Some Recent Developments On Statistical Convergence ( A New Type Of Convergence ) And Convergence In Statistics

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 2, Issue Ver. III (Ja. - Feb. 206), PP Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece ) Ad Covergece I Statistics A. G. K. Ali AND 2A. M. Broo Departmet of Mathematics, Kashim Ibrahim College of Educatio, Maiduguri, Boro State, Nigeria. 2 Departmet ofmathematics ad Statistics, Uiversity of Maiduguri, Boro State, Nigeria. Abstract: Oe of the most recet geeralizatios of cocept of classical covergece of sequeces ( a ew type of covergece ) is statistical covergece defied by Fast. Recetly, it became the cetre of attractio for may researchers. The objective of this review is to discuss fudametal cocepts ad results i statistical covergece alog with various geeralizatios which have bee subsequetly formulated. Ad fially expose the statistics i statistical covergece. Keywords: Statistical covergece, covergece i statistics ad sequeces. Mathematics subject classificatio: Primary 40F05, 40J05, 40G05. I. Itroductio The classical otio of Cauchy Covergece has bee geeralized i various ways through summability ad sequece-to-fuctio methods. The first geeralizatio was that of almost covergece which was iitiated as early as 932 by Baach (932). It was studied i details by Loretz (948). Oe of the most recet geeralizatios of the cocept of classical covergece of sequeces ( ew type of covergece ) has bee itroduced by Fast (95). He itroduced cocepts such as statistic al covergece, Lacuary statistical covergece ad λ-statistical covergece. It was defied almost fifty years ago but recetly it became the cetre of attractio for may researchers. The objective of this study is to discuss fudametal cocepts ad results i this field alog with various geeralizatios which have bee subsequetly formulated. With a view to expose the profoud lik betwee the classical theory ad statistical covergece theory. We will also look at some costructio from the theory of statistical covergece to cosider relatios betwee statistical covergece ad cocepts of covergece i statistics such as the mea (average) ad stadard deviatio. Before startig to discuss o statistical covergece, we cosider a very brief summary of cocepts of covergece of sequeces of real umbers ad some other importat defiitios related to sequeces. II. Notios ad Notatios Defiitio 2.: A sequece is a fuctio whose domai of defiitio is the set N of all atural umbers. Sequeces obtai differet ames with respect to their rage. If the rage of the sequece is R, the we call this sequece a real umber sequece (or real sequece). If the terms are all ratioal umbers, the, we called this sequece ratioal umber sequece (or ratioal sequece). Geerally, we use the otatio x = x k k = k N To represet sequeces. For each value of k, the term x k is kow as k t term of x. The space of all sequeces deoted by ω. The above defiitio ca be re-writte iformally as follows: A sequece x k k = of real umbers is a fuctio x: D(x) N R of a ifiite subset D(x) of the atural umbers N ito the real umbers R defied by x k = x k R k N Example 2.: The sequece x k k= where x k = is a ifiite sequece,,,,,. Formally, of course k 2 3 k this is the fuctio with domai N, whose value at each k is. The set of values is:,,,. k 2 3 Example 2.2: Cosider the sequece give by x k k = = k k =. This is also the ifiite sequece,,,,. This sequece represets a fuctio whose domai is N ad its set of values is,. This is a alteratig sequece (a sequece i which the cosecutive terms have opposite sigs). Example 2.3: The sequece x k k = = 3 k= is the costat sequece 3, 3, 3, whose set of values is the sigleto 3. Defiitio 2.2: Let {x k } be a sequece ad let k be a strictly icreasig sequece of atural umbers. The sequece x k is called a subsequece of {x k }. DOI: / Page

2 Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece )... Example 2.4: The sequece has subsequeces k k = 2k k= k 2 k= Example 2.5: The set of prime umbers 2, 3, 5,7,, 3, is a subsequece of the sequece k k = of all positive itegers. Defiitio 2.3: A sequece {x k } is called bouded from above if there exists M R, which satisfies the iequality x k M for all k N. I this case, we say M is a upper boud for x. Defiitio 2.4: we say {x k } is bouded from below if there exists m R which satisfies the iequality m x k for all k N. I this case we say that m is a lower boud of x. Defiitio 2.5: We say that a sequece {x k } is bouded if there exists a real costat U > 0, which satisfies the iequality x k U for all k N Lemma 2.: we say {x k } is bouded if ad oly if it is bouded from below ad bouded from above. Recall that, l = x ω: x k is bouded. Example 2.6: Sequeces defied by x k = k ad y k = are both bouded. k DOI: / Page 2k, Aalogous to the idea of ordiary bouded sequece, statistically bouded sequece is defied. Defiitio 2.6 [Fridy ad Orha, (997)]: A sequece x k is said to be statistically bouded if there exists a umber B such that δ k: x k > B = 0 Example 2.7: Let x = x k be give by k, if k is prime x k =, if k is odd but ot prime 0, if k is eve but ot prime It is easy to see that, x is ot statistically coverget ad ubouded above but it is statistically bouded. δ k: x k > = 0 Defiitio 2.7: A sequece x k k = of real umbers is said to coverge to a real umber l if ad oly if for each ε > 0, there exists a atural umber (ε) such that x k l < ε k (ε) I this we write: lim x k = l or lim x k = l or x k l as k k The umber l is called the limit of x k. A sequece which does ot coverge to some real umber is said to diverge. We use the otatio c, to represet the space of coverget sequeces, c = x ω: x k is coverget Aalogous to the idea of space of ordiary covergece sequece, the space of all statistically coverget sequeces is deoted by c. Defiitio 2.8: if a sequece x k of real umbers coverges to zero, it is called a ull sequece. The spaces of all ull sequeces is deoted by c 0,i. e. c 0 = x ω: x k 0. Example 2.8: The sequece x k defied by x k = k +3 k 2. is a ull sequece. Remark 2.: Obviously c 0 c ω. Aalogous to the idea of space of ull covergece sequece, the space of statistically ull coverget sequece is deoted by c 0. Theorem 2.: Every covergece sequece is bouded, however the coverse may ot be true. Remark 2.2: I geeral, a bouded sequece eed ot be coverget. I fact the sequece x k = k is bouded but ot coverget. is coverget to the limit l if ad oly if all of its subsequeces coverges to Theorem 2.2: A sequece x k k= the same limit l. Theorem 2.3: (Balzao-Weierstrass). Every bouded sequece i R has a coverget subsequece. Remark 2.3: For statistical aalogue of Bolzao-Weierstrass [see Fridy (993)] Theorem 2.4: If a sequece x k coverges to a limit, the the limit is uique. Proof: Suppose, for cotradictio, that the sequece x k coverges to two limits x ad y, x y. The give ay ε > 0, there exists a atural umber (ε) such that x k x < ε 2 Also, there exists (ε) N such that k (ε). x k y < ε k (ε). 2 Hece, for all k max ε, (ε), both iequalities implies x k x < ε ad x 2 k y < ε hold. The 2

3 Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece )... x y = x x k + x k y x x k + x k y < ε 2 + ε 2 = ε For all max ε, (ε). This implies that x = y. Cotradictio. Hece the theorem is proved. Defiitio 2.9: for each ε > 0, ad l R. The set K E l = x R: x l < ε Is called the ε eighbourhood of l. Lemma 2.2: Assume that x k l. The for every ε > 0, except fiitely may terms of x k, all other terms lie i K E l. i other word, N: x k l ε is fiite. Defiitio 2.0: A sequece x k is said to be (mootoe) icreasig if x k x k+ for all values of k ad (mootoe) decreasig if x k x k+ for all k. If x k < x k+ for all k, x k is said to be strictly icreasig ad if x k > x k+ for all k, x k is said to be strictly decreasig. A sequece satisfyig ay of the above coditios is said to be mootoe. Example 2.9: The sequece, 2, 3, k, is a strictly icreasig ubouded sequece. Example 2.0: The sequece is strictly decreasig sequece which coverges to zero. k Theorem 2.5: (mootoe covergece theorem): A mootoe sequece of real umbers is coverget if ad oly if it is bouded. Defiitio 2.: Give a bouded sequece x = x k of real umbers. The upper limit or limit superior of x k deoted by lim sup x k (or lim k x k ) ad is defied as lim sup x k = lim k x k = if sup x k : k N Aalogous to the idea of ordiary limit superior, statistical limit superior of x were defied as follows [Fridy ad Orha, (997)]: S lim sup x = sup B x, if B x, if B x = Defiitio 2.2: Give a bouded sequece x = x k of real umbers. The lower limit or limit iferior of x k deoted by lim if x k (or lim k x k ) ad is defied as lim if x k = lim k x k ) = sup if x k :k N Aalogous to the idea of ordiary limit iferior, statistically limit iferior were defied as follows: Defiitio 2.3: [Fridy ad Orha, (997)] The statistical limit iferior of x is defied as S lim if x = if A x, if A x +, if A x = Aalogous to the idea of ordiary covergece, statistically coverget sequeces were defied as follows: As it is well kow, the theory of statistical covergece ad other types of covergeces are all based o a desity fuctio. This is why we eed to start with the defiitio of destity fuctio. Defiitio 2.4 [Nive ad Zuckerma, (980)]: Let A be a subset of N. The atural desity δ(a) is defied as δ A = lim A, Where A = k :k A, provided the limit exists. The vertical bars deote the cardiality (or order) of the eclosed set. III. Statistical Covergece Sequece Defiitio 3.: [Fast, (95)] A sequece x = (x k ) is said to be statistically coverget to l if for every ε > 0, we have δ k N: x k l ε = 0. I this case, we write S limx = l, which is ecessarily uique ad S deotes the set of all statistically coverget sequeces. By a idex set, we mea a fiite subset k i of N with k i k i +. Thus, a ifiite set k i is precisely the sequece k i i N of idices. Note that a sequece x, is statistically coverget to l if ad oly if there is ifiite idex set K = k i so that δ K = ad lim k x k = l, Mursalee ad Edely (2002). Aalogous to the followig classical covergece otios; Defiitio 3.2: A sequece x k k = of real umbers is called a Cauchy sequece (a fudametal sequece) if it satisfies the followig: ε > 0, (ε) N such that x k x < ε k, (ε) DOI: / Page

4 Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece )... That is equivaletly x k x 0, as k,. Theorem 3.: Every real valued Cauchy sequece is bouded. Theorem 3.2: A sequece of real umber is coverget if ad oly if it is a Cauchy sequece. We have; Defiitio 3.3 [Fridy, (985)]: A sequece x = (x k ) is called statistically Cauchy sequece if for each ε > 0, there exists a atural umber = (ε) such that δ k N: x k x ε = 0, That is, lim k : x k x ε = 0 Theorem 3.3: It is also show that a real sequece is statistically Cauchy if ad oly if it is statistically coverget. Proof: Suppose that x = x k is statistically coverget to l. By defiitio there exists a subset K of atural umbers with δ K = ad x l o K i the ordiary sese. This meas that x is a Cauchy sequece o K or equivaletly x is statistical Cauchy sequece. Coversely, suppose that x = x k is statistical Cauchy sequece. Sice every sequece is statistically Cauchy if ad oly if there exists K N with δ K = ad so x is Cauchy o K. Therefore it is coverget o K, which shows that x is statistical Cauchy sequece. Defiitio 3.4: We say that the sequece x k of real umbers teds to the limit l R if ad oly if for each E > 0, there exists (ε) such that x k l < ε k l Aalogous to the defiitio of a limit poit, statistical limit poits were defied as follows: Defiitio 3.5 [Fridy, (993)]: A subsequece x k() of sequece x is called a thi subsequece if K = k : N has atural desity zero. If K does ot have zero atural desity, the the subsequece is said to be othi. We ote that x k() is a othi subsequece of x if either δ K is a positive umber or K fails to have atural desity. Defiitio 3.6 [Fridy, (993)]: The umber λ is a statistical limit poit of the umber sequece x provided that there is a othi subsequece of x that coverges to λ. Defiitio 3.7 [Fridy, (993)]: The umber γ is a statistical cluster poit or accumulatio poit of the umber sequece x provided that for each ε > 0, the set k N: x k γ < ε does ot have atural desity zero. If Λ x, Γ x ad L x deote the set of all statistical limit poits, set of all statistical cluster poits ad set of (ordiary) limit poits, respectively, the it has bee proved that Λ x Γ x L x. Example 3.: let x k =, if k = 2 ad x k = 0 otherwise L x = 0, ad Λ x = 0. It is clear that Λ x L x for ay sequece x. To show that Λ x ad L x ca be differet, cosider a sequece x for which Λ x = φ while L x = R, the set of real umbers. Example 3.2: let us cosider the sequece l = x k :k =,2,3, whose terms are k, we k is a square x k = /k, oterwise The, it is easy to see that l is diverget i the ordiary sese, while 0 is the statistical limit of l sice δ K = 0, where K = 2 for =,2,3,. Not all properties of coverget sequeces are true for statistical covergece. For example, it is kow that a subsequece of coverget sequece is coverget. However, for statistical covergece, this is ot so. Ideed, the sequece y = k: k =,2,3, is a subsequece of the statistical coverget sequece l from example 3.2. At the same time, y is statistically diverget. Defiitio 3.8 [Fridy ad Orha, (993)]: A sequece θ = k r satisfyig: i. k 0 = 0, ii. r = k r k r, r Is called a Lacuary sequece. For each Lacuary sequece θ, we defie the iterval I r (k r,k r ] ad the fractio q r = k r. Lacuary k r statistical covergece has bee itroduced by Fridy ad Orha i the followig way: Example 3.3: The sequece θ = 2 r is a Lacuary sequece with I r (2 r, 2 r ] ad q r 2. Defiitio 3.9 [Fridy ad Orha, 993)]: A sequece x is called Lacuary statistical coverget if for each ε > 0, we have DOI: / Page

5 Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece )... lim k I r r : x k l ε = 0 r Ad deoted by x k l(θ S) Lemma.3: For a lacuary sequece θ = k r, x k l implies x k l (θ S) if ad oly if lim if r q r >. Lemma.4: For a lacuary sequece θ = k r, x k l implies x k l (θ S) if ad oly if lim sup r q r <. As a cosequece of the above lammas, we have: Theorem 3.4: Let θ = k r be a lacuary sequece. The statistically covergece ad θ statistical covergece of x k l if ad oly if < limif r q r < limsup q r < r Obviously, example 3.3 above satisfies the coditios of the above theorem for r > 0. Remark 3.: To defied λ Statistical Covergece, first we eed a sequece λ r of positive o decreasig umbers such that λ r,as r, λ = ad λ r+ λ r +. Assume that ω is the space of all sequeces satisfyig these coditios. The for each {λ r } ω, ad for each r, we defied itervals M r = r λ r +, r. Defiitio 3.0 [Mursalee, (2000)]: A sequece x is said to be λ statistical coverget to l, if for each ε > 0, lim r λ r k M r : x k l ε = 0. The λ statistical covergece of x to l is represeted by the otatio x k l λ S. Remark 3.2: For λ r = r, λ statistical covergece coicides with the statistical covergece. Further remarks: The idea of statistical covergece goes back to the first editio (published i Warsaw i 935) of the moograph of Zygmud (979). Formally, the cocept of statistical covergece was itroduced by Steihaus (95) ad Fast (95) ad later reitroduced by Schoeberg (959). Differet mathematicias studied properties of statistical covergece ad applied this cocept i various fields such as Measure theory Miller, (995), Trigoometric series Zygmud, (979), Approximatio theory Duma et al., (2003), Locally covex spaces Maddox, (988), Fiitely additive set fuctios Coor ad Klie, (996), i the study of subsets of the Stoe-Chech compactificatio of the set of atural umbers Coor ad Swardso, (993) ad Baach spaces Coor et al., (2000). However, i geeral case, either limits or statistical limit ca be calculated or measured with absolute precisio. To reflect this imprecisio, ad to model it by mathematical structures, several approaches i mathematics have bee developed: fuzzy set theory, fuzzy logic, iterval aalysis, set valued aalysis, etc. Oe of these approaches is the eoclassical aalysis (Burgi, 995, 2000). I it ordiary structure of aalysis, that is, fuctios, sequeces, series ad operators are studied by meas of fuzzy cocepts: fuzzy limit, fuzzy cotiuity ad fuzzy derivatives. For example cotiuous fuctios which are studied i the classical aalysis become a part of the set of the fuzzy cotiuous fuctios i eoclassical aalysis. Neoclassical aalysis exteds methods of calculus to reflect ucertaities that arise i computatios ad measuremets. Bele ad Yildirim (205) defied the cocepts of A-statistical covergece ad A j -statistical covergece i a 2-ormed space ad preset a example to show the importace of geeralized form of covergece through a ideal. Ad they also itroduced some ew sequece spaces i a 2-Baach space ad examie some iclusio relatios betwee these spaces. Kukul (204) studied αβ-statistical covergece ad started with the discussio of statistical covergece. The cocept of αβ-statistical covergece which is the mai iterest of his study has bee cosidered i his work. He also showed that αβ-statistical covergece is a o-trivial extesio of statistical, λ-statistical ad lacuary statistical covergeces. Fially, he also itroduced boudedess of a sequece i the sese of αβ-statistical covergece. Nuray (999) derived algebraic ad order properties, preservatio uder uiform covergece, Cauchy properties, ad properties of cluster poits. Ad also showed that for certai types of statistical covergece stroger covergece properties also hold ad also he further discussed the relatioship betwee geeralized statistical covergece ad subsets of the Stoe-Cech compactificatio of the itegers. Bala (20) itroduced the cocept of weak statistical covergece of sequece of fuctioals i a ormed space. He also showed that i a reflexive space, weak statistically coverget sequeces of fuctioal are the same as weakly statistically coverget sequeces of fuctioals. Gumus (205) i his study, he provided a ew approach to I - statistical covergece. Ad he also itroduced a ew cocept with I - statistical covergece ad weak covergece together ad call it weak I - statistical covergece or WS(I ) - covergece. The he itroduced this cocept for lacuary sequeces ad obtaied lacuary weak I- statistical covergece i.e. WSq (I ) - covergece. Hazarika ad Sava (203) i their study itroduced the cocept of λ-statistical covergece i - ormed spaces. Some iclusio relatios betwee the sets of statistically coverget ad λ-statistically DOI: / Page

6 Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece )... coverget sequeces are established. They also fid its relatios to statistical covergece, (C,)-summability ad strog (V; λ )-summability i -ormed spaces. Kaya, Kucukasla ad Wager (203) ivestigated the properties of statistically coverget sequeces. Also they itroduced defiitio of statistical mootoicity ad upper (or lower) peak poits of real valued sequeces. They also studied the iterplay betwee the statistical covergece ad the above cocepts. Fially, the statistically mootoicity is g eeralized by usig a matrix trasformatio. Borgohai (20). Studied the cocept of desity for sets of atural umbers i some lacuary A- coverget sequece spaces. He also ivestigated some relatio betwee the ordiary covergece ad module statistical covergece for every ubouded modulus fuctio. Moreover he also studied some results o the ewly defied lacuary f -statistically A-coverget sequece spaces with respect to some Musielak-Orlicz fuctio. Dutta, (203) itroduces a ew cocept of strog summability ad statistical covergece of sequece of fu zzy umbers ad established relatios betwee them. Erkus ad Duma (2003) use the cocept of A- statistical covergece which is a regular (o-matrix) summability method, ad obtaied a geeral Korovki type approximatio theorem which cocers the problem of approximatig a fuctio f by meas of a sequece {L f } of positive liear operators. Viod et al (202) coducted a study to see how weak ideal covergece looks like i lp spaces ad exted the recetly itroduced cocept of weak statistical covergece to have a ew cocept of weak ideal covergece. Ad they gave the ecessary ad sufficiet coditios for a sequece of bouded liear fuctioal o a Baach space to be weak ideal coverget. Esi (summitted) defied the cocept of lacuary double m statistical coverget sequeces i probabilistic ormed space ad gave some results. The mai purpose of his study was to geeralize the results for double sequeces o statistical covergece i probabilistic ormed space give by Esi ad O zdemir (2003) earlier ad obtaied some results o lacuary statistical covergece for double geeralized differece sequeces o probabilistic ormed spaces. The results th ey obtaied are more geeral tha the results of Esi (2003). Miller ad Orha (2004) Studied the basis for comparig rates of covergece of two ull sequeces which shows that x = x coverges (stat T) faster tha z = z provided that (x = z ) is T-statistically coverget to zero" where T = t m is a mea. They also exteded the previously kow results either o the ordiary covergece or statistical rates of covergece of two ull sequeces. They also cosidered lacuary statistical rates of covergece. Based o the cocept of ew type of statistical covergece defied by Aktuglu, Srivastava (205) has itroduced the weighted αβ statistical covergece of order θ i case of fuzzy fuctios ad classified it ito poitwise, uiform ad equi-statistical covergece. He has checked some basic properties ad ivestigated covergeces i terms of their α cuts. The iterrelatios amog them are also established. He has also proved that cotiuity, boudedess etc are preserved i the equi-statistical sese uder some suitable coditios, but ot i poitwise sese. Bardaro et al (205) dealt with a ew type of statistical covergece for double sequeces, called Ψ-Astatistical covergece, ad proved a Korovki-type approximatio theorem with respect to this type of covergece i modular spaces. Fially, they gave some applicatios to momet-type operators i Orlicz spaces. Kostyrko et al (2000) itroduced the cocept of I-covergece of sequeces of real umbers based o the otio of the ideal of subsets of N. The I-covergece gives a uifyig look o several types of covergece related to the statistical covergece. I a sese it is equivalet to the cocept of μ-statistical covergece itroduced by J. Coor (μ beig a two valued measure defied o a subfield of 2 N ). Savas ad Esi (202) defied statistical aalogues of covergece ad Cauchy for triple sequeces o probabilistic ormed space. Ulusu ad Nuray (202) defied lacuary statistical covergece for sequeces of sets ad study i detail the relatioship betwee other covergece cocepts. Note: It should be oted that, Statistical covergece is a atural geeralizatio of the usual (classical) covergece of sequeces. The idea which is used to defie ew type of covergece was the followig; a sequece may have ifiitely may terms which are ot icludig i ε eighbourhoods of the limit poit for ε small eough but the set of idices of such terms have desity zero. As it is well kow this is ot possible i ordiary sese. Therefore, ew type of covergeces defied i this way give us a ew type of covergece which is differet from the ordiary covergece. I may years, researchers focused o covergeces which are obtaied from differet desity fuctios. But a careful observatio shows that all desity fuctios are based o differet class of itervals. For example, statistical covergece ad lacuary covergece are based o itervals [, ] ad (k,k ]. it should be oted that, ay covergece sequece is statistically coverget but the coverse is ot true. Recall that all fiite subsets of atural umbers have desity zero. If we combie this fact by that of ordiary covergece of a sequece to a real umber l, implies that k: x k l ε DOI: / Page

7 Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece )... is a fiite set which shows that ordiary covergece implies statistical covergece. Also the boudedess property does ot hold by statistical covergece. Recall that i the sese of ordiary covergece, coverget sequeces are all bouded. So this also shows that statistical covergece is differet from ordiary covergece. Every covergece sequece is statistically coverget with same limit. Sequeces which are statistically coverget may either be coverget or bouded. Example 3.4: Cosider the sequece x = x k defied by k, if k is a square x k = 0, oterwise It ca be see that this sequece is statistically coverget to zero but it either coverget or bouded ordiarily. Example 3.5: Cosider a sequece x = x k defied by, if k = 2 x k =, for =,2,3, 2, if k = 2 This sequece is ot statistically coverget IV. Statistics i Statistical Covergece Statistics is a brach of mathematics cocered with the collectio, orgaisig, aalysig, summarisig ad iterpretatio of umerical facts about a give situatio i order to draw valid coclusio. There are two braches of statistics: iferetial ad descriptive. Iferetial statistics is usually used for two tasks: to estimate the properties of a populatio give sample characteristics ad to predict property of a system give its past ad curret properties. To do this, specific statistical costructios were iveted. The most popular ad useful of them are the average or mea (or more precisely arithmetic mea) μ ad stadard deviatio σ (variace σ 2 ). To make predictio for future, statistics accumulates data for some period of time. To kow abo ut the whole populatio, samples are used. Normally, such ifereces (for future or for populatio) are based o some assumptios o limit process ad their covergece. Iterative processes are used widely i statistics. For example, the empirical approach to probability based o the law (or better to say suppositio) of large umbers, states that a procedure repeated agai ad agai, the relative frequecy probability approaches the actual probability. The foudatio for estimatig populatio parameters ad hypothesis testig is formed by the cetral limit theorem which tell us how sample mea chage whe the sample size grows. I experimets, scietists measure how statistical characteristics (e.g., meas or stadard deviatios coverge Harris ad Chiag ( 999). Bosheritza et al (2000) cosider geeralizatios of the poitwise ad mea ergodic theorems to ergodic theorems averagig alog differet subsequeces of the itegers or real umbers. The Birkhoff ad Vo Neuma ergodic theorems give coclusios about covergece of average measuremets of systems whe the measuremets are made at iteger times. They also cosidered the case whe the measuremets are made at times a() or ([a()]) where the fuctio a(x) is take from a class of fuctios called a Hardy field, ad also assume that a(x) goes to ifiity slower tha some positive power of x. A special, well-kow Hardy field is Hardy's class of logarithmico-expoetial fuctios. Their mai iterest is to poit out that for a fuctio a(x) as described above, a complete characterizatio of the ergodic averagig behaviour of the sequece ([a()]) is possible i terms of the distace of a(x) from (certai) polyomials. Chu (2008) studied weighted polyomial multiple ergodic averages. A sequece of weights is called uiversally good if ay polyomial multiple ergodic average with this sequece of weights coverges i L 2. They also foud a ecessary coditio ad show that for ay bouded measurable fuctio o a ergodic system, the sequece (T x) is uiversally good for almost every x. Host ad Kra (2003) studied the L 2 -covergece of two types of ergodic averages. The first is the average of a product of fuctios evaluated at retur times alog arithmetic progressios, such as the expressios appearig i Fursteberg's proof of Szemere di's Theorem. The secod average is take alog cubes whose sizes ted to +. For each average ad showed that it is sufficiet to prove the covergece for special systems, the characteristic factors. They built these factors i a geeral way, idepedet of the type of the average. To each of these factors they associated a atural group of trasformatios ad give them the structure of a ilmaifold. From the secod covergece result they derived a combiatorial iterpretatio for the arithmetic structure iside a set of itegers of positive upper desity. Covergece of meas/averages ad stadard deviatios have bee studied by may authors ad applied to differet problems (Akcoglu ad Suchesto, 975: Akcoglu ad Del Juco, 975: Assai, 2003, 2005: Duford ad Schwartz, 955: Fratzikiakis ad Kra, 2005: Host ad Kra, 2005: Joes ad Roseblatt, 992: Leibma, 2002, 2005, Vapik ad Chervoekis, 98). Covergece of statistical characteristics such as mea ad stadard deviatios are related to statistical covergece. DOI: / Page

8 Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece )... Note: Fially it is clear that, statistical covergece is ot a atural geeralizatio of covergece i statistics. But rather a coditioal equivalet to covergece i statistics as follows: A sequece l is statistically coverget if ad oly if its sequece of partial averages μ(l) coverges ad its sequece of partial stadard deviatios σ(l) coverges to zero. To each sequece l = {x k ; k =,2,3, } of real umbers, it is possible to correspod a ew sequece μ(l) = {μ = (/) k = x k ; =,2,3, } of its partial averages (meas). Here a partial average of l is equal to μ = (/) k = x k. Sequeces of partial averages/meas play a importat role i the theory of ergodic systems Billigsley (965). Ideed, the defiitio of a ergodic system is based o the cocept of the time average of the values of some appropriate fuctio g argumets for which are dyamic trasformatios T of a poit x from the maifold of the dyamical system. This average is give by the formula g(x) = lim (/) k = g(t k x). I other words, the dyamic average is the limit of the partial averages/meas of the sequece {T k x ; k =,2,3, }. Let l = {x k : k =,2,3, } be a bouded sequece, i.e., there is a umber m such that x k < m for all k N. This coditio is usually true for all sequeces geerated by measuremets or computatios, i.e., for all s equeces of data that come from real life. Theorem 4. [Burgi ad Duma, (2006)]: If x = S-lim l, the x = lim μ(l). Proof: Sice x = S lim l, for every ε > 0, we have lim (/) {k, k N: x k x ε} = 0 As x k < m for all k N, there is a umber p such that x k x < p for all k N. Namely, x k x x k + x = p. Takig the set L,ε x = {k, k N: x k x ε}, deotig L,ε x by u, ad usig the hypothesis x k < m for all k N, we have the followig system of iequalities: μ x = (/) (/) k= k = x k x x k x {ku + u ε} {ku + ε} = ε + k u From equatio () above, we get, for sufficietly large, the iequality μ x < ε + kε Thus, x = lim μ(l). Hece, the theorem is proved. Remark 4.. However, covergece of the partial averages/meas of a sequece does ot imply statistical covergece of this sequece as the followig example demostrates. Example 4.: Let us cosider the sequece l = {x k : k =,2,3, } whose terms are x k = ( ) k k. This sequece is statistically diverget although lim μ(l) = 0. Takig a sequece l = {x k : k =,2,3, } of real umbers, it is possible to costruct ot oly the sequece μ(l) = { μ = k = x k ; =,2,3, } of its partial averages (meas) but also the sequeces σ(l) = { σ = ( (x k = k μ ) 2 ) 2 2, =,2,3, } of its partial stadard deviatios σ ad σ l = {σ 2 = (x k = k μ)2, =,2,3, } of its partial variaces σ2. Theorem 4.2 [Burgi ad Duma, (2006)]: If x = S liml ad x k < m for all k N, the, lim σ l = 0. Proof: We will show that lim σ 2 l = 0. By the defiitio σ 2 = (x k = k μ ) 2 = x 2 2 k = k μ. Thus σ 2 l = lim x 2 k = k lim μ 2. If x k < m for all k N, the there exists a umber p such that x 2 k x 2 < p for all k N. Namely, x 2 k x 2 x 2 k + x 2 < m 2 + x 2 < m 2 + x 2 + m + x = p. Let us cosider the absolute value of the differece μ 2 x 2 k = k = σ 2. Takig the set L,ε x = k,k N: x k x ε, deotig L,ε x by u, ad usig the hypothesis x k < m for all k N, we have the followig iequalities: DOI: / Page

9 Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece )... = σ 2 = < p k = k = k= k = x k 2 μ 2 x k 2 x 2 (μ 2 x 2 ) x k 2 x 2 + μ 2 x 2 x k x + μ 2 x 2 < p u + u ε + μ 2 x 2 < p u + ε + μ 2 x 2 = p u + εp + μ 2 x 2 As x 2 k x 2 = x k x x k + x = p. x k x. By theorem 4., we have x = limμ l, which guaratees that lim μ 2 = x 2. Also by theorem 4., lim μ = 0. Sice ε > 0, was arbitrary, the right had side of the above iequality teds to zero as. Therefore, we have lim σ l = 0. Hece, the theorem is proved. Corollary 4.: If x = S lim l ad x k < m for all k N, the lim σ 2 (l) = 0. Theorem 4.3 [Burgi ad Duma, (2006)]: A sequece l is statistically coverget if its sequece of partial averages μ(l) coverges ad x k lim μ(l) (or x k lim μ(l) for all k =, 2, 3, ) Proof: Let us assume that x = lim μ(l), x k lim μ(l) ad take some ε > 0, the set L,ε x = k, k N: x k x ε, ad deote L,ε x by u. the we have x u = x k = x k = = k = k = x x k x x k Cosequetly, lim x μ lim u lim k, k N: x k x ε = 0, i.e, x = S lim l. x x k k = ε x x k u ε ε. as lim x μ = 0, ad ε is a fixed umber, we have The case whe x k lim μ(l) for all k =, 2, 3, is cosidered i a similar way. Hece, the theorem is proved as ε is a arbitrary positive umber. Theorem 4.4 [Burgi ad Duma, (2006)]: A sequece l is statistically coverget if ad oly if it sequece of partial averages μ(l) coverges ad its sequeces of partial stadard deviatio σ(l) coverges to zero. Proof: Necessary coditio follows from theorem 4. ad 4.2. Sufficiet coditio: let us suppose that x = lim μ(l), lim σ l = 0, ad take some ε > 0. This implies that for ay λ > 0, there exists a umber such that λ > x μ. the takig a umber such that it implies the iequality ε > λ, we have σ 2 = x k = k μ 2 k = k = x k μ 2 ; x k μ ε x k x + x μ 2 ; x k x ε x k x ± λ 2 k = ; x k x ε (4.) DOI: / Page

10 Some Recet Developmets O Statistical Covergece ( A New Type Of Covergece )... k = k = x k x 2 ± 2λ x k x + λ 2 ); x k x ε x k μ 2 ; x k x ε ± 2λ x k = k x ; x k x ε + λ 2 (4.2) As x k μ = x k x + (x μ ) ad we take λ or λ i the expressio (4.) accordig to the followig rules: ) If x k x 0 ad x μ 0, the x k x + x μ x k x > x k x λ, we take λ; 2) If x k x 0 ad x μ 0, x k x + x μ x k x x μ > x k x λ, ad we take λ; 3) If x k x 0 ad x μ 0, the x k x + x μ = x x k x x k > (x x k ) λ=xk x+λ, ad we take +λ; 4) If x k x 0 ad x μ 0, the x k x + x μ x x k > x k x + λ as x k x < ε, ad we take +λ. 2 I the expressio (4.2), it is possible to take a sequece λ p ;p =,2,3, such that the sequece λ k coverges to 0 because the sequece μ ; =,2,3, coverges to x whe teds to. The sum 2λ k x; xk x ε also coverges to 0 whe p teds to because λp coverges to 0 ad x k x < x k + x m + x. At the same time, the sequece σ : =,2,3, also coverges to 0. Thus, lim x k μ 2 ; x k x ε = 0 This implies that lim x k μ 2 ; x k x ε = 0 At the same time, lim x k μ 2 ; x k x ε ε lim k, k N; x k x ε = 0 For ay ε > 0 as ε is a arbitrary positive umber, that is, x = S lim l Hece, the theorem is proved. V. Coclusio We have reviewed the classical theory of ordiary covergece of sequeces which successfully shows the lik betwee ordiary covergece ad statistical covergece of sequeces. Fially, we exposed profoudly the lik betwee statistical covergece ad covergece i statistics such as mea (average) ad stadard deviatio. It should oted here that, all results o statistical covergece of sigle sequece have similar versios for statistical covergece of double sequeces. For further readig o results for double sequeces [see Mursalee ad Edely (2003)], [Mursalee (2004)], Caka, C. et al (2006), Asi, A. (203), Miller, H. I. Ad Miller, V. (2008), Sarabada, S. Ad Talabi, S. (202), Alotaibi, A. M. (200), Viod, C. T. Ad Sarma, B. (2006), Patterso, R. F. (999, 2002) ad may others. Refereces []. Abdullah M. A. (200). O statistical covergece of double sequeces i radom 2-ormed spaces. Joural of Iequalities ad Special Fuctios, (2), [2]. Akcoglu, M. A. & Del Juco, A. (975) covergece of averages of poit trasformatios. Proc. America Mathematics. Soc. 49: [3]. Ackcoglu, M. A. Suchesto, L. (975) weak covergece of positive cotractios implies strog covergece of averages, probability theory ad related fields, 32: [4]. Assai, I. (2005). Poitwise covergece of o covetioal averages, college mathematics 02, [5]. Bala, I. (20). O Weak Statistical Covergece of Sequece of Fuctioals. Iteratioal Joural of Pure ad Applied Mathematics. 70 (5): [6]. Bardaro, C. et al., (205) Korovki-Type Theorems for Modular Ψ-A-Statistical Covergece. Joural of Fuctio Spaces, Article ID [7]. Bele, C. & Yildirim, M. (205). Geeralized statistical covergece ad some sequece spaces i 2 -ormed spaces. Hacettepe Joural of Mathematics ad Statistics Volume 44 (3), [8]. Billigsley, P. (965) Egordic theory ad iformatio, Joh Wiley & sos, New York. [9]. Bosheritza, M. (2005). Ergodic Averagig Sequeces. ", J. d'aalyse Mathematique, 95, [0]. Buck, R. C. (953). Geeralized asymptotic desity. America Joural of Mathematics. 75: []. Burgi, M. S. (2000) theory of fuzzy limits, fuzzy sets ad systems 5, [2]. Burgi, M & Duma, O. (2006) statistical covergece ad covergece i statistics. [3]. Cakalli, H. (2009). A study o statistical covergece. Fuctioal Aalysis, Approximatio ad Computatio, (2), DOI: / Page k = x k

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