Recently, the concept of π½-statistical Convergence was introduced considering a sequence of infinite
matrices π½ = (πππ π ). Later, it was used to define and study π½-statistical limit point, π½-statistical cluster point,
π π‘π½ β πππππ‘ inferior and π π‘π½ β πππππ‘ superior. In this paper we analogously define and study 2π½-statistical
limit, 2π½-statistical cluster point, π π‘2π½ β πππππ‘ inferior and π π‘2π½ β πππππ‘ superior for double sequences.
Generalised Statistical Convergence For Double Sequences
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 01-04
www.iosrjournals.org
www.iosrjournals.org 1 | Page
Generalised Statistical Convergence For Double Sequences
A. M. Brono, Z. U. Siddiqui
Department of Mathematics and Statistics, University of Maiduguri, Nigeria
Abstract: Recently, the concept of π½-statistical Convergence was introduced considering a sequence of infinite
matrices π½ = (π ππ π ). Later, it was used to define and study π½-statistical limit point, π½-statistical cluster point,
π π‘π½ β πππππ‘ inferior and π π‘π½ β πππππ‘ superior. In this paper we analogously define and study 2π½-statistical
limit, 2π½-statistical cluster point, π π‘2π½ β πππππ‘ inferior and π π‘2π½ β πππππ‘ superior for double sequences.
Keywords: Double sequences, Statistical convergence, π½-statistical Convergence Regular matrices, RH-
regular matrices.
I. Introduction
A double sequence π₯ = [π₯ππ ]π,π=0
β
is said to be convergent in the pringsheim sense or p-convergent if for
every π > 0 there exist π β β such that π₯ππ β πΏ < π whenever j, k > N and L is called the Pringsheim limit
[1], denoted P-ππππ₯ = πΏ .
A double sequence x is bounded if there exist a positive number M such that π₯ππ < π for all j, k i.e if
π₯ = π π’πππ π₯ππ < β. Note that in contrast to the case for single sequences, a convergent double sequence
need not be bounded.
Let π΄ = (π ππ ) π,π=1
β
be a non-negative regular matrix. Then A-density of a set πΎ β β is defined if
πΏ π΄ πΎ = πππ π π πππβπΎ exists [2].
A sequence π₯ = π₯ π is said to be A-statistically convergent to L if for every π > 0 the set πΎ π =
π β πΎ βΆ |π₯ π β πΏ| β₯ π has A-density zero [3], (see also [4] and [5]).
Recently, Kolk [6] generalized the idea of A-statistical convergent to π½-statistical Convergence by using
the idea of π½-summability or πΉπ½ -convergence due to Stieglitz [7].
Let π΄ = [πππ
ππ
]π,π=0
β
be a regular doubly infinite matrix of real numbers for all m, n = 0, 1, β¦ . In the
similar manner as in [2], we define 2A-density of the set πΎ = { π , π β β Γ β} ππ for m, n = 0, 1, 2, β¦
πΏ2π΄ πΎ = lim ππββ πππ
πππ
π=0
π
π=0 (1.1)
exists and a double sequence π₯ = (π₯ππ ) is said to be 2A-statistically convergent to L if for every π > 0 the set
πΎ(π) has 2A-density zero.
Let π½ = (π½π ) be a sequence of infinite matrices with π½π = πππ π . Then π₯ = (π₯ππ ) β ββ
2
, the space of
bounded double sequences, is said to be F-convergent (2 π½-summable) to the value 2 π½ β ππππ₯ (denotes the
generalized limit) if
lim π (π½π π₯) π = lim π,πββ πππ
ππ
π = 2π½
π
π=0
π
π=0 -lim x (1.2)
uniformly for i > 0, m, n = 0, 1, β¦ . The method 2 π½ is regular if π½ = [πππ
ππ
]π,π=0
β
is RH-Regular. (see [8]).
Kolk [6] introduced the following:
An index set K is said to have π½-ππππ ππ‘π¦ πΏπ½ (πΎ) equal to d, if the characteristic sequence of K is π½-
summable to d, i.e.
lim π π ππ (π)πβπΎ = π, (1.3)
uniformly in i, where by index set we mean a set K= ππ β β, ππ < π1+π for all i.
Now we extend this definition as follows:
An indexed set K= (π, π) β β Γ β, ππ < ππ+1, ππ < π1+π for all i, is said to have 2π½-
ππππ ππ‘π¦ πΏ2π½ πΎ = π, if the characteristic sequence of K is 2π½-summable to d, i. e., if
lim ( )mn
mn jk
j K k K
b i d
ο ο
ο½ο₯ο₯ (1.4)
uniformly in i.
Let ββ
denote the set of all RH-regular methods 2π½ with πππ
ππ
(π) β₯ 0 for all j, k and i. let π½ π ββ
, A
double sequence π₯ = (π₯ππ ) is called 2π½-statistically convergent to the number L if for every π > 0 there exist a
subset K= π, π β β Γ β π, π = 1, 2, β¦ such that
2. Generalised Statistcal Convergence For Double Sequences
www.iosrjournals.org 2 | Page
πΏ2π½ π, π , π β€ π, π β€ π βΆ |π₯ππ β πΏ| β₯ π = 0 (1.5)
and we write π π‘2π½ -lim x = L.
We denote by π π‘(2π½), the space of all 2π½-statistically convergent sequences.
In particular, if π½ = (πΆ1), the Cesaro matrix, the π½-statistical convergence is reduced to C11-statistical
convergence.
II. ππ·-Statistical Cluster And Limit Points
We use the following examples to show that neither of the two methods, statistical convergence and
2π½-statistical convergence, implies the other.
Example 2.1: Consider the sequence of infinite matrices π½ = (π½π ) with
πππ
ππ
π =
1
π
+
1
ππ
, ππ π = π2
β π, π
1 β
π
π(π +1)
, ππ π = π2
+ 1 βπ, π
0, ππ‘ππππ€ππ π
It is clear that π½ π ββ
,
Now let the sequence π₯ = (π₯ππ ) πππ π¦ = (π¦ππ ) be defined by
π₯ππ =
0, ππ π = π2
, πππ πππ π πππ π πππ π β β
1
π
, ππ π = π2
+ 1, πππ πππ π πππ π πππ π β β
π, ππ‘ππππ€ππ π
and
π¦ππ =
π, ππ π = π2
, πππ πππ π πππ π πππ π β β
0, ππ π = π2
+ 1, πππ πππ π πππ π πππ π β β
1, ππ‘ππππ€ππ π
Then x is not statistically convergent to zero as πΏ (π, π): π₯ππ β₯ π β 0 but it is 2π½ -statistically
convergent to zero; and on the other hand y is statistically convergent but not 2π½-statistically convergent.
We now give some definitions for the method 2π½.
Definition 2.2: Let π½ π ββ
. Then number πΎ is said to be 2π½ βstatistical cluster point of a sequence π₯ππ if for
given π > 0, the set (π, π); π₯ππ β π < π does not have 2π½ β ππππ ππ‘π¦ π§πππ.
Definition2.3: Let π½ π ββ
. The number π is said to be 2π½ βstatistical limit point of a sequence π₯ππ if there is a
subsequence of π₯ππ which convergence to π such that whose indices do not have 2π½ β ππππ ππ‘π¦ π§πππ .
Denote by Ξ π₯ 2π½ , the set of 2π½ βstatistical cluster points and by Ξ π₯ (2π½) the set of 2π½ βstatistical
limit points of π₯ = (π₯ππ ) it is clear from the above examples, that Ξ π₯ 2π½ = {0} and Ξ π₯ 2π½ = 0 , Ξ π¦ 2π½ =
{0} and Ξ π¦ 2π½ = 0 . Throughout this paper we will consider π½ π ββ
.
Definition 2.4: Let us write πΊπ₯ = ππβ: πΏ2π½ ( (π, π): π₯ππ > π ) β 0 and πΉπ₯ = ππβ: πΏ2π½ ( π, π : π₯ππ < π ) β
0 for a double sequence π₯=π₯ππ . Then we define 2π½βstatistical limit superior and 2π½βstatistical limit inferior of
π₯ = π₯ππ as follows:
π π‘2π½ β lim ππ’ππ₯ =
ππ’ππΊπ₯ , ππ πΊπ₯ β β
ββ,, ππ πΊπ₯ = β
π π‘2π½ β πππ πΌπππ₯ =
πΌπππΉπ₯ , ππ πΉπ₯ β β
+β, ππ πΉπ₯ = β .
Definition 2.5: The double sequence π₯ = π₯ππ is said to be 2π½ βstatistically bounded if there is a number d
such that πΏ2π½ (π, π): π₯ππ > π = 0.
Definition 2.6: Consider the same π½ as defined in example 2.1, define the sequence π§ = π§ππ by
π§ππ =
0, ππ π = π2
, πππ πππ π πππ π πππ π β β
1, ππ π = π2
+ 1, πππ πππ π πππ π πππ π β β
π, ππ‘ππππ€ππ π.
Here we see that z is not bounded above but it is 2π½ βstatistically bounded for πΏ2π½ (π, π): π§ππ > 1 =
0. Also z is not statistically bounded. Thus πΊπ₯ = ββ,1 πππ πΉπ₯ = (0, β) so that π π‘2π½ β lim ππ’ππ§ =
3. Generalised Statistcal Convergence For Double Sequences
www.iosrjournals.org 3 | Page
1 πππ π π‘2π½ β ππππΌππ π§. Moreover Ξ π₯ 2π½ = 0,1 = π¬ π₯ (2π½) and z is neither 2π½ βstatistically nor statistically
convergent. In this example we see that z is 2π½ βstatistically bounded but not 2π½ βstatistically convergent. On
the other hand in example 2.1 y is statistically convergent and not 2π½ βstatistically bounded.
Also note that π π‘2π½ β lim ππ’ππ§ equals the greatest element of Ξ π₯ 2π½ while π π‘2π½ β ππππΌππ π§ is the
least element Ξ π₯ 2π½ . This observation suggests the following.
Theorem 2.7
(a) If π1 = π π‘2π½ β lim ππ’ππ₯ is finite, then for every positive number π
(i) πΏ2π½ (π, π): π§ππ > π1 β π β 0 and πΏ2π½ (π, π): π§ππ > π1 + π = 0.
Conversely, if (i) holds for every π > 0, then π1 = π π‘2π½ β lim ππ’ππ₯.
(b) if π2 = π π‘2π½ β lim πΌπππ₯ is finite, then from every positive number π
(ii) πΏ2π½ (π, π): π§ππ > π2 + π β 0 and πΏ2π½ (π, π): π§ππ > π2 β π = 0.
Conversely, if (i) holds for every π > 0 then π2 = π π‘2π½ β lim πΌπππ₯.
From definition 2.2 we see that the above theorem can be interpreted as showing that π π‘2π½ β
lim ππ’ππ₯ and π π‘2π½ β lim πΌπππ₯ are the greatest and the least 2π½ βstatistical cluster points of x.
Note that 2π½ βstatistical boundedness implies that π π‘2π½ β lim ππ’ππ₯ and π π‘2π½ β lim πΌπππ₯ are
finite, so that properties (i) and (ii) of Theorem 2.7 hold.
III. The Main Results
Throughout this paper by πΏ2π½ πΎ β 0; πΎ = {(π, π) β π Γ β} we mean that either πΏ2π½ πΎ > 0 ππ πΎ
does not have 2π½ β ππππ ππ‘π¦.
Theorem 3.1: For every real number sequence x, π π‘2π½ β lim πΌπππ₯ β€ π π‘2π½ β lim ππ’ππ₯.
Proof: First consider the case in which π π‘2π½ β lim ππ’ππ₯ = ββ, This implies that πΊπ₯ = β , Therefore for every
π β π , πΏ2π½ (π, π): π₯ππ > π = 0 , which implies that πΏ2π½ (π, π): π₯ππ β€ π = 1. So that for every π β π ,
πΏ2π½ (π, π): π₯ππ < π β 0. Hence π π‘2π½ β lim πΌπππ₯ = ββ,
Now consider π π‘2π½ β lim ππ’ππ₯ = +β. This implies that for every β π , πΏ2π½ (π, π): π₯ππ > π β 0. This
implies that πΏ2π½ (π, π): π₯ππ β€ π = 0. Therefore for every π β π , πΏ2π½ (π, π): π₯ππ < π = 0, which implies that
πΉπ₯ = β . Hence π π‘2π½ β lim πΌπππ₯ = +β.
Next assume that π1 = π π‘2π½ β lim ππ’ππ₯ < +β and let π2 = π π‘2π½ β lim πΌπππ₯. Given π > 0 we show
that π1 + π β πΉπ₯, so that π2 β€ π1 + π. By Theorem 2.7(a), πΏ2π½ (π, π): π₯ππ > π1 +
π
2
= 0, since π1 = ππ’π πΊπ₯ . This
implies that πΏ2π½ (π, π): π₯ππ β€ π1 +
π
2
= 1, which in turn gives πΏ2π½ (π, π): π₯ππ < π1 + π = 1. Hence π1 + π = πΉπ₯
and so that π2 β€ π1 + π i.e π2 β€ π1 since π was arbitrary.
Remark: For any double sequence π₯ = (π₯ππ )
π π‘2 β lim πΌπππ₯ β€ π π‘2π½ β lim πΌπππ₯ β€ π π‘2π½ β πππ ππ’ππ₯ β€ π π‘2 β πππ ππ’ππ₯
where
π π‘2 β πππ ππ’ππ₯ = π β πππππ’ππ₯
π π‘2 β lim πΌπππ₯ = π β ππππΌπππ₯
Theorem 3.2: For any double sequence π₯ = π₯ππ , 2π½ βstatistical boundness implies 2π½ βstatistical convergence
if and only if
π π‘2π½ β lim πΌπππ₯ = π π‘2π½ β πππ ππ’ππ₯.
Proof: Let π1 = π π‘2π½ β lim ππ’ππ₯ and π2 = π π‘2π½ β lim πΌπππ₯. First assume that π π‘2π½ β lim π₯ = πΏ and π > 0.
Then πΏ2π½ (π, π): π₯ππ β πΏ β₯ 0 = 0. So that
πΏ2π½ (π, π): π₯ππ > πΏ + π = 0,
which implies that π1 β€ πΏ. Also
πΏ2π½ (π, π): π₯ππ > πΏ β π = 0,
which implies that πΏ β€ π2. By theorem 3.1, we finally have π1 = π2
Conversely, suppose π1 = π2 = πΏ and x be 2π½ βstatistically bounded. Then for π > 0, by Theorem 2.7
we have πΏ2π½ (π, π): π₯ππ > π1 +
π
2
= 0 and πΏ2π½ (π, π): π₯ππ < π1 β
π
2
= 0. Hence π π‘2π½ β lim π₯ = πΏ
Theorem 3.3: If a double sequence π₯ = π₯ππ is bounded above and 2π½ β π π’ππππππ to the number πΏ = π π‘2π½ β
lim ππ’ππ₯, then x is 2π½ β π π‘ππ‘ππ π‘ππππππ¦ convergent to L.
Proof: Suppose that π₯ = π₯ππ is not 2π½ β π π‘ππ π‘ππππππ¦ convergent to L. Then by Theorem 3.2 π π‘2π½ β lim πΌπππ₯ <
πΏ. So there is a number M < L such that πΏ2π½ (π, π): π₯ππ < π β 0.
Let πΎ1 = (π, π): π₯ππ < π , then for every π > 0, πΏ2π½ (π, π): π₯ππ > πΏ + π = 0.
4. Generalised Statistcal Convergence For Double Sequences
www.iosrjournals.org 4 | Page
We write πΎ2 = (π, π): π β€ π₯ππ β€ πΏ + π and πΎ3 = (π, π): π₯ππ > πΏ + π , and let πΊ = ππ’πππ π₯ππ < β,
since πΏ2π½ (πΎ1) β 0, there are many n such that
lim ππ’ππ π ππππ π β₯ π > 0ββ
ππ =0,0
and for each n, i
π ππππ (π)π₯ππ < βββ
π,π=1,1 .
Now
π ππππ
ββ
ππ =1,1 π π₯ππ = +ππ βπΎ1
+Ξ£ππ βπΎ3ππ βπΎ2
π ππππ (π)π₯ππ
β€ π π ππππ π +ππ βπΎ1
(πΏ + π) π ππππ π + πΊππ βπΎ2
π ππππ πππ βπΎ3
= π π ππππ π +ππ βπΎ1
πΏ + π π ππππ πββ
ππ =1,1 β πΏ + π π ππππ π +ππ βπΎ1
π(π)
= β π ππππ π [βπ + (ππ βπΎ1
πΏ + π)] + πΏ + π π ππππ πββ
ππ =1,1 + π(1)
β€ πΏ π ππππ πββ
ππ =1,1 β π πΏ β π + π( π ππππ π β π) + π(1)ββ
ππ =1,1
Since π is arbitrarily, it follows that
lim πΌππ 2π½π₯ β€ πΏ β π(π β π) < πΏ.
Hence x is not 2π½ β summable to L
Theorem 3.4: If the double sequence π₯ = (π₯ππ ) is bounded below and 2π½ β π π’ππππππ to the number πΏ =
π π‘2π½ β πππ πΌπππ₯, then x is 2π½ β statisticallyconvergent to πΏ.
Proof. The proof follows on the same lines as that of Theorem 3.3.
Note: It is easy to observe that in the above Theorems 3.3 and 3.4 the boundedness of π₯ = (π₯ππ ) can not be
omitted or even replaced by the 2π½ β statistical boundedness.
For example consider the matrix π½ = π΄ = (π ππ ) π.π=1
β
and define π₯ = (π₯ππ ) by
π₯ = π₯ππ =
2π β 1, if k is an odd square for all π
2, if k is an even square for all π
1, if k is an odd nonsquare for all π
0, if k is an even nonsquare for all π
Then, ππ’π π₯ππ = β, ππ’π‘ πΊπ₯ = 2, β , πΉπ₯ = ββ,0 ,
π π‘2π½ β πππ ππ’ππ₯ = 2, and π π‘2π½ β πππ πΌπππ₯ = 0. [see [9])
This makes it clear that every bounded double sequence is π π‘2 β bounded and every π π‘2 β
bounded sequence is π π‘2π½ β bounded, but not conversely, in general.
IV. Conclusion
The double sequence which is bounded above and 2π½-summable to the number L = 2 limsupst xο’ ο ,
then it is 2π½-statistically convergent to L. Similarly, a double sequence which is bounded below and 2π½-
summable to the number ο¬ = 2 liminfst xο’ ο , then it is 2π½-statistically convergent to ο¬ .
References
[1] A. Pringsheim, Zur theoreie der zweifach unendlichen zahlenfolgen, Mathematical Annals. 53, 1900, 289β321.
[2] A. R. Freedman and J. J. Sember, Densities and summability, Pacific Journal of Mathematics, 95, no. 2, 1981, 293β305.
[3] R. C. Buck, Generalized asymptotic density, American. Journal of Mathematics. 75, 1953, 335 β 346
[4] J. S. Connor, The statistical and strong p-CesΓ‘ro convergence of sequences, Analysis, 8, no. 1-2, 1988, 47β63.
[5] E. Kolk, Matrix summability of statistically convergent sequences, Analysis, 13, 1981, 77β83.
[6] E. Kolk, Inclusion relations between the statistical convergence and strong summability, Acta et. Comm. Univ. Tartu. Math. 2, 1998,
39β54.
[7] M. Steiglitz, Eine verallgemeinerung des begriffs der fastdonvergenz, Math. Japon, 18, 1973, 53β70.
[8] M. Mursaleen and O. H. H. Edely, Generalized statistical convergence, Information Science, no. 162, 2004, 287 β 294.
[9] A. M. Alotaibi, CesΓ‘ro statistical core of complex number sequences, International. Journal of Mathematics and Mathematical
Science, Hindawi Publishing Corporation, 2007, Article ID 29869, 2007, pp 1β9.