Mathematica Slovaca. Tibor Šalát On statistically convergent sequences of real numbers. Terms of use: Persistent URL:

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1 Mathematica Slovaca Tibor Šalát O statistically coverget sequeces of real umbers Mathematica Slovaca, Vol. 30 (1980), No. 2, Persistet URL: Terms of use: Mathematical Istitute of the Slovak Academy of Scieces, 1980 Istitute of Mathematics of the Academy of Scieces of the Czech Republic provides access to digitized documets strictly for persoal use. Each copy of ay part of this documet must cotai these Terms of use. This paper has bee digitized, optimized for electroic delivery ad stamped with digital sigature withi the project DML-CZ: The Czech Digital Mathematics Library

2 Math. Slovaca 30,1980, No. 2, ON STATISTICALLY CONVERGENT SEQUENCES OF REAL NUMBERS T. SALAT The otio of the statistical covergece of sequeces of real umbers was itroduced i papers [1] ad [5]. I the preset paper we shall show that the set of all bouded statistically coverget sequeces of real umbers is a owhere dese subset of the liear ormed space m (with the sup-orm) of all bouded sequeces of real umbers ad the set of all statistically coverget sequeces of real umbers is a dese subset of the first Baire category i the Frechet space s. 1. Itroductio I this part of the paper we shall itroduce some defiitios, otatios ad two auxiliary results. If AcziV={l, 2,...,,...} the we put A()= ^ 1- ^ lere exists a^, ae.a lim -, it will be called the asymptotic desity of the set A ad will be deoted -»«> by 5(A). Obviously we have 5(A) = 0 provided that A is a fiite set of positive itegers. Defiitio 1.1. The sequece x = { *}r=i of real umbers is said to coverge statistically to the real umber (this fact will be deoted by (1) lim stat & = k-*<*> or briefly x stat > ) // for each e>0 we have 6(A e ) = 0, where A e = {en; Ik-SlSe}. I paper [5] istead of (1) the otatio D-lim fc = 5 is used ad the statistical covergece is called the D-covergece. It is easy to see that if (1) holds, the the umber is determied uiquely. If lim * =, the (1) holds, too, sice the set A e is i this case fiite for each 139

3 e >0. The coverse is ot true (see example 1,1). Thus the statistical covergece is a atural geeralizatio of the usual covergece of sequeces. Let us further observe that the coditio b(a e ) = 0 is equivalet to the coditio b(a' e ) = 1, where A e = {en;\% -Z\<e}( = N-A ). The sequece which coverges statistically eed ot be bouded. This fact ca be see from the followig Example 1,1. It is easy to see that the set A = {l 2,2 2,..., 2,...} has the asymptotic desity 0. Sice the set of all ratioal umbers is coutable, there exists such a sequece {rj/-}r=i that the set of all terms of this sequece coicides with the set of all ratioal umbers. Put r\ = for ± j 2 (/= 1, 2,...). The lim stat r\ k = 0 ad simultaeously each real umber is a limit poit of the sequece {r\ k } k=lm The aalysis of the structure of the sequece {r] k } k=1 from the previous example suggests the cojecture that the structure of each statistically coverget sequece is aalogous to the structure of this sequece, i.e. if (1) holds, the there exists such a setxcn that b(k) = 1 ad lim t; k = (lim fc = <f meas that for each e>0 k-*oo kek \fc K» kek there exists such a k 0 that for each k>k 0y kek we have * -t;\<e). The followig lemma is a affirmative solutio of the metioed cojecture. Lemma 1,1. Statemet (1) holds if ad oly if there exists such a set K = {k 1 <k 2 <...<k <...}czn that b(k) = 1 ad lim & =. -*oo Proof. 1. If there exists a set with the metioed properties ad e is a arbitrary give positive umber, the we ca choose such a umber 0 en that for each > 0 we have (2) l&,-5l<*. Put A e = {en; - ^e}. The from (2) we get A e czn {Ac 0+ 1, kq+2,.../ ad o the right-had side there is a set the asymptotic desity of which is 0. Therefore 6(A C ) = 0, hece (1) holds. 140

4 2. Let (1) hold. Put K, = {GN;,- <f} 0=L2,...). The accordig to defiitio 1,1 we have b(k,)= 1 (j = l, 2,...). It is evidet from the defiitio of K, (j= 1, 2,...) that (3) K l z>k 2 =>...=>K i z>k i+1 =>..., (3') 6(^)=1 0=L2,...). Let us choose a arbitrary umber v 1 ek 1. Accordig to (3') there exists such 1C (\ 1 av 2 >v l9 v 2 ek 2 that for each ^v 2 we have >^- Further, accordig to (3') there exists such a v 3 >v 29 v 3 ek 3 that for each ^v 3 we have Thus we ca costruct by iductio such a sequece v 1 <v 2 <...<v i <... of positive itegers that v i ek, (j = 1, 2,...) ad (4) * 0>Lzi j 3 >- a.s. 0. for each rc^i;, (/ = 1> 2,...). Let us costruct the set K as follows: Each atural umber of the iterval (1, v t ) belogs to K 9 further, ay atural umber of the iterval (v i9 v i+1 ) belogs to K if ad oly if it belogs to K t (/ = -U 2,...). Accordig to (3), (4) for each 9 v,^<v i+1 we get KW^KMu-l From this it is obvious that 6(JC) = 1. Let e >0. Choose a ; such that -<. Let ^v i9 ek. The there exists such a umber l^j that Vi^<v i+1. But the o the basis of the defiitio of K 9 e K t, hece Thus! - <e for each e K 9 ^v i9 i.e. lim * =. AceK The followig result ca be obtaied (cf. [1], [5]) directly from the defiitio 1,1. Ac >oo 141

5 Lemma 1,2. If lim stat * = a, lim stat rj k =b ad c is a real umber, the (i) lim stat (&+r/*) = a+ 6, Ac-»oo (ii) lim stat (c -% k ) = ca. k-* o It follows from lemma 1,2 that the set of all bouded statistically coverget sequeces of real umbers is a liear subspace of the liear ormed space m of all bouded sequeces of real umbers (with the orm x = sup J, x = k-l, 2,.. {%k}k=iem). 2. Bouded statistically coverget sequeces of real umbers We deote by m 0 the set of all bouded statistically coverget sequeces of real umbers. Theorem 2,1. The set m 0 is a closed liear subspace of the liear ormed space m. Proof. Let x () em 0 ( = 1, 2,...), jc () >JC em. We shall show that x em 0. Accordig to the assumptio for each there exists such a real umber a that i.e. if jc <) ={ l, r, the J*-i x i) >a (AT = 1,2,...) stat lim stat i"' = A (/i = 1,2,...). We shall prove that a) the sequece (of real umbers) {a } =i coverges to a real umber a ; b) x >a (i.e. if x = {&}*=-- the lim stat fc = a). stat k-> o From a), b) the assertio follows o accout of Lemma 1,2. Proof of a). Sice {jt () },7=i is a coverget sequece of elemets from m, for each e >0 there exists such a 0 en that for each /, > 0 v/e have (/>_ *.(">.! (5) II* 1 "-JГЧK;, 142

6 Further, accordig to Lemma 1,1 there exist such sets A,, A, A,, A cat that S(A,) = 5(A ) = 1 ad (6) limlípfc kєaj = */ (7) Иm Й">fc»«> fcea = «The set AjA is ifiite sice the asymptotic desity of this set is equal to 1. Hece we ca choose such a keajc\a that we have (see (6), (7)): (8) \&~a,\<\, \^-a \< -. Accordig to (5) ad (8) we get for each /, > 0 + ^>-a <f + H =. Sice the sequece {a k }Z= x fulfils the Cauchy coditio for covergece, it must coverge to a real umber a, hece (9) a = lima*. fc-»<» Proof of b). Let r\ >0. It suffices to prove that there exists such a set A czn that 5(A) =1 ad for each kea the iequality \^k a\<r\ holds (see Lemma 1,1). Sice JC 0) >JC, there exists such apen that (10) * W -JC <3. The umber p ca be chose i such a way that together with (10) also the iequality (ii) k-«l<? holds (see (9)). Sice JC (P) >a p, there exist such a set A c:n that 5(A) = 1 ad for each kea stat we have d2) i?r-«p i<f. 143

7 Now accordig to (10), (11), (12) we get for each \^k-a\^k-^\ + \^-a \ + + k-a <f =. Hece b) follows. Usig the previous theorem we ca easily prove the followig result o the structure of the set m 0. Theorem 2,2. The set m 0 is a owhere dese set i m. Proof. It is a well-kow fact that every closed liear subspace of a arbitrary liear ormed space E, differet from E, is a owhere dese set i E (cf. [2], p. 37, Exercise 4; [3]). Hece o accout of Theorem 2,1 it suffices to prove that m 0 i=m. But this is evidet, sice the sequece x = {(-1)* } =i e m does ot belog to m 0. kea 3. Statistically coverget sequeces of real umbers ad the space s Deote by s 0 the set of all statistically coverget sequeces of real umbers. I what follows s deotes the Frechet metric space of all real sequeces with the metric g, ( \ v- 1 %~T\ k \ x = { *}r=i e s, y = {rj fc }r=i e s. I this part of the paper we shall describe some fudametal properties ad the structure of s 0 i the space s. Theorem 3,1. (i) The set s 0 is dese i the space s. (ii) The set s 0 is a set of the first Bake category i the space s. Corollary. The set s-s 0 (of all real sequeces which are ot statistically coverget) is a residual set (of the secod Baire category) i the space s. For the proof of Theorem 3,1*) we shall use the followig Lemma 3,1. Let g k (k = 1, 2,...) be complex valued cotiuous fuctios o R=( oo 9 +oo). Let us suppose that there are two distict complex umbers c u c 2 such that for each sufficietly large k we have c u c 2 eg k (R). *) The author is idebted to Professor M. Novoty for his suggestio, which led to a improvemet of the origial versio of the proof of this theorem. 144

8 Let (a k ) be a triagular matrix with the followig (Pi) For each fixed k we have lim a k = 0; -*o (P 2 ) lim i> fc = l. properties: Tfte fhe set Si of all such x = {% k }Z=\ e s for which there exists a (fiite) limit lim Tfl^f^) is a set of the first Baire category i s. -+ *=i Proof. For * = { *}r=igsi we put f(x)=jza k g k ( k ) ( = l,2,...), f(x) = \im f (x). k=\ -> We shall prove that a) /. ( = 1, 2,...) is a cotiuous fuctio o s,; b) / is discotiuous at each poit of s x. a) Let a = {a k } k=x es u x <J) = {%V}k=\ s x (j = 1, 2,...), jc a) >a. Sice from the covergece i the space s the covergece "by coordiates" follows, for each fixed k we have lim ^ = a k. But the o accout of the cotiuity of fuctios g k ( = 1,2,...) we get lim/ (jc 0) ) = limia ^fc (^)) = /-»» /_oo k = l = ^a k \img k (t; k i) ) = '^a k g k (a k ) = f (a). k=l /» k=l Thus f ( = 1, 2,...) is cotiuous o s t. b) Let b = {ft}r=i esi. Deote by v such a umber c, (i = 1 or 2) that differs from f(b). It suffices to prove that i each sphere S(b, d) = {x esi; g(b, x)<6} (6>0) there exists such a elemet x = { *}*=i that f(x) = v. Let (5>0. Choose a atural umber m such that 2 2~*<<5. Accordig to fc=m + l the assumptio there exists such a m' that for each k>m' there exists such r\ k er that g k (t] k ) = v. Put m 0 = max {m, m'} ad defie the sequece x = { *}r=i i the followig way: % k = ft for k^m 0 ad fc = rj* for A: >m 0. TheJC = { * }r=i e s ad by choice of m 0 we get g(b, x)<6. Further for >m 0 we have /.(*) = E^(^(j3 fc )-v))-hvx«fc. k=i k=i Now from the properties (Pi), (P 2 ) it follows at oce that f(x) = lim f (x) = v. 145

9 Accordig to previous cosideratios the fuctio / is i the first Baire class ad is discotiuous at each poit of s u But it is a well-kow fact that the set of discotiuity poits of a arbitrary fuctio of the first Baire class is a set of the first Baire category (cf. [6], p. 185). Hece s x is a set of the first Baire category i s, ad so a set of the first Baire category i s, too. The proof is fiished. Remark. Every regular triagular matrix (a k ) has the properties (Pi), (P 2 ) from Lemma 3,1 (cf. [4], p. 8). The coverse is ot true. Puttig e.g. _J J_ 1_ ^l / > -22,..., a-2 /, a-1 /, a 1 V V V V for odd ad _J J J _ 1 #l / 9 Q2,..., a - 3, a-2 - /, V V V V 1 a \ i* _ for eve we get the triagular matrix with the properties (Pi), (P 2 ), for which Ekfcl-^ + ^C"^00 )- k = \ Hece this matrix is ot regular. Proof of Theorem 3,1. (i) If x = {^k} k=l es 0 ad the sequece y = {t] k } k =\ of real umbers differs from x oly i a fiite umber of terms, the evidetly y e s 0, too. From this statemet (i) follows at oce o the basis of the defiitio of the metric i s. (ii) For the proof of (ii) we shall use the followig result from [5] (Theorem 4). Theorem A. Statemet (1) holds if ad oly if for each real umber t we have lim ± _>'**= e**. 1 " Deote by s[ the set of all such x = {% k } k =i es for which the fiite lim y\e* k»-~ k=i exists. Puttig i Lemma 3,1 g k (t) = e u (k = 1,2,...), a k =-(k = l,2,...,\ = l,2,...) we see that si is a set of the first Baire category i s. Accordig to Theorem A we have s 0 csl, hece s 0 is a set of the first Baire category i s, too. The proof is fiished. 146

10 I what follows deote by s* the set of all such x = {t= k } k=1 es sequece for which the G 21*0" \ k=\ ) =i is bouded. The set s* will be cosidered as a subspace of the space s. For e R deote by s*( ) the set of all such x = {% k } k=l e s for which lim stat & =. We shall fc»oo show that s*( ) is a set of the secod Borel class i the space s*. Theorem 3,2. The set s*( ) is a Fas-set i s*. For the proof of the theorem we shall use the followig lemma. Lemma 3,2. The sequece x = {t; k } k=1 es* coverges statistically to the real umber if ad oly if for each ratioal umber t we have (13) lim-i> l ** = e'*. ^ k=i Proof. 1) If (1) is true, the accordig to Theorem A (cf. [5]) the equality (13) holds for each real / ad so for each ratioal umber t. 2) Let (13) hold for each ratioal umber t. Let t 0 be a arbitrary real umber. We shall prove that (14) lim- Ve'^ = ^. «~ k=i From this statemet (1) follows accordig to Theorem A. For t G R put A-(/o,0 = i Іe, л --І^. k=1 fc= Sice e iv * = cos v% 4- i si v (v e R), we get 1 JL A (t 0,t) -S- X V(cosf 0& - cos*&) 2 + (sito?* - si/&) 2. W fc = i Usig the mea value theorem we get (15) ia.(.-o,oi-s-^k-*.iii&. By the assumptio x = {^k}k=\ e s*. Hece there exists such a K>0 that for each = 1, 2,... we have 06) ^il&l-sic. Al fc = l 147

11 It follows from (15), (16) that (17) \A (to,t)\z%y/2k\t-t 0 \. Further, we have (18) ^ie^^i^+a.co.o. k=i k=i From this we get (19) i v p«<ft k _ pi'fi < II V f" e " - P" S I + \- ifti ^ = \^C V + \e i,s -e" *\ + \A (t 0,t)\. Let e > 0. Accordig to the cotiuity of the fuctio h (x) = e 1 ** (x e R ) ad (17) we ca choose a ratioal umber t such that (20) \e u *-e^\< -, (20') A.(fo,0l<f ( = l,2,...). By our assumptio there exists such a 0 that for each > 0 the first summad o the right-h; right-had side of (19) is less tha -. The with respect to (20) ad (20') we get from (19) I- Te l ' r^-e'^ <f \ A for each > 0. Hece (14) is valid. Proof of Theorem 3,2. Deote by Q the set of all ratioal umbers. From Lemma 3,2 we get (2i)»*g)=u #(*,/)> teqj=\ p = l =p + \ where H(,j) = \x = {&}:. l es*; 1 ^ - ^ l - s l V l k=\ I J ) It ca be easily checked that the set H(,j) (for each,j) is closed i s*. But the the assertio of the theorem follows at oce from (21). 148

12 Problems. From the defiitio of the set s* it follows that (22) s* = U»?, where / = - -T = {* = {&}:-.e»;v^i t g/l 0 = L2,...). I " k=l J It ca be easily checked that s* (/ = 1, 2,...) is a closed set i s. This is a simple cosequece of the fact that the covergece i the space s is equivalet to the covergece "by coordiates". Hece accordig to (22) the set s* is a F a -set i s. Sice the set s*(f) is a F ao -set i s* ad s* is a F a -set i s, the set s*(f) is a Fas-set i s. I coectio with the foregoig fact the questio arises whether the set s(<f) of all such x = {^k}k=i es, for which lim stat fc = is a F^-set i s, too. Further, the followig questio remais ope: Is the set s 0 csa Borel set i s ad if the aswer is affirmative, i which Borel class is the set s 0? REFERENCES [1] FAST, H.: Sur la coveгgece statistique. Coll. Math., 2, 1951, [2] KOLODZIEJ, W.: Wybгae rozdzialy aalizy matematyczej. PWN, Warszawa [3] NEUBRUNN, T. SMÍTAL, J. ŠALÁT, T: O the structure of the space M(0,1). Rev. Roumaie Math. pures et appl., 13, 1968, [4] PETERSEN, G. M.: Regular Matrix Tгasformatios. McGraw-Hill, New Yoгk Toroto Sydey [5] SCHOENBERG, I. J.: The itegгability of ceгtai fuctios ad related summability methods. Amer. Math. Mothly, 66, 1959, [6] SIKORSKI, R.: Fukcje rzeczywiste I. PWN, Waгszawa Received Decembeг 13, 1977 Katedгa algebгy a teóríe čísel Pгíгodovedecká fakulta UK Mlyská dolia Bгatíslava 149

13 O CTATИCTИЧECKИ CXOДЯЩИXCЯ ПOCЛEДOBATEЛЬHOCTЯX ДEЙCTBИTEЛЬHЫX ЧИCEЛ T. Шaлaт Peзюмe Пocлeдoвaтeльнocть {a }ñ=i дeйcтвитeльныx чиceл нaзывaeтcя cтaтиcтичecки cxoдящяяcя к чиcлy Ű, ecли для кaждoгo >0 acимптoтичecкaя плoтнocть мнoжecтвa {; \a a =е} pавняeтcя нyлю. B pабoтe пoказанo, чтo мнoжecтвo вcex cтатиcтичecки cxoдящиxcя пocлeдoватeльнocтeй пpocтpанcтва вcex oфаничeнныx пocлeдoватeльнocтeй нигдe нe плoтнo, мнoжecтвo вcex cтатиcтичecки cxoдяu иxcя пocлeдoватeльнocтeй пpocтpанcтва Фpeшe s являетcяя мнoжecтвoм пepвoй катeгopии Бэpа. 150