On asymptotically statistical equivalent sequences of interval numbers

In this paper we introduce and study the concepts of asymptotically s  statistical equivalent and strongly asymptotically   equivalent sequences for interval numbers and prove some inclusion relations. In the last section we introduce the concept of Cesaro asymptotically   equivalent sequences defined by Orlicz function of multiple L and we have established some relation between those classes.

The idea of statistical convergence for single sequences was introduced by Fast (1952).Schoenberg (1959) studied statistical convergence as a summability method and listed some of elementary properties of statistical convergence.Both of these authors noted that if bounded sequence is statistically convergent, then it is Cesaro summable.
A set consisting of a closed interval of real numbers x such that a x b   is called an interval number.A real interval can also be considered as a set.We denote the set of all real valued closed intervals by I Any elements of I is a closed interval and denoted by Hence an interval number x is a closed subset of real numbers (CHIAO, 2002).Let l x and r x be first and last points of x interval number, respectively.For 1 2 , x x  I , we have, The set of all interval numbers I is a complete metric space defined by , 2002) In the special case x  i w denotes the set of all interval numbers with real terms and the algebraic properties of i w can be found in Chiao (2002).
Now we give the definition of convergence of interval numbers: Definition 1.1.Chiao (2002)

Main results
In this paper, we introduce and study the concepts of asymptotically s   statistical equivalent and strongly   asymptotically equivalent sequences for interval numbers. Let and let  denote the set of all non-decreasing sequences  .
In Esi (2011), introduced the concept of statistical   convergence of interval numbers as follows: In this case we write lim .
then   statistically convergence reduces to statistically convergence as follows: In this case we write lim .
Following this result we introduce two new notions, namely asymptotically   statistical equivalent of multiple L and strong asymptotically The next definition is natural combination of Definition 2.1 and 2.2.
of interval numbers with 0 k y  for all k  are said to be asymptotically s   statistical equivalent of multiple L provided that for every for all k  are said to be asymptotically statistical equivalent of multiple L provided that for every  for all k  are said to be strongly Cesaro asymptotically equivalent of multiple L provided that

If we take
and simply strongly Cesaro asymptotically equivalent if x y x y


Then we can assume that x y  Further, we have x y  (iii) Follows from (ii) and (iii).
Proof.For given 0   , we have Taking limit as n   and using lim inf where r>M.Then we have: Finally we conclude this section by stating a definition which generalizes Definition 2.5. of this section and Theorem 2.2 related to this definition.
equivalence reduces to strong Cesaro asymptotically p   equivalence as follows: The following theorem is similar to that of Theorem 2.1.,so the proof omitted.  equivalent of multiple L with respect to an Orlicz function provided that are Cesaro strong asymptotically   equivalent of multiple L with respect to an Orlicz function provided that for some constant  Proof.Proof of the proposition is similar to that given in Theorem 2.3, using into consideration that M is non-decreasing function, so it is omitted.

Conclusion
In this paper are given and studied the concepts of asymptotically s   statistical equivalent and strongly asymptotically   equivalent sequences for interval numbers and are proved some inclusion relations.This results are extension of the known results from asymptotically equivalent real numerical sequences.Also are given the concept of Cesaro asymptotically   equivalent sequences defined by Orlicz function of multiple L and we have established some relation between those classes.

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for all k  are said to be asymptotically equivalent if which is continuous, nondecreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and ( )M x   as .x   It is well known that if M is a convexfunction and M (0RUTITSKY, 1961).We will define the following asymptotically   statistical equivalences: Definition 3.1.Two sequences ( ) all k  are Cesaro asymptotically Acta Scientiarum.Technology Maringá, v. 35, n. 3, p. 515-520, July-Sept., 2013 Now taking into consideration 2  conditions of Orlicz functions, we get the following estimation: s R 