ
,
θ
,
f
y. Let ∑
1 denote the sum over k
∈
I
r and x
σ
k
(
i
)
/
y
σ
k
(
i
)
-
L
≥
ɛ.
Then(35)
1
h
r
∑
k
∈
I
r
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≥
1
h
r
∑
1
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≥
f
ɛ
the
number
of
k
∈
I
r
:
x
σ
k
i
y
σ
k
i
-
L
≥
ɛ
.
Therefore,x
∽
S
σ
,
θ
y.
Part(
2
): suppose that f is bounded and x
∽
S
σ
,
θ
y. Since f is bounded, there exists an integer T such that f
(
x
)
≤
T for all x
≥
0. Given ɛ
>
0,
(36)
1
h
r
∑
k
∈
I
r
‍
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≤
T
1
h
r
the
number
of
k
∈
I
r
:
x
σ
k
i
y
σ
k
i
-
L
≥
ɛ
+
f
(
ɛ
)
.
Therefore,x
∽
N
σ
,
θ
,
f
y.
Part(
3
): follows from (
1
) and (
2
).
For(
i
)
=
i
+
1 we write [
S
θ
] for S
σ
,
θ. Hence the two nonnegative sequences [
x
] and [
y
] are said to be [
S
θ
]-asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ
>
0
(37)
lim
⁡
r
⁡
1
h
r
the
number
of
k
∈
I
r
:
x
σ
k
i
y
σ
k
i
-
L
≥
ɛ
=
0
,
uniformly
in
i
=
1,2
,
3
,
…
,
(
denoted
by
x
∽
[
S
θ
]
y
)
and [
S
θ
]-asymptotically lacunary statistical equivalent, if L
=
1. Hence we have the following.Proposition 27.
Letf be any modulus. Then(1)
ifx
∽
M
θ
,
f
y, then x
∽
[
S
θ
]
y;
(2)
iff is bounded and x
∽
[
S
θ
]
y, then x
∽
M
θ
,
f
y;
(3)
iff is bounded, then x
∽
M
θ
,
f
y
⇔
x
∽
[
S
θ
]
y.Proof.
It follows from Theorem26 for σ
(
i
)
=
i
+
1 for all i
=
1,2
,
3
,
….Proposition 28.
x
∽
M
θ
y implies x
∽
[
S
θ
]
y.Proof.
It follows from Proposition27
(
1
) for f
(
x
)
=
x for all x
∈
[
0
,
∞
).Theorem 29.
Letf be bounded; then x
∽
S
σ
,
θ
y implies x
∽
w
σ
,
f
y for every lacunary sequence θ.Proof.
Letn be any integer with n
∈
I
r; then
(38)
1
n
∑
k
=
1
n
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
1
n
∑
p
=
1
r
-
1
∑
k
∈
I
p
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
+
1
n
∑
k
=
1
+
k
r
-
1
n
f
x
σ
k
i
y
σ
k
i
-
L
.
Consider the first term on the right in (38):
(39)
1
n
∑
p
=
1
r
-
1
∑
k
∈
I
p
f
x
σ
k
i
y
σ
k
i
-
L
≤
1
k
r
-
1
∑
p
=
1
r
-
1
∑
k
∈
I
p
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
=
1
k
r
-
1
∑
p
=
1
r
-
1
h
p
1
h
p
∑
k
∈
I
p
‍
f
x
σ
k
i
y
σ
k
i
-
L
.
Sincef is bounded and x
∽
S
σ
,
θ
y, it follows from Theorem 26
(
2
) that
(40)
1
h
p
∑
k
∈
I
p
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
⟶
0
.
Hence(41)
1
k
r
-
1
∑
p
=
1
r
-
1
h
p
1
h
p
∑
k
∈
I
p
f
x
σ
k
i
y
σ
k
i
-
L
⟶
0
.
Consider the second term on the right in (38); since f is bounded, there exists an integer T such that f
(
x
)
≤
T for all x
≥
0. We split the sum for k
r
-
1
<
k
≤
n into sums over x
σ
k
(
i
)
/
y
σ
k
(
i
)
-
L
≥
ɛ and x
σ
k
i
/
y
σ
k
i
-
L
<
ɛ. Therefore, we have for every ɛ
>
0 that
(42)
1
n
∑
k
=
1
+
k
r
-
1
n
f
x
σ
k
(
i
)
y
σ
k
(
i
)
-
L
≤
T
1
h
r
the
number
of
k
∈
I
r
:
x
σ
k
i
y
σ
k
i
-
L
≥
ɛ
+
f
(
ɛ
)
.
Sincex
∽
S
σ
,
θ
y
,
f is continuous from the right at 0 and ɛ is arbitrary; the expression on left side of (42) tends to zero as r
→
∞, uniformly in i. Hence (38), (41), and (42) imply that x
∽
w
σ
,
f
y.Proposition 30.
Letf be bounded; then x
∽
[
S
θ
]
y implies x
∽
[
w
f
]
y for every lacunary sequence θ.Proof.
It follows from Theorem29 for σ
(
i
)
=
i
+
1 for all i
=
1,2
,
3
,
….



---

*Source: 945902-2014-12-03.xml*