Fuzzy Statistical Limits

Fuzzy Statistic al Limits Mark Burgin a and Oktay Dum an b a Department of Mat hematics, University of California, Los Angeles, California 90095-1555, USA b TOBB Economics an d Technology University, Fa culty of Arts and Sciences, Departmen t of Mathematics, Sö ğ ütözü 06530, Ankara, Turkey Abstract Statistical limits are defined relaxing conditions on conventional convergence. The main idea of the statistical convergence of a sequence l is that t he majority of el ements from l converge and we do not care what is going on with other elements. At the sam e time, it is known that sequences that com e f rom real life sources, such as measurement and computation, do not allow, in a g eneral case, to test whe ther they converge or statistically converge in the strict m athematical sense. To overcome t hese limitations, fuzzy convergence was introduced ea rlier in the context of neoclassical analysis and fuzzy sta tistical convergen ce is introduced and studied i n this paper. We find relations between fuz zy statistical convergence of a sequence an d fuzzy statistical convergence of its subsequences (Theorem 2.1), as well as between fuzzy statistical convergence of a sequence and convention al convergence of its subsequences ( Theorem 2.2). I t is demonstrated what operations with fuzzy stat istical limits a re i nduced by o perations on sequences (T heorem 2.3) and h ow fuzzy st atistical limits of different sequences i nfluence one another (Theorem 2.4). In Sect ion 3, r elations between fuzzy statistical convergence and fuzzy converg ence of statistical characteristics, such as the m ean (average) and standard deviation, are studied ( Theorems 3.1 and 3.2). Keywords : statistical convergence, fuzzy sets, fuzzy limits, statistics, mean, standard deviation, fuzzy conv ergence Mark Burg in and Oktay Duman 2 1. Introduction Statistical limits are defined relaxing c ondition s on conventional convergence. Conventional convergence in anal y sis d emands almost all elements of the sequence to satisfy the conver gence condition. Namely, almo st all elements of the sequence h ave to belong to arbitraril y small neighborhood of the lim it. The main idea of stati stical convergence is to relax thi s condition and to demand validi ty of the convergence condition onl y for a majorit y of elements. As alwa ys, stati stics cares onl y about big quantities and majorit y is a surro gate of the con cept "almost all" in pure mathematics. The reason is that statisti cs works with finite populations and sampl es, while pure mathematics is mostl y interested in infini te sets. However, the idea o f statisti cal convergence, which eme rged in the first edition (published in W arsaw in 1935) of the mo nograph of Zygmund [ 19], stemmed no t from statistics but from problems of series s ummation. The concept of statist ical convergence was formali zed by Steinhaus [18] and Fast [10] and later reintroduced b y Schoenber g [16]. Since that time, statistical conver gence ha s become an area of active r esearch. Researchers studied properties of statistical convergence and applied this concept in various areas: measure t heor y [13], trigonometric series [ 19], approxi mation theory [9], locally convex spaces [12] , summ ability theo ry and th e limit point s of sequences [7], [15], densi ties of natural numbers [14] , in the study of subsets of t he Stone- Č hech compactification of the set of natural numbe rs [6], and Banach spaces [8] . However, as a rule, neither l imits nor statisti cal limits can be calcul ated o r measured with absolute precision. To reflect this imprecisio n and to model it by mathematic al structures, sev eral appro aches in mathematics have been developed: fuzz y set theory, fuzz y logic, interval ana lysis, set valued analysis, etc. One of these app roaches is the neoclassical anal ysis (cf., for example, [2, 3] ). In i t, ordinar y st ructures o f anal y sis, that is, functions, sequences, series, and operators, are st udied by m eans of f uzz y conc epts: fuzz y limits, fuzzy continuit y, and fuzz y derivatives. For exampl e, continuous functions, which are stu died in the classical anal ysis, become a part of the set of the fuzz y continuous functions studied in neoclassical a nalysis. Neoclassi cal anal ysis extends Fuzzy Sta tistical Converg ence of Nu mber Seq uences 3 methods of classi cal calculu s t o reflect uncertaint ies t hat arise in computations and measurements. The aim o f the present paper is to extend and study the concept of statistical convergence utiliz ing a f uzz y logic appro ach and principles of the neoclas sical analysis , which is a new branch of fuzz y mathematics and extends possibilities provided b y the classical anal ysis [2, 3]. Ideas of fuzz y lo gic and fuz zy set theor y have been used not onl y in man y applicati ons, s uch as, in bifurcation of non-lin ear d ynamical systems, in the control of chaos, in the comp uter programming, in the quantum ph y sics, but also in various branches of m athematics, such as, the ory o f metric and topo logical sp aces, studies of convergence of sequences and fun ctions, in the theor y of linear systems, etc. In the second section of this paper, going after int roduction, we introduce a new t ype of statistical convergence, th e conc ept of fuzz y st atistical convergence, and give a useful characterization of this type of convergence. In the third section, we consider relations between fuzz y statistical convergence and fuzz y c onvergence of statistical characteristics such as the mean (average) and stand ard deviation. For simplicity, we consider here only sequences of real n umbers. However, in a similar wa y, it is possible to define statistical fuzz y convergence for sequences of complex numbers and obtain s imilar properties. In what follows, N denot es the set of all natural num bers and R denotes the set of all natural numbers. 2. Fuzzy statistical convergenc e Here we extend the concept of statis tical convergence to the concept of fu zz y statistical convergence, which, as we h ave discussed, is mo re realistic for real life applications. For conveni ence, throughout t he paper, r denotes a non-ne gative re al number and l = { a i ; i = 1, 2, 3, …} represents a sequen ce of real numbers. Consider a subset K of the set N . Let K n = { k ∈ K ; k ≤ n }. Mark Burg in and Oktay Duman 4 The ( asymptotic ) density d ( K ) of a set K is equal to lim n →∞ (1/ n ) | K n | whenever this limit exists (cf., for example, [11] , [14]). Changing the conventional lim it in this formula to the r -limi t [3], we obtain fuzz y asymptotic densit y. Definition 2 .1. The asymptotic r - density d r ( K ) of the set K is equal to r -lim n →∞ (1/ n ) | K n |, whenever the fuzz y limit exists; here | B | denotes the cardinalit y of the set B . As a rule, r -limi t o f a sequence is not uniquel y defined, i.e., one sequence can have many r -lim its. Consequently, several num bers ca n be equal to the asy mptotic r -density d r ( K ) of the same set K . Example 2.1. Let us tak e K = { 2 i ; i = 1, 2, 3, …}. Then ½ = d ½ ( K ), ¼ = d ½ ( K ), and 3/5 = d ½ ( K ). However, in some cases, the as ymptotic r -density d r ( K ) is unique. Lemma 2.1. The as ymptotic 0-densit y o f any subset K of N (if it exists) coincides with the asymptotic densit y of K. This shows that as y mptoti c r -density i s a natural extension of the c oncept of asymptotic densit y. Lemma 2.2. If x = d r ( K ), then x = d q ( K ) for an y q > r . Corollary 2.1. If the asymptotic d ensity d ( K ) of a su bset K of N is equal to x , then x = d r ( K ) for an y r > 0. Fuzzy Sta tistical Converg ence of Nu mber Seq uences 5 However, the fol lowing example s hows that th e converse of C orollary 2.1 is not always true. Moreover, asymptotic r -densit y can exis t in such cases when as ymptotic density does not exist. Example 2.2. Let K be the set of all even positive integers with an even number of digits to base t en. Then it i s known from [ 14; pp: 248-249] that the densit y d ( K ) does not exists. However, Definition 2.1 d irectly implies d ½ ( K ) = 0. Furthermore, as 0 ≤ | K n | ≤ n , Definition 2.1 yields the following resul ts. Lemma 2.3. If k = r -lim n →∞ (1/ n ) | K n | for a set K ⊆ N , then - r ≤ k ≤ 1 + r . Lemma 2.4. 0 = 1-lim n →∞ (1/ n ) | K n | a nd 1 = 1-lim n →∞ (1/ n ) | K n | f or an y set K ⊆ N . It me ans that for r = 1, the concept of asymptoti c r -density, i n some sense, degenerates as an y numb er between 0 and 1 b ecomes the as ymptotic r -de nsity of an y set K ⊆ N . Let K and H be subsets of N , t hen properties of fuzz y lim its impl y the following result. Lemma 2.5. If k = d r ( K ) and h = d q ( H ), then d r + q ( K ∪ H ) exists and d r + q ( K ∪ H ) ≤ k + h Corollary 2.2. If k = d ( K ) and h = d ( H ), then d ( K ∪ H ) exi sts and d ( K ∪ H ) ≤ k + h Corollary 2.3. If k = d r ( K ) and h = d ( H ), then d r ( K ∪ H ) exists and d r ( K ∪ H ) ≤ k + h Mark Burg in and Oktay Duman 6 As in the case of asymptotic density [5], there are connections between as ymptotic r - density and fuzz y Cesáro sum mability. As usual, th e well-know n Cesáro m atrix C = { c nk ; n,k = 1,2,3,...} is defin ed b y its coefficients 1 / n , if 1 ≤ k ≤ n c nk = 0, otherwise. Using summabilit y methods, we obtain the following result. Proposition 2 .1. If th e as ymptotic r -densit y d r ( K ) of a subset K of N exists, then w e have d r ( K ) = r -lim n →∞ (1/ n ) ∑ 1 ≤ k ≤ n χ K ( k ) = r -lim n →∞ ( C χ K ) n where χ K denotes the characteristic fun ction of the set K , and C χ K denotes the sequenc e of Cesáro matrix transform ation. Let us consid er a sequen ce l = { a i ; i = 1, 2, 3, … } o f real numbers, real num ber a , and the set L ε ( a ) = { i ∈ N ; | a i – a | ≥ ε }. Definition 2.2. The asymptot ic r - density , or simpl y, r - d ensity d r ( l ) of t he sequence l with respect to a and ε is equal to d r ( L ε ( a )). Lemma 2.6. If x = d r ( L ε ( a )), then x = d q ( L ε ( a )) for any q > r . This asymptotic densit y allows us to define statistical converg ence. Definition 2.3. A sequence l r - stati stically converges to a number a i f 0 = d r ( L ε ( a )) for eve ry ε > 0. The number (point) a is called an r - statistical li mit of l . We denote this limit b y a = r - stat -lim l . Fuzzy Sta tistical Converg ence of Nu mber Seq uences 7 Then, Definition 2.3 impli es the following result. Lemma 2.7. (a) a = r - stat -lim l ⇔ ∀ ε > 0, 0 = r -lim n →∞ (1/ n ) |{ i ∈ N ; i ≤ n ; | a i - a | ≥ ε }|. (b) a = r - stat -lim l ⇔ ∀ ε > 0, 1 = r -lim n →∞ (1/ n ) |{ i ∈ N ; i ≤ n ; | a i - a | < ε }|. Remark 2.1. We know from [ 3] that if a = l im l (in t he ordinar y sense), then for an y r ≥ 0, we have a = r -lim l . In a simi lar wa y, using D efinition 2 .3, we c an easi ly see that if a = st at -lim l , then we hav e a = r - stat -lim l for an y r ≥ 0. However, its converse is not true. It is demonstrated in the following example of a sequence that i s r -statistically convergent but not convergent and also not statistically convergent. Example 2.3. Let us consider the sequen ce l = { a i ; i = 1,2,3,…} whose term s are i , if i = 2 n ( n = 1,2,3,…) a i = 1, otherwise. Then, it i s easy to see that the sequence l is divergent in the ordinary sense. Even more, the sequence l has no r -limits for any r since it is unbounded from above (se e Theorem 2.3 from [3] ). Likewise, the sequence l is not statisticall y convergent. At the same time, 1 = (½)- stat -li m l since 0 = d ½ ( K ) where K = {2 n ; n ∈ N }. Remark 2.2. Example 2.3 also demonstrates a difference betw een the three types of convergence: conventi onal fuzz y and fuzz y statist ical convergence. It is k nown t hat ever y Mark Burg in and Oktay Duman 8 convergent (in the ordin ary sens e) or r -convergent sequence is bounded. However, this situation is not valid for r -statistically convergent sequences. Lemma 2.1 implies the followi ng result. Lemma 2.8. 0-statistical convergence coincides with s tatistical convergence. This result shows t hat fuz zy statistical conv ergence is a natural ext ension of statistical convergence. As in the case of statistical conver gence [5] , there are connections bet ween fuz zy statistical convergence and fuz zy Cesáro summ ability. Namely, Proposi tion 2.1 and Definition 2.3 impl y the foll owing result. Proposition 2.2. a = r - st at -lim l if and onl y if 0 = r -lim n →∞ ( C χ L ε ( a ) ) n Using t he d efinitions of statistical convergence and r -statistical convergence, it is possible to prove the following result . Lemma 2.9. If a = r - stat -lim l , then a = q - stat -lim l for an y q > r . Corollary 2.3. If a = stat -li m l , then a = r - stat- lim l for an y r ≥ 0. Proposition 2.3. An y sequence l is 1-statistically convergent. Indeed, b y Lemma 2.4, 0 = 1-lim n →∞ (1/ n ) | K n | and 1 = 1-lim n →∞ (1/ n ) | K n | for an y set K ⊆ N . Cons equently, 0 = d 1 ( L ε (0)) for ever y ε > 0 and any sequence l of real numbers as the num ber of elements i n the set L n ε (0) = { k ∈ L ε (0); k ≤ n } is alwa ys less than n . Thus, the sequence l 1-statisticall y converges to zero. Fuzzy Sta tistical Converg ence of Nu mber Seq uences 9 The result of Propositi on 2.3 shows that r -statistical conver gence i s interesting onl y for small r . Let us take a sequence l = { a i ; i = 1, 2, 3, …} and its s equence h . Definition 2.4. The as ymptotic density , or simply, density d l ( h ) of the subsequence h of the sequence l is equal t o d ({ i ∈ N ; a i ∈ h }). Lemma 2.10. If h is a s ubsequence of the seque nce l and k is a su bsequence o f th e sequence h , then d l ( k ) ≤ d l ( h ). Note that it is possi ble that d h ( k ) > d l ( h ). Definition 2.5. Th e asymptoti c r - density , or simp ly, r - density d r,l ( h ) of a s ubsequence h of the sequence l is equal t o d r ({ i ∈ N ; a i ∈ h }). Lemma 2.11. If h is a s ubsequence of the seque nce l and k is a su bsequence o f th e sequence h , then d r,l ( k ) ≤ d r,l ( h ) for any r > 0. Note that it is possi ble that d r,h ( k ) > d r,l ( h ). Definition 2 .6 [4] . A subsequence h of a sequence l is called statisticall y dense in l if d l ( h ) = 1. It is known that a subsequence of a fuzz y convergent s equence is fuz zy convergent [3]. However, for statis tical convergence th is is not t rue. Indeed, the sequence h = { i ; i = 1, 2, 3,…} is a subsequence of the fuz zy statisti cally convergent sequence l from Example 2.3. At th e same time, h is statisticall y fuz zy divergent. However, if we consid er statisticall y dens e subsequences of st atistically fuzz y convergent sequences, it is possible to prove the following result. Mark Burg in and Oktay Duman 10 Theorem 2.1. A sequence is r -statis tically conver gent if and onl y if any its statisticall y dense subsequence is r -statisti cally convergent. Proof. Since the sufficiency is obvious, it i s enough to prove the necessity. So, assume that a sequence l = { a i ; i = 1, 2, 3 ,…} is r -statisticall y convergent, say a = r - stat- lim l , and that h i s an y statisticall y dense subsequence o f l . W e will show that a = r - stat- lim h . B y hypothesis, w e may write fo r ever y ε > 0, th ere exi sts an increasing index subset K of N with d r ( K )=1 such that | a i - a | < ε f or sufficientl y l arge i ∈ K (see Lemma 2.6.b). Als o, since h is a ny statisticall y dense subsequence of l , there exi sts an increasing index subset M with d ( M )=1 such that h = { a i ; i ∈ M }. Now w e c an see that d r ( K ∩ M ) = 1, and so we have | a i - a | < ε for suffici ently lar ge i ∈ K ∩ M . Th is impl ies t hat a = r - stat- lim h and completes the proof. An r -statisticall y convergent sequence contains no t only dense r -st atistically convergent subsequences , but also dense convergent subsequences. The following theorem gives us a useful characteriz ation of r -statisticall y convergent sequences. Theorem 2.2. Let r ∈ [ 0,1]. Then, a = r - stat- lim l if and only if th ere exists an increasing index sequence K = { k n ; k n ∈ N , n = 1, 2, 3 , …} of the natural numbers such that d ( K ) = 1 - r and a = l im l K where l K = { a i ; i ∈ K }. Proof . N ecessity . Let us consider a sequenc e l = { a i ; i = 1, 2, 3, …} of real numbers, for which a = r - stat- lim l . Then for an y n atural nu mber n , we can build the set L 1/2 n ( a ) = { i ∈ N ; | a i – a | ≥ 1/2 n }. As a = r - stat- lim l , we have 0 = d r ( L ε ( a )). It means that there is a number p su ch that (1/ p ) | L 1/2 n ( a ) | < r + 1/2 n . As t he num ber p depends o n n , we denot e it by p ( n ). Using the sequence of n umbers p ( n ), we build a sequence { l 0 , l 1 , … , l n , … } of subsequences of the sequ ence l . He re l 0 = l , l 1 = l 0 \ { a i ; i ∈ L 1/2 n ( a ) }, … , l n = l n -1 \ { a i ; i ∈ L 1/2 n ( a ) }, … , n = 1, 2, 3, … Let us put h = ∩ n =1 ∞ l n . Fuzzy Sta tistical Converg ence of Nu mber Seq uences 11 By our constru ction, th e ratio (1/ p ( n )) | L 1/2 n ( a ) | conver ges to r when n tends to in finity. Let us denote b y I ( h ) indices of all those element f rom l that belong to h . Then d ( I ( h )) = 1 – r and a = stat- lim h . By Th eorem 3.2 from (see Burgin and Duman [4]) a = stat-r -lim l if and only if there exists an increasing index s equence K = { k n ; k n ∈ N , n = 1,2,3,…} o f the natural numbers such that d ( K ) = 1 and a = r -lim l K where l K = { a i ; i ∈ K }. Thus, in the c ase of r = 0, th ere is a subsequen ce k = { a i ; i ∈ J } of h such that J ⊆ I ( h ) and d I ( h ) ( J ) = l im n →∞ | J n | / n = 1, where J n are those numbers of the elements from h that belong to k and to the first n elements from h . B y our construction, we have a = lim k . Necessity is proved. Sufficiency. Let us consider a sequence l = { a i ; i = 1, 2, 3,…} of real n umbers and its subsequence k = { a j ; j ∈ J ⊆ N } of h such t hat d ( J ) = 1 - r and a = lim k . Then d ( N \ J ) = 1 – (1 – r ) = r . This means t hat 0 is an r -statistical limi t of l . Theorem is proved. Let l = { a i ∈ R ; i = 1, 2, 3, …}and h = { b i ∈ R ; i = 1, 2, 3, …}. Then their sum l + h is equal to the s equence { a i + b i ; i = 1, 2, 3, …} and their difference l - h is equal to the sequence { a i - b i ; i = 1, 2, 3, … }. Lemma 2.6 allows us to prove the follow ing result. Theorem 2.3. Let a = r - stat- li m l and b = q - stat- lim h . Th en we have (a) a + b = ( r+q )- stat -lim( l+h ); (b) a - b = ( r+q )- stat -lim( l - h ); (c) ka = ( r |k|)- stat -lim ( k l ) for any k ∈ R where kl = { ka i ; i = 1,2,3,…}. Proof . (a) Let l = { a i ; i = 1,2,3,…} and h = { b i ; i = 1,2,3,…}. Then, b y hypotheses , we get, for a given ε > 0, that 0 = r -lim n →∞ (1/ n ) |{ i ∈ N ; i ≤ n ; | a i - a | ≥ ε /2}| and 0 = q - lim n →∞ (1/ n ) |{ i ∈ N ; i ≤ n ; | b i - b | ≥ ε /2}|. Set u n = |{ i ∈ N ; i ≤ n ; | a i - a | ≥ ε /2}| and v n = |{ i ∈ N ; i ≤ n ; | b i - b | ≥ ε /2}|. Now observe that |{ i ∈ N ; i ≤ n ; |( a i + b i ) - ( a + b) | ≥ ε }| ≤ u n + v n , which implies Mark Burg in and Oktay Duman 12 (1/ n ) |{ i ∈ N ; i ≤ n ; |( a i + b i ) - ( a + b) | ≥ ε }| ≤ (1/ n ) u n + (1/ n ) v n . Thus, for any α > 0 and for sufficientl y large n, we ma y write that (1/ n ) |{ i ∈ N ; i ≤ n ; |( a i + b i ) - ( a + b) | ≥ ε }| ≤ r + α + q + α = ( r+q ) + 2 α . Therefore, we obtain that a + b = ( r+q )- stat -lim( l+h ). Proofs of the statements (b ) and (c) are similar.  Corollary 2.4 [16] . If b = stat -lim l and c = stat -lim h , then: (a) a + b = stat -lim ( l+h ); (b) a - b = stat -lim ( l - h ); (c) ka = stat -lim ( kl ) for an y k ∈ R . Now we can obtain the following “squeeze ru le” for r- statisticall y convergent sequences. Let l = { a i ∈ R ; i = 1, 2, 3, … }, h = { b i ∈ R ; i = 1, 2 , 3, …} and k = { c i ∈ R ; i = 1, 2, 3, …}. Theorem 2.4. If a i ≤ b i ≤ c i ( i = 1, 2, 3, … ), and if a = r-stat - lim l = r-stat- lim k , then we have a = r-stat- li m h . Proof . For a given ε > 0, l et u n = |{ i ∈ N ; i ≤ n ; - ε < a i - a }|, v n = |{ i ∈ N ; i ≤ n ; | b i – a | < ε }|, y n = |{ i ∈ N ; i ≤ n ; c i - a < ε }|. Since a i ≤ b i ≤ c i ( i = 1, 2, 3, … ), we easily see th at u n / n ≤ v n / n ≤ y n / n for each n ∈ N . From the initial conditi ons and Lemma 2.6 (b), we know that 1 = r -lim ( u n / n ) = r -lim ( y n / n ). Thus, for an y α > 0 and su fficiently large n , we ma y write that - r - α < ( u n / n ) – 1 ≤ ( v n / n ) -1 ≤ ( y n / n ) - 1 < r + α , which guarantees that |( v n / n ) -1| < r + α for an y α > 0 and suffici ently large n . Consequently, we get 1 = r -li m ( v n / n ), which yields, by Lemma 2.6, that a = r-stat- lim h . Fuzzy Sta tistical Converg ence of Nu mber Seq uences 13 The proof is completed. 3. Fuzzy convergence of statistical ch aracteristics To each sequen ce l = { a i ; i = 1, 2, 3, …} of real numbers, it is possible to correspond a new sequence µ ( l ) = { µ n = (1/ n ) Σ i =1 n a i ; n = 1,2,3,…} of its partial averages (means). Here a partial avera ge of l is equal to µ n = (1/ n ) Σ i =1 n a i . Partial averages are also called Cesáro means. The y are used to determine Cesáro s ummabilit y in the theor y of divergent series. Cesáro summabil it y is often applied to Fourier series as it i s more powerful than traditional summabili ty. Sequences o f partial ave rages/means pl ay an imp ortant role in the th eor y of ergodic systems [1]. I ndeed, t he definition of an ergodic system i s based on the concept of the “time average” of th e values of some approp riate function g arguments for which are dynamic t ransformations T of a point x from the manifold of the dynamical system. This average is given b y the formula ĝ ( x ) = lim (1/ n ) Σ k =1 n -1 g ( T k x ). In other words, th e d ynamic avera ge is the limit of the p artial averages/means of the sequence { T k x ; k =1,2,3,…}. Properties of th e av erage ĝ ( x ) whe n the par ameter k is a discrete time are described in the famous Birkhoff-Kh inchin theorem, which is one of the most important results in ergodic theory [17]. Let l = { a i ; i = 1, 2, 3,…} be a boun ded sequ ence, i.e., there i s a num ber m such th at | a i | < m for all i ∈ N . This condition is usuall y true for all sequences generated b y measurements or computations, i.e., for all sequences of data that come fro m real life. Theorem 3.1. If a = r - st at -lim l , then a = u -lim µ ( l ), where u = ( m + | a |) r . Mark Burg in and Oktay Duman 14 Proof. Since a = r-stat -lim l , for every ε > 0, we have 0 = r -lim n →∞ (1/ n ) |{ i ≤ n , i ∈ N ; | a i - a | ≥ ε }| (3.1) If | a i | < m for all i ∈ N , then t here i s a number k such that | a i - a | < k for all i ∈ N . Namel y, | a i - a | ≤ | a i | + | a | ≤ m + | a | = k . Taking the set L n , ε ( a ) = { i ≤ n , i ∈ N ; | a i - a | ≥ ε }, denoting | L n , ε ( a )| by u n , and using the hypothesis | a i | < m for all i ∈ N , we have the following system of inequalit ies: | µ n - a | = |(1/ n ) Σ i =1 n a i - a | ≤ (1/ n ) Σ i =1 n | a i - a | ≤ (1/ n ) { ku n + ( n - u n ) ε } ≤ (1/ n ) ( ku n + n ε ) = ε + k ( u n / n ). From (3.1), we get the i nequality | µ n - a | < ε (1 + k ) + kr for sufficiently large n . Since m + | a | = k , we have a = u- lim µ ( l ), where u = ( m + | a |) r . Theorem is proved. Taking r = 0, we have th e following result. Let l = { a i ; i = 1,2,3,…} be a bounded sequence such t hat | a i | < m for all i ∈ N . Corollary 3.1. If a = stat -li m l , then a = lim µ ( l ). Remark 3.1 . Howev er, (fuzzy) conver gence of the partial averages/means of a sequence does not impl y (fuzzy) statistical convergence of this sequ ence. Remark 3.2. For unbounded sequ ences, the res ult of Theorem 3.1 can be invalid. Thus, the condition of boundedness i s essential in this theorem. Taking the sequence l = { a i ; i = 1, 2, 3,…} of rea l numbers, it is possi ble to construct not onl y the sequ ence µ ( l ) = { µ n = (1/ n ) Σ i =1 n a i ; n = 1, 2, 3,… } o f its partial av erages Fuzzy Sta tistical Converg ence of Nu mber Seq uences 15 (means) but also the sequences σ ( l ) = { σ n = ((1/ n ) Σ i =1 n ( a i - µ n ) 2 ) ½ ; n = 1,2,3,…} of it s partial s tandard deviation s σ n and σ 2 ( l ) = { σ n 2 = (1/ n ) Σ i =1 n ( a i - µ n ) 2 ; n = 1,2,3,…} of its partial variances σ n 2 . Let us find how fuzz y statistical convergence of a sequence is related to fuzzy statistical convergence of thi s sequence of its partial standard deviati ons. Theorem 3.2. If a = r -stat- lim l and | a i | < m for a ll i = 1, 2, 3, … , then 0 = [ p (2 r + u)] ½ -lim σ ( l ) where p = max { m 2 + | a | 2 , m + | a |} and u = ( m + | a |) r . Proof . At first, we show that lim σ 2 ( l ) = 0. B y t he definition, σ n 2 = (1/ n ) Σ i =1 n ( a i - µ n ) 2 = (1/ n ) Σ i =1 n ( a i ) 2 - µ n 2 . Thus, lim σ 2 ( l ) = lim n →∞ (1/ n ) Σ i =1 n ( a i ) 2 - lim n →∞ µ n 2 . S ince | a i | < m for all i ∈ N , th ere is a number p su ch that | a i 2 - a 2 | < p for all i ∈ N . Namel y, | a i 2 - a 2 | ≤ | a i | 2 + | a | 2 < m 2 + | a | 2 < max { m 2 + | a | 2 , m + | a |} = p . Takin g the set L n , ε ( a ) = { i ∈ N ; i ≤ n and | a i - a | ≥ ε }, denoting | L n , ε ( a ) | b y u n , and using th e hypothesis | a i | < m for all i ∈ N , we have the followin g system of inequalities: | σ 2 n | = |(1/ n ) Σ i =1 n ( a i ) 2 - µ n 2 | = |(1/ n ) Σ i =1 n ( a i 2 - a 2 ) - ( µ n 2 – a 2 )| ≤ (1/ n ) Σ i =1 n | a i 2 - a 2 | + | µ n 2 – a 2 | < ( p / n ) Σ i =1 n | a i - a | + | µ n – a | | µ n + a | < ( p / n ) ( u n + ( n - u n )( r + ε )) + | µ n – a | (| µ n | + | a |) ≤ ( p / n ) ( u n + n ( r + ε )) + | µ n – a | ((1 /n ) Σ i =1 n | a i | + | a |) < p ( u n / n ) + p ( r + ε ) + p | µ n – a |. Now b y h ypothesis and Theorem 3.1, we have a = u- lim µ ( l ), where u = ( m + | a |) r . Also, for every ε > 0 and suffici ently large n , we may write that | σ 2 n | < p ε + pr + p ( r + ε ) + p (u + ε ) = p (2 r + u) + 3 p ε (3.2) As ( x + y ) ½ ≤ x ½ + y ½ for any x, y > 0, the inequalit y (3.2) implies the inequalit y | σ n | ≤ [ p (2 r + u)] ½ + (3 p ε ) ½ , which yields that 0 = [ p (2 r + u)] ½ -lim σ ( l ). The proof is completed. Mark Burg in and Oktay Duman 16 Taking r = 0, we have th e following result for bounded sequences. Corollary 3.2. If a = stat -li m l , then 0 = lim σ ( l ). 4. Conclusion We have developed the concept of fuzzy statistical convergence and studied its properties. In pa rticular, it is demonst rated t hat fuzzy st atistical convergence of a sequence impl ies fuz zy convergence of p artial averages (means) of this sequence (Theorem 3.1 ) and fuz zy convergence of partial standard deviati ons of this sequence (Theorem 3.2). Some relations between fuzz y statistical convergence and fuz zy summabilit y are obtained for Cesáro su mmabilit y . 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