On some generalizations of statistical boundedness
© Bhardwaj and Gupta; licensee Springer. 2014
Received: 19 September 2013
Accepted: 18 December 2013
Published: 9 January 2014
Fridy and Orhan (Proc. Am. Math. Soc. 125(12):3625-3631, 1997) introduced the concepts of statistical boundedness, statistical limit superior, statistical limit inferior, and they established an analog of Knopp’s Core Theorem. In the present paper, we examine the concept of statistical boundedness to establish statistical analogs of various well-known results concerning boundedness and generalize the concept of statistical boundedness by introducing the concepts of statistical boundedness of order α, λ-statistical boundedness, and λ-statistical boundedness of order α.
MSC:40C05, 40A05, 46A45.
Keywordssequence space statistical boundedness statistical convergence Köthe-Toeplitz duals
The idea of statistical convergence was given by Zygmund  in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was formally introduced by Steinhaus  and Fast  and later reintroduced by Schoenberg . Over the years and under different names statistical convergence has been applied in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory, and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Šalàt , Fridy , Connor , Fridy and Orhan , Rath and Tripathy , Mòricz , Malkowsky and Savas , Mursaleen , Et , Bhardwaj and Bala [14, 15], Savas [16–19], Çolak , Savas and Mohiuddine , Mohiuddine et al. , and many others.
where the vertical bars indicate the total number of elements in the enclosed set . It is clear that any finite subset of ℕ has zero natural density and , where . A set A is said to be statistically dense  if . A subsequence of a sequence is said to be statistically dense if the set of all indices of its elements is statistically dense. A sequence (of real or complex numbers) is said to be statistically convergent to some number L if for every , . In this case, we write . We shall denote the set of all statistically convergent sequences by .
Throughout this paper, we deal with sequences of scalars. We recall  that if is a sequence such that satisfies property P for all k except a set of natural density zero, then we say that satisfies P for ‘almost all k’ and we abbreviate this by ‘a.a.k’.
The following concept is due to Fridy . A sequence is said to be statistically Cauchy if for each , there exists a number such that , for a.a.k, that is, . It was shown that a number sequence is statistically convergent if and only if it is statistically Cauchy.
Fridy  enriched the theory of statistical convergence by introducing the concept of statistical limit point and statistical cluster point as follows: If is a subsequence of and , then abbreviate by . In case , is called a nonthin subsequence of x. The number ζ is a statistical limit point of the number sequence x provided that there is a nonthin subsequence of x that converges to ζ. The number γ is a statistical cluster point of the number sequence x provided that for every , the set does not have density zero.
where and is a non-decreasing sequence of positive numbers tending to ∞ such that , . We shall denote the set of all such sequences by Γ and the set of all λ-statistically convergent sequences by .
provided the limit exists. It is clear that any finite subset of ℕ has zero α-density. Note that the α-density of any set reduces to the natural density of the set in the case .
for every . The set of all statistically convergent sequences of order α will be denoted by .
where , and . The concept of λ-statistical convergence of order α is a generalization of the concepts of statistical convergence, λ-statistical convergence, and statistical convergence of order α.
Quite recently, the concepts of ‘λ-statistical convergence of order α for the sequences of function’ and ‘the pointwise and uniform statistical convergence of order α’ have been introduced by Et et al.  and Çinar et al. , respectively.
In 1997, a statistical analog of a very basic property of convergent sequences was given by Fridy and Orhan  by the formal introduction of the concept of statistical boundedness as follows: ‘The real number sequence x is statistically bounded if there is a number B such that ’.
They developed the statistical analogs of properties of the ordinary limit superior and inferior, and a natural analog of Knopp’s core. In the same year, Tripathy  proved the Decomposition Theorem for statistically bounded sequences and established the necessary and sufficient condition for a sequence to be statistically bounded.
In the present paper, we take the opportunity to contribute to the theory of statistical convergence, by introducing the concepts of λ-statistical boundedness, statistical boundedness of order α, and λ-statistical boundedness of order α. In the second section of this paper, we study and examine the concept of statistical boundedness to establish statistical analogs of various well-known results concerning bounded sequences. The third section is devoted to the introduction of the concept of statistical boundedness of order α where . The relationship between statistical boundedness of order α and statistical boundedness of order β for has been studied. In the fourth section of this paper, we introduce the concept of λ-statistical boundedness and prove inclusion relations between , and under different conditions. In the last section, we introduce the concept of λ-statistical boundedness of order α and establish the condition for a statistically bounded sequence to be λ-statistically bounded of order α. It is observed that the concept of λ-statistical boundedness of order α is more general than the concepts of statistical boundedness, λ-statistical boundedness, and statistical boundedness of order α as it includes these concepts for some particular choices of α and λ.
normal (or solid) if whenever , , for some ,
monotone if it contains the canonical preimages of all its stepspaces,
sequence algebra if whenever ,
symmetric if implies where π is a permutation on ℕ.
Obviously , where ϕ is the well-known sequence space of finitely non-zero scalar sequences. Also if , then for or β. For any sequence space X, we denote by where or β. It is clear that where or β.
For a sequence space X, if then X is called a Köthe space or a perfect sequence space.
2 Some more results on statistical boundedness
In the present section, we examine the concept of statistical boundedness in greater detail by giving statistical analogs of some of the results concerning boundedness.
Definition 2.1 
By , we shall denote the linear space of all statistically bounded sequences.
Bounded sequences are obviously statistically bounded as the empty set has zero natural density. However, the converse is not true, as the following example demonstrates.
Clearly is not a bounded sequence. However, , as the set of squares has zero natural density and hence is statistically bounded.
Proposition 2.3 Every convergent sequence is statistically bounded.
Although a statistically convergent sequence does not need to be bounded, the following proposition shows that every statistical convergent sequence is statistically bounded.
Proposition 2.4 Every statistical convergent sequence is statistically bounded.
Proof Let converges to L statistically. So for each , . As , so a.a.k. □
The following example illustrates that not every sequence is statistically bounded.
Example 2.5 Consider the sequence . Then for any , we have , where S is a finite subset of N and so . Thus is not statistically bounded.
It is well known that every subsequence of a bounded sequence is bounded. However, for statistically bounded sequences this is no longer true. For example, the sequence is a subsequence of the statistically bounded sequence , as given in Example 2.2 but is not statistically bounded.
A useful characterization of a statistically bounded sequence in terms of its bounded subsequence has been given by Tripathy  as follows.
Theorem 2.6  is statistically bounded if and only there exists such a set such that and , the space of all bounded scalar sequences.
We give another characterization of a statistically bounded sequence in terms of its statistically dense subsequences as follows.
Theorem 2.7 A sequence is statistically bounded if and only if every statistically dense subsequence of it is statistically bounded.
The proof is similar to that of the corresponding result for statistical convergence in [, Theorem 2.1] and hence is omitted.
We now give a statistical analog of the well-known result that ‘If is a null sequence and is a bounded sequence, then is a null sequence.’
Proposition 2.8 If converges statistically to 0 and , then converges statistically to 0.
Fridy  has shown that a sequence is statistically convergent if and only if there exists a convergent sequence such that a.a.k. We now establish a similar result for statistically bounded sequences. Our proof is quite different from that of Fridy.
Theorem 2.9 is a statistically bounded sequence if and only if there exists a bounded sequence such that a.a.k.
Then and a.a.k.
Conversely, as so there exists such that for all . Let . Clearly . This gives a.a.k because . □
Remark 2.10 Tripathy  has shown that where and denote the spaces of statistically null and bounded sequences, respectively. We observe that . But the representation of every element of as the sum of an element of and of is not unique, i.e., as . In fact, where is the space of null sequences.
Our next result is a statistical analog of a well-known result stating that every Cauchy sequence is bounded but not conversely.
Theorem 2.11 Every statistically Cauchy sequence is statistically bounded, but the converse does not need to be true.
Proof Let be a statistically Cauchy sequence. Then for each , there exists a number N () such that a.a.k. This implies that a.a.k, where .
The sequence is bounded and hence statistically bounded but is not statistically Cauchy. □
Proposition 2.12 
Suppose x is a number sequence that has a monotone subsequence . If and x is bounded on M, then x is statistically convergent.
Corollary 2.13 Every monotone and statistically bounded sequence is statistically convergent.
Theorem 3 of  can be restated in terms of statistical boundedness as follows.
Theorem 2.14 Every statistically bounded sequence has a statistical cluster point.
is not symmetric.
is normal and hence monotone.
is a sequence algebra.
Let and be a rearrangement of terms of the sequence . Clearly for any , .
Let and be such that for all . As , there exists such that . Clearly as . In view of the fact  that every normal space is monotone, is monotone.
Let . Then there exists such that and . The proof follows from the fact that .
Proposition 2.16 , the space of finitely non-zero scalar sequences.
Then and so . But , which is a contradiction to the fact that . Hence .
In view of the fact  that for a monotone sequence space α- and β-dual spaces coincide, we have . □
The proofs of the following Tauberian theorems are similar to the proofs of Theorem 3 and Theorem 5 in  and hence are omitted.
Theorem 2.17 If is a sequence such that and , then .
Theorem 2.18 Let be an increasing sequence of positive integers such that and let x be a corresponding gap sequence: if for ; if , then .
3 Statistical boundedness of order α
In the present section, we introduce and examine the concept of statistical boundedness of order α, where α is any real number such that .
By , we shall denote the linear space of all statistically bounded sequences of order α. For , the statistical boundedness of order α is the same as the statistical boundedness. We do not talk of statistical boundedness of order α, , since in this case , the space of all scalar sequences, i.e., statistical boundedness, becomes trivial.
Theorem 3.2 for all α, and the inclusion is strict.
Then . But , for and so for . □
Now, one may ask a natural question: Is there any unbounded sequence which is not statistically bounded of order α, for any α ()? The answer is yes. In fact the sequence is an unbounded sequence but there does not exist any α, such that is statistically bounded of order α.
Theorem 3.3 Let . Then and the inclusion is strict for .
and this gives . The inclusion is strict in view of the example cited in Theorem 3.2 as for , but for . □
if and only if .
i.e., every statistically bounded sequence of order α () is statistically bounded. In fact if and only if .
The next theorem shows that a statistically convergent sequence of order α is statistically bounded of order α ().
Theorem 3.5 For , , the inclusion being strict.
4 λ-Statistical boundedness
In the present section, we introduce and examine the concept of λ-statistical boundedness and obtain the condition on for which the concept of λ-statistical boundedness coincides with the concept of statistical boundedness.
where . For , λ-statistical boundedness is the same as statistical boundedness. By , we shall denote the linear space of all λ-statistically bounded sequences.
Theorem 4.2 , the inclusion being strict.
and for . □
If , then .
If , then .
If , then .
- (b)Let . Then
The proof follows from parts (a) and (b). □
If , then .
If , then and hence .
Theorem 4.5 Every λ-statistically convergent sequence is λ-statistically bounded.
The proof is easy and therefore is omitted.
5 λ-Statistical boundedness of order α
In this last section, we generalize the concepts of statistical boundednes, statistical boundedness of order α, and λ-statistical boundedness by introducing the concept of λ-statistical boundedness of order α.
for is the same as the statistical boundedness of order α,
for is the same as the λ-statistical boundedness,
for , is the same as the statistical boundedness.
Theorem 5.2 , the inclusion being strict.
The proof is easy and therefore is omitted.
The proof of the following theorem is similar to the proof of Theorem 3.3 in  and therefore is omitted.
If , then .
If and , then .
If , then .
If , then and so .
Theorem 5.5 Every λ-statistically convergent sequence of order α is λ-statistically bounded of order α.
The proof is easy and therefore is omitted.
The authors wish to thank the referees for their valuable suggestions, which have improved the presentation of the paper.
- Zygmund A: Trigonometric Series. 2nd edition. Cambridge University Press, Cambridge; 1979.Google Scholar
- Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 2: 73-74.MathSciNetGoogle Scholar
- Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241-244.MathSciNetMATHGoogle Scholar
- Schoenberg IJ: The integrability of certain functions and related summability methods. Am. Math. Mon. 1959, 66: 361-375. 10.2307/2308747MathSciNetView ArticleMATHGoogle Scholar
- Šalàt T: On statistically convergent sequences of real numbers. Math. Slovaca 1980,30(2):139-150.MathSciNetMATHGoogle Scholar
- Fridy JA: On statistical convergence. Analysis 1985, 5: 301-313.MathSciNetView ArticleMATHGoogle Scholar
- Connor JS: The statistical and strong p -Cesàro convergence of sequences. Analysis 1988, 8: 47-63.MathSciNetView ArticleMATHGoogle Scholar
- Fridy JA, Orhan C: Lacunary statistical convergence. Pac. J. Math. 1993, 160: 43-51. 10.2140/pjm.1993.160.43MathSciNetView ArticleMATHGoogle Scholar
- Rath D, Tripathy BC: On statistically convergent and statistically Cauchy sequences. Indian J. Pure Appl. Math. 1994,25(4):381-386.MathSciNetMATHGoogle Scholar
- Mòricz F: Statistical convergence of multiple sequences. Arch. Math. 2003, 81: 82-89. 10.1007/s00013-003-0506-9View ArticleMATHGoogle Scholar
- Malkowsky E, Savas E: Some λ -sequence spaces defined by a modulus. Arch. Math. 2000,36(3):219-228.MathSciNetMATHGoogle Scholar
- Mursaleen M: λ -Statistical convergence. Math. Slovaca 2000,50(1):111-115.MathSciNetMATHGoogle Scholar
- Et M: Strongly almost summable difference sequences of order m defined by a modulus. Studia Sci. Math. Hung. 2003,40(4):463-476.MathSciNetMATHGoogle Scholar
- Bhardwaj VK, Bala I: On weak statistical convergence. Int. J. Math. Math. Sci. 2007., 2007: Article ID 38530Google Scholar
- Bhardwaj VK, Bala I: On lacunary generalized difference sequence spaces defined by Orlicz functions in a seminormed space and -lacunary statistical convergence. Demonstr. Math. 2008,41(2):415-424.MathSciNetMATHGoogle Scholar
- Savas E: Strong almost convergence and almost λ -statistical convergence. Hokkaido Math. J. 2000, 29: 531-536. 10.14492/hokmj/1350912989MathSciNetView ArticleMATHGoogle Scholar
- Savas E: On statistically convergent sequences of fuzzy numbers. Inf. Sci. 2001,137(1-4):277-282. 10.1016/S0020-0255(01)00110-4MathSciNetView ArticleMATHGoogle Scholar
- Savas E: On asymptotically λ -statistical equivalent sequences of fuzzy numbers. New Math. Nat. Comput. 2007,3(3):301-306. 10.1142/S1793005707000781MathSciNetView ArticleMATHGoogle Scholar
- Savas E:On -statistically convergent double sequences of fuzzy numbers. J. Inequal. Appl. 2008., 2008: Article ID 147827Google Scholar
- Çolak R: On λ -statistical convergence. Conference on Summability and Applications, İstanbul, Turkey, May 12-13 2011.Google Scholar
- Savas E, Mohiuddine SA: -Statistically convergent double sequences in probabilistic normed spaces. Math. Slovaca 2012,62(1):99-108. 10.2478/s12175-011-0075-5MathSciNetView ArticleMATHGoogle Scholar
- Mohiuddine SA, Alotaibi A, Mursaleen M: A new variant of statistical convergence. J. Inequal. Appl. 2013., 2013: Article ID 309Google Scholar
- Niven I, Zuckerman HS: An Introduction to the Theory of Numbers. 4th edition. Wiley, New York; 1980.MATHGoogle Scholar
- Burgin, M, Duman, O: Statistical convergence and convergence in statistics. arXiv:math/0612179 [math.GM]Google Scholar
- Fridy JA: Statistical limit points. Proc. Am. Math. Soc. 1993, 118: 1187-1192. 10.1090/S0002-9939-1993-1181163-6MathSciNetView ArticleMATHGoogle Scholar
- Çolak R: Statistical convergence of order α . In Modern Methods in Analysis and Its Applications. Anamaya Pub., New Delhi; 2010:121-129.Google Scholar
- Çolak R, Bektaş ÇA: λ -Statistical convergence of order α . Acta Math. Sin. Engl. Ser. 2011,31(3):953-959.MATHGoogle Scholar
- Et M, Çinar M, Karakaş M: On λ -statistical convergence of order α of sequences of function. J. Inequal. Appl. 2013., 2013: Article ID 204Google Scholar
- Çinar M, Karakaş M, Et M: On pointwise and uniform statistical convergence of order α for sequences of functions. Fixed Point Theory Appl. 2013., 2013: Article ID 33Google Scholar
- Fridy JA, Orhan C: Statistical limit superior and limit inferior. Proc. Am. Math. Soc. 1997,125(12):3625-3631. 10.1090/S0002-9939-97-04000-8MathSciNetView ArticleMATHGoogle Scholar
- Tripathy BC: On statistically convergent and statistically bounded sequences. Bull. Malays. Math. Soc. 1997,20(1):31-33.MathSciNetMATHGoogle Scholar
- Cooke RG: Infinite Matrices and Sequence Spaces. Macmillan & Co., London; 1950.MATHGoogle Scholar
- Kamthan PK, Gupta M: Sequence Spaces and Series. Dekker, New York; 1981.MATHGoogle Scholar
- Köthe G, Toeplitz O: Lineare Räume mit unendlich vielen Koordinaten und Ringe unendlicher Matrizen. J. Reine Angew. Math. 1934, 171: 193-226.MathSciNetGoogle Scholar
- Chillingworth HR: Generalized ‘dual’ sequence spaces. Nederl. Akad. Wetensch. Proc. Ser. A 61 - Indag. Math. 1958, 20: 307-315.MathSciNetView ArticleGoogle Scholar
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