International Journal of Mathematics Research.
ISSN 0976-5840 Volume 6, Number 1 (2014), pp. 27-30
© International Research Publication House
http://www.irphouse.com
On Statistical Limit Points
Sumita Gulati
Assistant Professor of Mathematics S. A. Jain College, Ambala City (Haryana)
Mini
Assistant Professor of Mathematics S. A. Jain College, Ambala City (Haryana)
Abstract:The purpose of this paper is to study the statistical analogue of the set of limit
points or cluster points of number sequences.
Key words:- Natural density, statistical convergence, statistical limit points,
statistical cluster points.
1. Introduction:The idea of statistical convergence was introduced by Fast [3] and later on studied by
several authors [1], [2], [4], [5], [6], [10], [13], [14]. In the whole paper IN stands for
the set of natural numbers and we shall consider the sequences of real numbers.
The natural density [11], of a set A⊆IN is defined as
(A)=lim ∞ | a є A: a ≤ n |
n
where the vertical bars indicates the number of elements in the enclosed set.
∞
If
is a sequence such that
satisfies property P for all k except for a
set of natural density zero, then we write satisfies P for “ almost all k” (a.a.k.).
∞
The sequence
is said to converge statistically to the real number
(denoted by
lim ∞
) if for each
we have
0,
: ξ
ξ
ε.
is a sequence and if
If
is a subsequence of
and
0,
is called a
:
then we denote
by x}K . If
subsequence of density zero or a thin subsequence. On the other hand
is a nonthin subsequence of if K does not have density zero. Moreover, it should be noted
that x}K is a non-thin subsequence of if either
0 or
is not defined
(i.e. K does not have natural density).
In [5] Fridy, introduced the concept of statistical limit points and statistical cluster
points of real number sequences. Recall that the number is a statistical limit point of
a sequence
provided there is a non-thin subsequence of
that
converges to and a number is a statistical cluster point if a set
: ξ
γ
does not have density zero for every
0. It was established that for a bounded
sequence the set of statistical limit points may be empty while the set of statistical
cluster points is non empty.
28
Sumita Gulati and Mini
2. Statistical limit points and statistical cluster points
In this section we study the basic properties of statistical limit points and statistical
cluster points. We also discuss the similarities and differences between these points
and ordinary limit points.
• Definition (a): The number L is an ordinary limit point of a sequence if there
is a subsequence of x that converges to L.
• Definition (b): The number is a statistical limit point of a sequence if there
is a non-thin subsequence of that converges to λ.
For any sequence ,
denotes the set of statistical limit points and
the set of ordinary limit points respectively.
∞
Let us consider a sequence x
:
:
,
0, 1
0 . So, we have
⊆
then
different for this let us consider another sequence,
. But
and Lx can be very
whose range is the set
∞
r ;if k=
, , ,…
n
∞
of all rational numbers and define
by k
k ; otherwise.
Since the set of squares have density zero it follows, Λ
. While, the fact the
:
implies that
Ι (i.e. set of real
every real number is a limit point of
numbers).
• Definition (c): The number γ is a statistical cluster point of a sequence if for
|
every
0, the set k є IN:|
does not have density zero.
For a given sequence , let Γ denotes the set of all statistical cluster points of .
It is clear that Γ ⊆
for every sequence .
3. Main Results:
The following proposition gives the inclusion relation between Γ
Proposition 1. For any number sequence ,
.
⊆Γ .
∞
say
}
Proof: Suppose
Λ . Then a non-thin subsequence of
that converges to
and δ K
d 0, where K={k(1), k(2), k(3), …}. As,
, so for each
0 a positive integer jo , such that ξ
λ
. Hence for each
k j :j
Therefore,
0, j: ξ
λ
ε is a finite set and so,
: ξ
λ
IN ~ finite set .
n
k
n: ξ
λ
n
|k j
n|
n
0 1
and this implies
|
| d. Hence δ k IN: ξ
limn→∞ | k ≤ n: |
λ
0. Thus λ is a
n
statistical cluster point of x. Hence Λ ⊆ Γ .
The completes the proof.
Athough our experience with ordinary limit points may lead us to expect that Λ
and Γ are equivalent. The next example shows that this is not always the case.
On Statistical Limit Points
29
2
, where
Example 2: Define a sequence
2
1 ; .
1 is the number of factors of 2 in the prime factorization of k.
It is easy to see that for each p, δ k: ξ
2
0. Thus
Λ . Also,
δ k: 0
2
ξ
so, 0 Γ . Hence, Γ
O
a subsequence that has limit zero then we have δ K
. But 0
lim
k n: ξ
,
is
0.
Proposition 3: If x is a statistically convergent sequence say
and Γ are both equal to singleton set
.
Proof:- As
lim ∞
Λ for if,
lim
then Λ
0,
λ
0.
ε
This implies δ Aε
: ξ
λ
ε and so δ
0 where A
1
|
|
|
|
where
: ξ
λ
and thus δ
: ξ
λ
0.
Therefore, λ Γ . Similarly, we can show that λ Λ . Further, it can be proved that
λ is the only statistical limit point and statistical cluster point of the sequence x.
This completes the proof..
The converse of the above proposition need not hold. For this consider the
following example.
Example 4: Consider the sequence
defined by
1
1
. We
have, Λ
0 ,Γ
0 . But
does not converges statistically to zero.
From example 2, we see that Λ need be a closed point set, but the next result
shows that Γ like
is always a closed set.
Propositions 5: For any sequence x, the set Γ of statistical cluster points of x is a
closed point set.
Proof: Let p be an accumulation
point γ in
,
. Choose
,
⊆
,
|
Since є Γ , , δ k: ξ
γ|
i.e. δ k: ξ
,
Therefore, δ k: ξ
,
Hence
Γ
This completes the proof.
Theorem 6: If
. . ., then Λ
Λ and Γ
point of Γ .Then for every
so that,
.
є
0,
0.
0.
.
and
Γ .
, Γ contains some
are sequences such that
0. Let λ Λ , so a non thin
Proof: As
. . ., therefore, δ k: ξ
η
subsequence of x say
that converges to . Since δ k: k K and ξ
η
0, it
follows that the :
does not have density zero. Therefore, the
30
Sumita Gulati and Mini
set :
yields a non-thin subsequence
that
converges to . Hence λ Λ . This implies Λ ⊆ .
By symmetry, we see that Λy ⊆ Λx .
Therefore, Λx Λy .
The assertion that Γx Γy can be proved by a similar argument.
This completes the proof.
Following theorem establishes a connection between statistical cluster points and
ordinary limit points.
Theorem 7: If
is a number sequence then a sequence
. .
Γ and
. . ,. moreover, the range of y is a subset of the range of x.
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