1 Introduction and Some De…nitions

A-Statistical Convergence And A-Statistical Monotonicity

Maya Alt¬nok

, Mehmet Küçükaslan

Received 24 September 2013

Abstract

The aim of the present paper is to give some properties of A-statistical con- vergence of sequences. We give de…nition of A-statistical monotonicity, upper and lower peak points of sequences. The relation between these concepts and A-statistical monotonicity is investigated. Also, some results given in [11] are generalized.

1 Introduction and Some De…nitions

Statistical convergence of real or complex valued sequences was …rstly introduced by Fast [5] and Steinhaus [16] in the journal Colloquim Math. independently in 1951.

Since then some properties of statistical convergence have been studied in [6, 7, 9, 15].

The idea of this subject depends on asymptotic density of the subset K of natural numbers N(see [3, 4]).

LetKbe a subset of natural numbers Nand K(n) :=fk2K:k6ng: Then, the asymptotic density of the set K Nis de…ned by

(K) := lim

n!1

1 njK(n)j;

if the limit exists. The vertical bars above indicate the cardinality of the setK(n).

A real or complex valued sequencex= (xn)is said to be statistically convergent to the numberl, if for every" >0, the set

K(n; ") =fk:k6nandjxk lj>"g has asymptotic density zero, i.e.,

nlim!1

1

njK(n; ")j= 0

Mathematics Sub ject Classi…cations: 40A05,40C05,10H25.

yDepartment of Mathematics, Faculty of Sciences and Arts, Mersin University, Mersin, Turkey

zDepartment of Mathematics, Faculty of Sciences and Arts, Mersin University, Mersin, Turkey

249

and it is denoted by x_n!l(S)orst lim_n!1x_n=l.

Statistically convergence is deeply connected to the strongly Cesàro summability and uniform summability (see [10]).

LetA= (ank)be a matrix. If the matrixA= (ank)transforms convergent sequences to convergent sequences with the same limit, then it is called regular. The following theorem gives the conditions for a matrix to be regular:

THEOREM 1.1. ([18], p.165) A = (ank) is a regular matrix if and only if the following conditions hold

(i) sup_nP

jankj<1;

(ii) ank!0 (n! 1; k …xed);and (iii) P

ank!1 (n! 1):

A-density of a subsetK of the natural numbersNis de…ned as

A(K) := lim

n!1

X

k2K

a_nk; if the limit exists.

The sequence x= (xn) is A-statistically convergent to l 2 R, if for every " >0, the set K(n; ") :=fk : k6n; jxk lj>"g has A-density zero [8]. It is denoted by xn!l(A st).

2 Some Results About A-Statistical Convergence

The space of all complex valued sequences x= (xn) will be denoted byC^N. In many circumstances we refer toC^Nas the space of arithmetical functionsf :N!C, specially, when f re‡ects the multiplicative structure of N. This is the case for additive and multiplicative functions.

Throughout this article, the matrixA= (a_nk)is non-negative and regular.

De…ne the functiond_A:C^N C^N![0;1)as follows, dA(x; y) := lim sup

n!1

X

k6n

ank'(jxk ykj)

forx= (xn),y= (yn)2C^Nand'is the function': [0;1)![0;1)where '(t) := t ift61;

1 ift >1:

It is clear that, the functiond_A is a semi-metric and it is called A-semi-metric on C^N.

THEOREM 2.1. The sequence x = (x_n)is A-statistically convergent to l 2R if and only ifdA(x; y) = 0where yn=l for alln2N.

PROOF. Let us assumed_A(x; y) = 0wherey_n =l for alln2N. Then, if" >0,

lim sup

n!1

X

k6n jxk lj>"

ank

8<

:

1

"lim sup

n!1

P

k6nank'(jxk lj); " jxk lj 1;

lim sup

n!1

P

k6nank'(jxk lj); jxk lj>1;

6 max 1;1

" lim sup

n!1

X

k6n

ank'(jxk lj) = max 1;1

" dA(x; l):

This calculation implies thatxn!l(A st).

Now, assume that x= (xn)is A-statistically convergent to l 2R. Then for every

" >0, X

k6n

a_nk'(jx_k y_kj) = X

k6n jxk lj<"

a_nk'(jx_k lj) + X

k6n jxk lj>"

a_nk'(jx_k lj)

6 "X

k6n

ank+ X

k6n jxk lj>"

ank:

Then, sinceA= (a_nk)is a regular matrix, we have d_A(x; y) = lim sup

n!1

X

k6n

a_nk'(jx_k y_kj)6"lim sup

n!1

X

k6n

a_nk+ lim sup

n!1

X

k6n jxk lj>"

a_nk6"

and this implies that

d_A(x; y)6"

for any" >0 wherey= (yn)andyn =l (n2N). This ends the proof.

COROLLARY 2.1. If the sequencex= (xk)is convergent tol (in the usual case) thendA(x; y) = 0, whereyn=lfor alln2N.

PROOF. Let us assumex= (xk)is convergent tol, i.e., for each" >0, there exists at least ann0=n0(")2Nsuch thatjxn lj< ^"₂ holds for alln > n0. Therefore, since A is a regular matrix

dA(x; y) = lim sup

n!1

X

k6n

ank'(jxk lj)

= lim sup

n!1

2 4X

k6n0

ank'(jxk lj) + X

n0+16k6n

ank'(jxk lj) 3 5

6 lim sup

n!1

X

k6n0

a_nk+ lim sup

n!1

Xn k=n0+1

a_nkjx_k lj

6 n0lim sup

n!1

ank+"lim sup

n!1

Xn k=n0+1

ank< "

2+"

2 =":

REMARK 2.1. The converse of Corollary 2.1 is not true in general. To see this, let us consider the sequencex= (xn)where

x_n=

pn forn=m²andm= 1;2; :::;

0 otherwise;

and the regular matrixA=C₁, the Cesàro matrix. It is clear that dA(x;0) = lim sup

n!1

1 n

X

k6n

'(jxk 0j) = lim sup

n!1

1 n

Xn k=1

pk= 0:

But the sequence above is not convergent to 0in the usual case.

Letf be an arithmetical function. WithMA(f), we denote A-value off, M_A(f) := lim

n!1

X

k6n

a_nkf(k)

if the limit exists.

THEOREM 2.2. Assume thatf :N!Cis bounded and A-statistically convergent to Land H Nis an arbitrary subset of Nwhich has …nite A-density A(H). Then, MA(1Hf) =L A(H).

PROOF. From the following inequality the proof can be obtained easily:

X

k2H

a_nkf(k) X

k2H

a_nkL 6 X

k2H jf(k) Lj<"

a_nkjf(k) Lj+ X

k2H jf(k) Lj "

a_nkjf(k) Lj

6 "X

k2H

ank+ X

k2H jf(k) Lj "

ankjf(k) Lj< " A(H) +" < "( A(H) + 1)< ":

3 A-Statistical Monotone Sequence

Statistical monotonicity for real valued sequences has been de…ned and studied in [11]

In this section, A-statistical monotonicity will be de…ned and its relation between A-statistical convergence will be investigated.

DEFINITION 3.1. A sequencex= (x_n)is called A-statistical monotone increasing (decreasing), if there exists a subsetH of the natural numbersNwith _A(H) = 1such that the sequencex= (x_n)is monotone increasing (decreasing) onH. A sequencex= (x_n)is called A-statistical monotone sequence if it is A-statistical monotone increasing or A-statistical monotone decreasing.

Now, we list some results about A-statistical monotonicity:

PROPOSITION 3.1. Ifx= (x_n)is monotone sequence thenx= (x_n)is A-statistical monotone. The converse is not true.

PROOF. Assume x= (x_n) is a nondecreasing sequence. That is, for all n 2 N, xn xn+1. So, we can consider the set H = N. Since, A(H) = 1, we see that x= (xn)is A-statistical monotone increasing. The proof can be obtained by the same way when the sequence is monotone decreasing.

Let us consider the sequencex= (x_n)as

xn= 1 forn=m² andm= 1;2; :::;

n otherwise,

and the matrixA=C1. It is clear thatx= (xn)is A-statistical monotone increasing but it is not monotone increasing.

THEOREM 3.1. If the sequencex= (x_n) is A-statistical monotone increasing or A-statistical monotone decreasing, then the set

fk2N:xk+1< xkg orfk2N:xk+1> xkg has A-density zero respectively.

PROOF. Let us assume thatx= (x_n)is A-statistical monotone increasing. That is, there exist a subsetH ofNwith _A(H) = 1such that (x_n)is monotone increasing onH, i.e.,

xn6xn+1 for alln2H:

Therefore, the inclusion

fk2N:x_k+1< x_kg N H and the inequality

A(fk2N:xk+1< xkg)6 ^A(N H) = 0 hold. From this argument the assertion is satis…ed.

REMARK 3.1. The converse of Theorem 3.1 is not true in general. This can be seen by looking at the example given in (page 264, [11]).

THEOREM 3.2. Letx= (xn)be a sequence. Ifx= (xn)bounded and A-statistical monotone, thenx= (xn)is A-statistical convergent.

PROOF. We shall give the proof only for A-statistical monotone increasing se- quence. From the de…nition of A-statistical monotonicity of x= (xn), there exists a subsetH of Nsuch that A(H) = 1andx= (xn)n2H is monotone increasing. Let us denote the element ofH bykn.

Without lost of generality, we may assume thatk_nis a monotone increasing sequence of natural numbers. Then (xkn)is the monotone increasing subsequence of(xn).

Since the sequencex= (x_n)is bounded, we see that the subsequence(x_kn)is also bounded. Therefore, the subsequence (xkn) is convergent to supxkn. It means that, for every " >0there exists a positive natural numberkn0 =kn0(")2Nsuch that

jxkn supxknj< "

holds for allkn> kn0.

Since all convergent sequence is A-statistical convergent, we see that x_kn!supx_kn (A st)

and

K(n) : =fk6n:jxk supxkj>"g

= fk6n:k6=k_n and jx_k supx_kj>"g [ fk6n:k=kn and jxk supxkj>"g

= K¹(n)[K²(n):

SinceK¹(n) N Handxkn !l(A st), we see that (K¹(n)) = 0and (K²(n)) = 0 respectively. Therefore,xn!l(A st).

REMARK 3.2. Boundedness of A-statistical monotone sequence is su¢ cient but not necessary for A-statistical convergence in general. To see this let us consider the matrixA=C1and sequencex= (xn)where

xn= m forn=m² andn2N,

1

n forn6=m²:

It is easy to see that the sequencex= (xn)is not bounded but it is statistical monotone decreasing and statistically convergent to zero.

REMARK 3.3. For a bounded sequence, A-statistical monotonicity is su¢ cient but not necessary for A-statistical convergence. Let us consider the matrix A = C1 and the sequencex= (xn)de…ned by

x_n:=

1

n fornis odd,

1

n fornis even.

It is clear that x = (x_n) is bounded and statistical convergent to zero but it is not statistical monotone.

DEFINITION 3.2. The real number sequencex= (x_n) is said to be A-statistical bounded if there is a numberM >0 such that

A(fn2N:jxnj> Mg) = 0:

REMARK 3.4. For A-statistical convergence, boundedness and A-statistical monotonic- ity is su¢ cient but not necessary.

Let us consider the Cesàro matrix and the sequencex= (x_n)de…ned by

xn= 8>

><

>>

:

n fornis an odd square, 2 fornis an even square,

1

n fornis an odd nonsquare,

1

n fornis an even nonsquare.

(1)

Obviously, x = (xn) is statistical convergent to zero but the sequence is neither A- statistical bounded nor statistical monotone.

By using De…nition 3.2, we may give the weak converse of Theorem 3.2 without proof as follows:

THEOREM 3.3. A-statistical monotone sequencex= (x_n)is A-statistical conver- gent if and only ifx= (x_n)is A-statistical bounded.

4 Peak Points and A-Statistical Monotonicity

In [11] the concept of peak points of real valued sequences has been de…ned and its relation between statistical monotonicity and statistical convergence has been investi- gated.

Let us recall the de…nitions of upper and lower peak points:

DEFINITION 4.1.([11]). The element xk is called an upper (or lower) peak point of the sequencex= (xn)ifxk >xl(respectively xl>xk) holds for alll>k.

The elementx_k is called a peak point of the sequence if it is an upper peak point or lower peak point.

THEOREM 4.1. If the index set of peak points of the sequence x = (xn) has A-density1, then the sequencex= (xn)is A-statistical monotone.

PROOF. Let us denote the index set of upper peak points of the sequencex= (x_n), H:=fk:xk is an upper peak point of (xn)g:

There exist a monotone increasing sequence(k_n)of positive natural number such that the setH can be represented as

H :=fk1< k2< k3< :::g with _A(H) = 1.

Since,x_kn is an upper peak point for alln2N, the following xk1 > xk2 > xk3 > ::: > xkn> :::

inequalities hold. So, x_n is A-statistical monotone decreasing. By using the same arguments for lower peak points the proof is obtained easily.

Letn_k be a strictly increasing sequence of positive natural numbers and x= (x_n) be a real valued sequence. De…nex~= (xnk)and Kx~=fnk :k2Ng.

DEFINITION 4.2. The subsequence x~ = (xnk) of x= (xn) is called (I) A-dense subsequence if A(Kx~) = 1,

(II) A-empty subsequence if A(Kx~) = 0.

DEFINITION 4.3. The sequences x= (x_n) and y = (y_n)are called A-statistical equivalent if there exists a subset M of Nwith _A(M) = 1 such thatx_n =y_n for all n2M. A-statistical equivalence is denoted byx^(A)y.

In the following we list some properties of A-statistical monotonicity and peak points:

(I) Every A-dense subsequence of an A-statistical monotone sequence is A-statistical monotone.

(II) Let x = (xn) and y = (yn) be A-statistical equivalent sequences, i.e. x ^(A) y. Then, x= (xn) is A-statistical monotone if and only if y = (yn) is A-statistical monotone.

(III) If the sequence x = (xn) is A-statistical monotone, then it has at least an A-dense and an A-empty subsequence.

The converse is not true. Let us consider the Cesàro matrix and the sequence x= (x_n)given in (1). The subsequence

(yn) = (x2; x3; x4; x5; x6; x7; x8; x10; x11; x12; x13; x14; x15; x16; ::)

= 1

2 ;1 3;2;1

5; 1 6 ;1

7; 1 8 ; 1

10; 1 11; 1

12; 1 13; 1

14; 1 15;2; :::

and

(zn) = (x1; x9; x25; x49; x81; :::) = (1;9;25;49;81; :::)

areC1-dense andC1-empty subsequence ofx= (xn). But it is not statistical monotone.

(IV) If the index set of peak points of the sequence has A-density 1, then it has monotone A-dense and A-empty subsequences.

THEOREM 4.2. Under the condition of Theorem 4.1, if the sequence xn is A- statistical bounded then it is A-statistical convergent.

PROOF. From the assumption of Theorem 4.1, we can assume x = (x_n) is A- statistical monotone increasing and A-statistical bounded. Therefore, the proof is obtained by using Theorem 3.3.

5 Inclusion Results for C and D

Let = (n) be a strictly increasing sequence of positive natural numbers such that (0) = 0. C and D asymptotic density of a subset of K of natural numbers N is

de…ned by

nlim!1

1

(n)jfk2K:k6 (n)gj and

nlim!1

1

(n) (n 1)jfk2K: (n 1)< k6 (n)gj respectively.

The more detailed knowledge about the densitiesC and D can be found in [2], [14, 17] respectively.

THEOREM 5.1. If the sequence x= (x_n) is D -statistical monotone, then it is C -statistical monotone.

PROOF. We shall apply the technique which was used by Agnew in his paper [1]. Assume, x = (xn) is a D -statistical monotone. That is, there is a subset H of N such that D (H) = 1, and (xn) is monotone on H. Let us denote the set fk : k6 (n); k2HgbyH(n). The setH(n)can be represented as

H(n) = fk2H : (0) + 16k6 (1)g [ fk2H : (1) + 16k6 (2)g [ :::[ fk2H : (n 1) + 16k6 (n)g

= [n j=1

fk2H : (j 1) + 1 k (j)g;

for an arbitrary n2N. From this representation we have

jH(n)j = jfk2H : 16k6 (1)gj+jfk2H : (1) + 16k6 (2)gj+ :::+jfk2H : (n 1) + 16k6 (n)gj

= Xn j=1

jfk2H : (j 1) + 16k6 (j)gj

and 1

(n)jH(n)j= Xn j=1

(j) (j 1) (n)

1

(j) (j 1)jfk2H : (j 1) + 16k6 (j)gj: Let us consider the matrixT = (tn;k)de…ned by

tn;k:=

( (k) (k 1)

(n) fork= 1;2; :::; n;

0 otherwise.

Clearly, the matrix T is regular, and

C (H) =T D (H):

Since, lim_n!1( _D (H))_n= 1andT is regular,

C (H) = 1:

It means that the sequencex= (x_n)isD -statistically monotone.

THEOREM 5.2. LetE=f (n)gbe an in…nite subset ofNwith (0) = 0. Then, a C -statistical monotone sequence is alsoD -statistical monotone sequence if and only if

lim inf

n!1

(n) (n 1) >1:

PROOF. Let x be a C -statistical monotone sequence. That is, there exists a subsetH ofNsuch that _C (H) = 1 and(x_n)is monotone onH. From the de…nition of _D (H)we have

nlim!1

jfk2N: (n 1) + 16k6 (n)gj (n) (n 1)

= (n)

(n) (n 1)

jfk2H : 16k6 (n)gj (n)

(n 1) (n) (n 1)

jfk2H: 16k6 (n 1)gj

(n 1) :

If we let the matrixT = (t_n;k)de…ned by

tn;k= 8>

<

>:

(n)

(n) (n 1) fork=n;

(n 1)

(n) (n 1) fork=n 1;

0 otherwise,

we obtain( D (H))n= (T( C (H)))n.

Therefore, D (H)is obtained by C (H)if and only ifT is regular. Thus, T will be regular if and only if the sequence

(n)

(n) (n 1) + (n 1)

(n) (n 1) _n2N (2)

is bounded. After simple calculation (2) is bounded if and only iflim inf_{n!1 (n}⁽ⁿ⁾₁₎>

1.

The following corollaries are simple consequences of Theorem 6.2 in (page 208, [11]).

COROLLARY 5.1. E = f (n)g and F = f (n)g be an in…nite subset of N. If F E is …nite, thenC -statistical (D -st.) monotonicity impliesC -statistical (D - st.) monotonicity.

COROLLARY 5.2. AssumeF E is …nite. ThenC (D ) statistical monotonicity implies if C (respectively D ) statistical monotonicity and vice versa.

COROLLARY 5.3. LetE=f (n)g be an in…nite subset ofNand lim sup

n!1

(n+ 1) (n) = 1:

Then, the sequence x= (xn) is C -statistical monotone if and only if it is statistical monotone.

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