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Kyungpook Mathematical Journal 2016; 56(2): 357-366

Published online June 1, 2016

Copyright © Kyungpook Mathematical Journal.

On Deferred Statistical Convergence of Sequences

Mehme Küçükaslan, Müjde Yilmaztürk

Department of Mathematics, Mersin University, Mersin, 33343, Turkey

Received: September 26, 2012; Accepted: March 28, 2016

Abstract

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In this paper, deferred statistical convergence is defined by using deferred Cesàro mean instead of Cesàro mean in the definition of statistical convergence. The obtained method is compared with strong deferred Cesàro mean and statistical convergence under some certain assumptions. Also, some inclusion theorems and examples are given.

Keywords: statistical convergence, deferred statistical convergence, summability of sequences, strongly summability.

1. Introduction and Definitions

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The concept of statistical convergence was introduced by I.J. Steinhaus in [17] and H. Fast in [6] independently in the same year. Nowadays, this subject has became one of the most active research area in the theory of summability. It was applied in different areas of mathematics such as number theory by P. Erdös-G. Tenenbaum [5] and summability theory by A. R. Freedman-J. J. Sember-M. Raphael [7].

Furthermore, this subject was studied in [3], [4], [8], [9], [10], [15], [16] etc.

Statistical convergence is also closely related to the subject of asymptotic density( or natural density) of the subset of natural numbers (see, [2]) and its root goes back to A. Zygmund [19].

In 1932, R.P. Agnew [1] defined the deferred Cesaro mean as a generalization of Cesàro mean of real (or complex) valued sequence x = (xk) by

(Dp,qx)n:=1q(n)-p(n)∑k=p(n)+1q(n)xk,   n=1,2,3,…,

where p = {p(n) : n ∈ ℕ} and q = {q(n) : n ∈ ℕ} are the sequences of nonnegative integers satisfying

p(n)<q(n) and limn→∞q(n)=∞.

R.P. Agnew also showed that the method in (1.1) has some important properties besides regularity(see for reqularity [11, Theorem 3]).

A sequence x = (xk) is said to be strong Dp,q–convergent to l if

limn→∞1q(n)-p(n)∑k=p(n)+1q(n)∣xk-l∣= 0,

holds and it is denoted by

limn→∞xn=l (D [p,q]).

Recall that a sequence x = (xk) is said to be statistically convergent to l if for every ɛ > 0,

limn→∞1n∣{k:k≤n,∣xk-l∣ ≥ɛ}∣=0,

satisfied where the vertical bars indicate the numbers of elements inside the set and it is denoted by limn→∞xn=l(S).

There is a natural relationship between statistical convergence and strong summability of sequences. This relation has been investigated in [3], [4], [12], [13], [14] and etc.

Definition 1.1

(Deferred Statistical Convergence (DS)) A sequence x = (xk) is said to be deferred statisticaly convergent to l ∈ ℝ if for every ɛ > 0,

limn→∞1q(n)-p(n)∣{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}∣=0,

holds and it is denoted by

limn→∞xn=l (DS [p,q]).

It is clear that;

  • (i) If q (n) = n and p (n) = 0, then Definition 1.1. is coincide with the definition of statistical convergence,

  • (ii) If we consider q (n) = kn and p (n) = kn−1 (for any lacunary sequence of nonnegative integers with kn–kn−1 → ∞ as n → ∞), then Definition 1.1. is turned to Lacunary Statistical convergence [9],

  • (iii) If q(n) = λn and p(n) = 0 (where λn is a strictly increasing sequence of natural numbers such that limnλn=∞), then Definition 1.1. is coincide λ–statistical convergence of sequences which is given by Osikievich [18] and Mursaleen [13].

2. Inclusion Theorems

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Throughout the paper, we consider the sequence of nonnegative natural numbers p = {p(n) : n ∈ ℕ} and q = {q(n) : n ∈ ℕ} satisfying (1.1). Any other restrictions on (if needed) p(n) and q(n) will be given in related theorems.

2.1 Comparison of D with DS

In this section, strong deferred Cesàro mean D[p, q] and deferred statistical convergence DS [p, q] will be compared. It is going to show that these two methods are equivalent only for bounded sequences.

Theorem 2.1.1

If xn → l (D[p, q]), then xn → l (DS [p, q]).

Proof

Assume xn → l (D[p, q]). For an arbitrary ɛ > 0, following inequality

1q(n)-p(n)∑k=p(n)+1q(n)∣xk-l∣=1q(n)-p(n)(∑k=p(n)+1∣xk-l∣≥ɛq(n) +∑k=p(n)+1∣xk-l∣<ɛq(n) )∣xk-l∣ ≥1q(n)-p(n)∑k=p(n)+1∣xk-l∣≥ɛq(n)∣xk-l∣≥ɛ1q(n)-p(n)∣{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}∣

holds. After taking limit when n → ∞, we have

limn→∞1q(n)-p(n)∣{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}∣=0.

Therefore, desired result is obtained.

Corollary 2.1.2

If xn → l (n → ∞), then xn → l (DS [p, q]).

Remark 2.1.3

The converse of Theorem 2.1.1 and Corollary 2.1.2 are not true, in general.

For this, consider a sequence x = (xk) as

xk:={k2,[|q(n)|]-m0<k≤[|q(n)|], n=1,2,3,…,0,otherwise,

where q(n) is a monotone increasing sequence and m0 ≠ 0 is an arbitrary fixed natural number.

If we consider D[p, q] for the sequence p(n) satisfying

0<p(n)≤[|q(n)|]-m0,

then for an arbitrary ɛ > 0 we have

1q(n)-p(n)∣{k:p(n)<k≤q(n), ∣xk-0∣ ≥ɛ}∣=m0q(n)-p(n)→0,

when n → ∞, i.e., xk → 0 (DS [p, q]).

On the other hand,

1q(n)-p(n)∑p(n)+1q(n)∣xk-0∣≥m0 ([|q(n)|]-m0)2q(n)-p(n)→m0,

when n → ∞, i.e., (xk) is not D[p, q] convergent to zero. It is also clear that the sequence does not convergent to zero in usual case.

Let us recall that l∞ is the set of all bounded sequences.

Theorem 2.1.4

If x = (xn) ∈ l∞and xn → l (DS [p, q]) then xn → l (D[p, q]).

Proof

Suppose that x = (xn) ∈ l∞ and xn → l (DS [p, q]). Under the assumption on (xn) there exists positive reel number M such that |xn− l| ≤ M hods for all n.

So, the inequality

1q(n)-p(n)∑k=p(n)+1q(n)∣xk-l∣= 1q(n)-p(n)(∑k=p(n)+1∣xk-l∣≥ɛq(n) +∑k=p(n)+1∣xk-l∣<ɛq(n) )∣xk-l∣≤1q(n)-p(n)(M∑k=p(n)+1∣xk-l∣≥ɛq(n)1+ɛ∑k=p(n)+1∣xk-l∣<ɛq(n)1)≤M1q(n)-p(n)∣{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}∣+ɛ1q(n)-p(n)∣{k:p(n)<k≤q(n), ∣xk-l∣ <ɛ}∣,

is hold. From the limit relation we have

limn→∞1q(n)-p(n)∑k=p(n)+1q(n)∣xk-l∣= 0.

So, the proof is completed.

2.2. Comparison of S with DS [p, q]

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In this section, statistical convergence and deferred statistical convergence will be compared under some restrictions on p(n) or q(n).

Theorem 2.2.1

If the sequence {p(n)q(n)-p(n)}n∈ℕ is bounded, then xn → l (S) implies xn → l (DS [p, q]).

Proof

Let’s give a note about the sequences of positive natural numbers (an)n∈ℕ and (bn)n∈ℕ without proof: if limn→∞an=a, a ∈ ℝ and limn→∞bn=+∞, then limn→∞abn=a.

From the assumption on (xn), the limit relation

limn→∞1n∣{k:k≤n, ∣xk-l∣ ≥ɛ}∣=0,

holds for every ɛ > 0. Since the sequence q(n) satisfies (1.2), then the sequence

{∣{k:k≤q(n), ∣xk-l∣ ≥ɛ}∣q(n)}n∈ℕ

is convergent to zero.

Therefore, the inclusion

{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}⊂{k:k≤q(n), ∣xk-l∣ ≥ɛ},

and the inequality

∣{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}∣ ≤ ∣{k:k≤q(n), ∣xk-l∣ ≥ɛ}∣,

are hold. From the last inequality we have

1q(n)-p(n)∣{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}∣≤(1+p(n)q(n)-p(n))·1q(n)∣{k:k≤q(n), ∣xk-l∣ ≥ɛ}∣,

and from the limit relation we get

xk→l(DS [p,q]).

So, desired result is obtained.

Corollary 2.2.2

Let {q(n)}n∈ℕbe an arbitrary sequence with q(n) < n for all n ∈ ℕ and {nq(n)-p(n)}n∈ℕ be a bounded sequence. Then, xn → l (S) implies xn → l (DS [p, q]).

Remark 2.2.3

The converse of Theorem 2.2.1 is not true even if {p(n)q(n)-p(n)}n∈ℕ is bounded.

Example 2.2.4

Let us consider p(n) = 2n, q(n) = 4n and a sequence x = (xn) as

xn={n+12,n is odd,-n2,n is even.

It is clear that the assumption of Theorem 2.2.1 is fulfilled and xn → 0 (D[2n, 4n]).

From Theorem 2.1.1 we get xn → 0 (DS[2n, 4n]). But, for an arbitrary small ɛ > 0,

limn→∞1n∣{k:k≤n, ∣xn-0∣ ≥ɛ}∣≠0.

Definition 2.2.5

A method DS[p, q] is called properly deferred when {p(n)} and {q(n)} satisfy in addition to (1.2) the condition {p(n)q(n)-p(n)} is bounded for all n.

Remark 2.2.6

Two properly deferred statistically convergence method must not be include each other. Let

xn:={k+1,n=2k+1,-k,n=2k.

It is clear that xn → 0(DS[2n, 4n]) and xn→12(DS[2n-1,4n-1]).

Theorem 2.2.7

Let q(n) = n for all n ∈ ℕ. Then, xn → l (DS [p, n]) if and only if xn → l (S).

Proof

(⇒) Let us assume that xk → l(DS [p, n]). We shall apply the technique which was used by Agnew in [1]. Then, for any n ∈ ℕ,

p(n)=n(1)>p(n(1))=n(2)>p(n(2))=n(3)>…,

and we may write the set {k ≤ n : |xk− l| ≥ ɛ} as

{k≤n:∣xk-l∣ ≥ɛ}={k≤n(1):∣xk-l∣ ≥ɛ}∪{n(1)<k≤n:∣xk-l∣ ≥ɛ},

and the set {1 < k ≤ n(1) : |xk− l| ≥ ɛ} as

{1<k≤n(1):∣xk-l∣ ≥ɛ}={k≤n(2):∣xk-l∣ ≥ɛ}∪{n(2)<k≤n(1):∣xk-l∣ ≥ɛ},

and the set {k ≤ n(2) : |xk− l| ≥ ɛ} as

{k≤n(2):∣xk-l∣ ≥ɛ}={k≤n(3):∣xk-l∣ ≥ɛ}∪{n(3)<k≤n(2):∣xk-l∣ ≥ɛ},

and if this process is continued we obtain

{k≤n(h-1):∣xk-l∣ ≥ɛ}={k≤n(h):∣xk-l∣ ≥ɛ}∪{n(h)<k≤n(h-1):∣xk-l∣ ≥ɛ}

for a certain positive integer h > 0 depending on n such that n(h) ≥ 1 and n(h+1) = 0. From the above discussion, the relation

1n∣{k≤n:∣xk-l∣ ≥ ɛ}∣=∑m=0hn(m)-n(m+1)n1n(m)-n(m+1)|{n(m+1)<k≤n(m):∣xk-l∣ ≥ɛ}|

holds for every n. This relation gives that statistical convergency of the sequence (xn) to l is a linear combination of following sequence.

{1n(m)-n(m+1)|{n(m+1)<k≤n(m):∣xk-l∣ ≥ɛ}|}m∈ℕ.

Let us consider the matrix

bn,m:={n(m)-n(m+1)n,m=0,1,2,…,h,0,otherwise.

where n(0) := n.

The matrix (bn,m) is satisfied the Silverman Toeplitz theorem (see in [11]). So, we have

limn→∞1n∣{k≤n:∣xk-l∣ ≥ɛ}∣=0

since

1n(m)-n(m+1)|{n(m+1)<k≤n(m):∣xk-l∣ ≥ɛ}|→0,

when n → ∞.

(⇒) Since q(n) = n is satisfied (1.2), then the inverse of the theorem is a simple consequence of Theorem 2.2.1.

Corollary 2.2.8

Assume that {q(n)}n∈ℕcontains almost all positive integers. Then, xn → l(DS [p, q]) implies xn → l(S).

Proof

Let xn → l(DS [p, q]) for an arbitrary {p(n)} and choose sufficiently large positive integer m such that the set {q(n)} contains all positive integers which is greater than m. Then, it can be constructed a sequence (kn) as follows:

k1=k2=⋯=km=1

and for each n > m an index kn such that qkn = n.

It is clear from the construction that (kn) is a monotone increasing sequence. So, from the assumption xn → l(DS [pkn, qkn]). Hence, the proof of Corollary follows from Theorem 2.2.5.

Corollary 2.2.9

Let us assume {q(n)} contains almost all positive integers. If x = (xn) is a sequence such that xn → l(DS [p, q]) for an arbitrary {p(n)}n∈ℕand Δxn=O(1n) then xn → l (n → ∞).

Proof

It is clear from Corollary 2.2.6 and the assumption on q(n) that xn → l(S). Therefore, if we use [9, Theorem 3] we obtained the proof.

2.3. Comparison of DS [p′, q′] with DS [p, q]

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In this section, the methods DS [p, q] and DS [p′, q′] will be compared under the following restriction

p(n)≤p′(n)<q′(n)≤q(n)

for all n ∈ ℕ.

Theorem 2.3.1

Let p′ = {p′(n)}n∈ℕand q′ = {q′(n)}n∈ℕbe sequences of positive natural numbers satisfying (3.1) such that the sets

{k:p(n)<k≤p′(n)}   and   {k:q′(n)<k≤q(n)}

are finite sets for all n ∈ ℕ. Then, xk → l(DS [p′, q′]) implies xk → l(DS [p, q]). Proof. Let us consider the sequence x = (xk) such that xk → l(DS [p′, q′]). For an arbitrary ɛ > 0, the equality

{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}={k:p(n)<k≤p′(n), ∣xk-l∣ ≥ɛ}∪{k:p′(n)<k≤q′(n), ∣xk-l∣ ≥ɛ}∪{k:q′(n)<k≤q(n), ∣xk-l∣ ≥ɛ},

and

1q(n)-p(n)∣{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}∣≤1q′(n)-p′(n)∣{k:p(n)<k≤p′(n), ∣xk-l∣ ≥ɛ}∣+1q′(n)-p′(n)∣{k:p′(n)<k≤q′(n), x∣xk-l∣ ≥ɛ}∣+1q′(n)-p′(n)∣{k:q′(n)<k≤q(n), ∣xk-l∣ ≥ɛ}∣,

are hold.

On taking limits when n → ∞, we obtain

limn→∞1q(n)-p(n)∣{k:p(n)<k≤q(n), ∣xk-l∣ ≥ɛ}∣=0.

This proves our assertion.

Theorem 2.3.2

Let {p(n)}n∈ℕ, {q(n)}n∈ℕand {p′(n)}n∈ℕ, {q′(n)}n∈ℕbe sequences of positive natural numbers satisfying (3.1) such that

limn→∞q(n)-p(n)q′(n)-p′(n)=d>0.

Then, xk → l(DS [p, q]) implies xk → l(DS [p′, q′]).

Proof

It is easy too see that the inclusion

{k:p′(n)+1≤k≤q′(n), ∣xk-l∣ ≥ɛ}⊂{k:p(n)+1≤k≤q(n), ∣xk-l∣ ≥ɛ}

and the inequality

∣{k:p′(n)+1≤k≤q′(n), ∣xk-l∣ ≥ɛ}∣ ≤ ∣{k:p(n)+1≤k≤q(n), ∣xk-l∣ ≥ɛ}∣

are true. So, we have

1q′(n)-p′(n)∣{k:p′(n)+1≤k≤q′(n), ∣xk-l∣ ≥ɛ}∣≤q(n)-p(n)q′(n)-p′(n)1q(n)-p(n)∣{k:p(n)+1≤k≤q(n), ∣xk-l∣ ≥ɛ}∣.

Taking limits as n → ∞ the desired results is obtained.

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