ON STATISTICAL CONVERGENCE AND STATISTICAL MONOTONICITY

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1 Aales Uiv. Sci. Budapest., Sect. Comp. 39 (203) ON STATISTICAL CONVERGENCE AND STATISTICAL MONOTONICITY E. Kaya (Mersi, Turkey) M. Kucukasla (Mersi, Turkey) R. Wager (Paderbor, Germay) Dedicated to the 70th birthday of Professor Karl-Heiz Idlekofer Commuicated by Imre Kátai (Received December 28, 202; accepted Jauary 07, 203) Abstract. The mai aim of this paper is to ivestigate properties of statistically coverget sequeces. Also, the defiitio of statistical mootoicity ad upper (or lower) peak poits of real valued sequeces will be itroduced. The iterplay betwee the statistical covergece ad these cocepts are also studied. Fially, the statistically mootoicity is geeralized by usig a matrix trasformatio.. Itroductio The cocept of statistical covergece for real or complex valued sequece was itroduced i the joural Colloq. Math. by H. Fast i [5] ad H. Steihaus i [7] idepedetly i the same year 95. The idea of this cocept is based o the otatio of asymptotic desity of a set K N (see for example [], [2]). Key words ad phrases: Statistical covergece, Lower-Upper peak poit, statistical mootoe. 200 Mathematics Subject Classificatio: 40A05, 40C05.

2 258 E. Kaya, M. Kucukasla ad R. Wager Statistical covergece has may applicatios i differet fields of mathematics such as: approximatio theory [9], measure theory [5], probability theory [8], trigoometric series [8], umber theory [4], etc. Let K be a subset of N ad K() := {k : k, k K}. The, the asymptotic desity of K, deoted by δ(k), is defied by (.) δ(k) := lim if the limit exists. eclosed set. K(), I (.), the vertical bars idicate the cardiality of the A real or complex valued sequece x = (x ) is said to be statistically coverget to the umber L, if for every ε > 0, the set has asymptotic desity zero, i.e. ad it is deoted by x L(S). K(, ε) := {k : k, x k L ε}, K(, ε) lim = 0, Deeply coected with this defiitio is the cocept of strogly Cesàro summability ad uiform summability (see Idlekofer [3]). A sequece x = (x ) is called strogly-[c,,α] summable (α > 0) to the mea L i case (.2) lim ad is writte x k L α = 0 x L[C,, α]. By w α we deote the space of strogly Cesàro summable sequeces. A sequece x = (x ) is called uiformly summable i case (.3) lim lim sup K x k K x k = 0. The space of uiformly summable sequeces is deoted by L.

3 O statistical covergece ad statistical mootoicity 259 Remark.. The space of all complex valued sequeces x = (x ) will be deoted by C N. I may circumstaces we refer to C N as the space of arithmetical fuctios f : N C, especially, whe f reflects the multiplicative structure of N. This is the case for additive ad multiplicative fuctios. We recall the defiitio. A arithmetical fuctio f is called additive or multiplicative, if for every pair m, of positive coprime itegers the relatio f(m) = f(m) + f() or f(m) = f(m)f(), respectively, is satisfied. 2. Some results about statistical covergece Defie the fuctio d : C N C N [0, ) for all x, y C N as follows, d(x, y) := lim sup ϕ( x k y k ) where ϕ : [0, ] [0, ) ϕ(t) = { t, if t ;, otherwise. It is clear that d is a semi-metric o C N. With these otatios we have Theorem 2.. The sequece x = (x ) is statistically coverget to L if ad oly if d(x, y) = 0 where y = (y ) ad y = L for all N. Proof. Let us assume d(x, y) = 0 where y = L for all N. The, if ε > 0, { lim sup max, } lim sup ϕ( x k L ) = ε x k L ε { = max, } d(x, y) = 0 ε ad x L(S). Now, assume that x is statistically coverget to L. The, for ay ε > 0, ϕ( x k L ) = ϕ( x k L ) + ϕ( x k L ) ε + x k L <ε x k L ε x k L ε

4 260 E. Kaya, M. Kucukasla ad R. Wager which implies immediately d(x, y) ε for ay ε > 0 where y = (y ) ad y = L ( N). This eds the proof of Theorem 2.. By the first part of this proof we coclude Corollary 2.. If x is strogly Cesàro summable to L the x is statistically coverget to L. Remark 2.. The iverse of Corollary 2. is ot true i geeral. To see this, it is eough to cosider the sequece x : N C as {, = m x := 2, m =, 2,...; 0, otherwise. O the other had the followig holds. Corollary 2.2. If x = (x ) is a bouded sequece ad statistically coverget to L, the x is strogly Cesàro summable to L. The ext result is well-kow (see J.A. Fridy [7]). But for the sake of completeess we give a proof of which is differet from that of [7]. Theorem 2.2. A sequece x is statistically coverget to L if ad oly if there exists H N with δ(h) = such that x is coverget to L i H,i.e. lim H x = L. Proof. Assume that x is statistically coverget to L. There is m j N such that 2 j 2 j < x k L < 2 j is satisfied for all m j. Deote the set The H j := { k N : 2 j x k L < } 2 j ad k < m j. 2 j < x k L < 2 j ad k N\H j < 2 j

5 O statistical covergece ad statistical mootoicity 26 holds for all N. If we cosider the set H := H j {k : x k = L} the x L ε holds oly to fiitely may H. This meas that x is coverget to L i the usual case. Now, let us show that δ(n \ H) = 0. Let ε > 0 be give ad choose a arbitrary r N such that j=r+ 2 j < ε 2 holds. For this r, there exists a l r N such that j= 2 j < x k L < 2 j ad k N\H < r +. ε 7 ad x k L ad k N\H < r +. ε 4 for all > l r ad j {, 2,..., r}. Therefore, k N\H < ε 2 + ε 2 holds for all l r. The iverse of theorem is obviously obtaied. 3. Some results for multiplicative ad additive fuctios I this sectio, we will give some results for multiplicative ad additive fuctios. With M(f) we deote the mea-value of the arithmetical fuctio f, if the limit M(f) := lim f(k) exists. Theorem 3.. Assume that f : N C is bouded ad statistically coverget to L ad H N is a arbitrary set which possesses a asymptotic desity δ(h). The, M( H f) exists ad equals L δ(h).

6 262 E. Kaya, M. Kucukasla ad R. Wager Proof. The proof is obtaied obviously by the followig iequality f(k) L k H, f(k) L <ε k H f(k) L + k H k H, f(k) L ε f(k) L. Theorem 3.2. If a multiplicative fuctio f is bouded ad statistically coverget to L 0, the f. Proof. Let p 0 P, P is the set of primes, k 0 N ad let H := { N : p k0 0 }, be the set of all elemets of N divisible exactly by p k0 0, i.e. ca be writte i the form = p k0 0 m where p 0 m. It is clear from Theorem 3. that M( H f) = Lδ(H) = L p k0 0 ( p 0 ) holds. Sice f is multiplicative, we have ( ) f(k) = f(p k0 0 ) f k p k0 0 for k H. Therefore, M( H f) = lim f(k) = lim f(pk0 k H = lim f(pk0 0 ) 0 ) p k0 0 = f(p k0 0 ) L p k0 0 m p k 0 0 p 0 m p k 0 0 f(m) = ( p 0 ). m p k 0 0 p 0 m f(m) = This implies f(p k0 0 ) =. Sice p 0 is a prime, k 0 N have bee chose arbitrarily it follows f = (ad L = ).

7 O statistical covergece ad statistical mootoicity 263 Remark 3.. If f is bouded ad statistically coverget to L, the f is Cesáro summable, i.e. (.2) holds (for every α > 0). The Idlekofer proved i [3] (Theorem 2 ). Propositio 3.3. Let f be multiplicative ad α > 0. The the followig assertios are valid: (i) If (.2) holds for L 0, the L = ad f() = for all N. (ii) (.2) holds for L = 0 if ad oly if f α L ad oe of the series diverges or p f(p) 2 ( f(p) ) 2, p p x f(p) p p f(p) > 2 as x. f(p) α p Assertio (i) correspods to Theorem 3.2. Theorem 3.4. If a additive fuctio f : N C is statistically coverget, the f 0. Proof. It is eough to prove Theorem 3.4 for real-valued additive fuctios. Let us defie the multiplicative fuctio g : N C by g() := e iαf() where α R. Obviously, g is bouded ad also statistically coverget to some L,where L 0. So, g. Therefore, αf() = 2πik where k Z. For a arbitrary β R we have also βf() = 2πi k where k Z. If we assume f() 0, the β α Q. This is a cotradictio because of β is a arbitrary real umbers. So, f Statistical mootoicity ad related results I this sectio we cosider oly real-valued sequeces ad itroduce the cocept of statistical mootoicity.

8 264 E. Kaya, M. Kucukasla ad R. Wager Defiitio 4.. (Statistical mootoe icreasig (or decreasig) sequece.) A sequece x = (x ) is statistical mootoe icreasig (decreasig) if there exists a subset H N with δ(h) = such that the sequece x = (x ) is mootoe icreasig (or decreasig) o H. A sequece x = (x ) is statistical mootoe if it is statistical mootoe icreasig or statistical mootoe decreasig. I the followig we list some (obvious) properties of statistical mootoe sequeces. (i) If the sequece x = (x ) is bouded ad statistical mootoe the it is statistically coverget. (ii) If x = (x ) is statistical mootoe icreasig or statistical mootoe decreasig the (4.) lim {k : k : x k+ < x k } = 0 or (4.2) lim {k : k : x k+ > x k } = 0 respectively. The iverse of these assertios is ot correct because of the followig example: Defie x = (x ) by {, if 2 x = k < 2 k+ for eve k, 0, otherwise. The the relatios (4.) ad (4.2) hold but x = (x ) is ot statistical mootoe (ad ot statistically coverget). Defiitio 4.2. The real umber sequece x = (x ) is said to be statistical bouded if there is a umber M > 0 such that δ({ N : x > M}) = 0. Let { k } be a strictly icreasig sequece of positive atural umbers ad x = (x ), defie x = (x k ) ad K x := { k : k N}. With this otatio Defiitio 4.3. (Dese Subsequece) The subsequece x = (x k ) of x = = (x ) is called a dese subsequece, if δ(k x ) =.

9 O statistical covergece ad statistical mootoicity 265 Two more simple properties are give by (iii) Every dese subsequece of a statistical mootoe sequece is statistical mootoe. (iv) The statistical mootoe sequece x = (x ) is statistically coverget if ad oly if x = (x ) is statistical bouded. Defiitio 4.4. The sequece x = (x ) ad y = (y ) are called statistical equivalet if there is a subset M of N with δ(m) = such that x = y for each M. It is deoted by x y. With this defiitio we formulate (v) Let x = (x ) ad y = (y ) be statistical equivalet. The x = (x ) statistical mootoe if ad oly if y = (y ) is statistical mootoe. For additive fuctios we have Theorem 4.. If a real-valued additive fuctio f is statistical mootoe, the there exists c R such that f() = c log ( N). I. Kátai [4] ad B.J. Birch [2] showed idepedetly, that it is eough to assume i Theorem 4., that f is mootoe o a set havig upper desity oe. A immediate cosequece is give i Corollary 4.. Let f be a (real-valued) multiplicative fuctio, which is statistical mootoe. The f() = c ( N) with some c Peak poits ad related results I this sectio, for real-valued sequeces upper ad lower peak poits will be defied ad its relatio with statistical covergece ad statistical mootoicity will be give. Defiitio 5.. (Upper (or Lower) Peak Poit) The poit x k is called upper (or lower) peak poit of the sequece x = (x ) if x k x l (or x l x k ) holds for all l k. Theorem 5.. If the idex set of peak poits of the sequece x = (x ) has asymptotic desity, the the sequece is statistical mootoe.

10 266 E. Kaya, M. Kucukasla ad R. Wager Proof. Let us deote the idex set of upper peak poits of the sequece x = (x ) by H := {k : x k upper peak poit of (x )} N. Sice δ(h) =, ad x = (x ) is mootoe o H, the sequece x = (x ) is statistical mootoe. Remark 5.. The iverse of Theorem 5. is ot true. Cosider x = (x ) where { x = m, = m2, m N;, m 2, i.e. x = (x ) = (, 2, 3, 2, 5, 6, 7, 8, 3,...). Sice the set H = { m 2 : m N } possesses a asymptotic desity δ(h) = 0, the sequece x = (x ) is statistical mootoe icreasig. But, it has ot ay peak poits. Corollary 5.. If x = (x ) is bouded ad the idex set of upper (or lower) peak poits H = {k : x k upper(lower) peak poit of (x )} possesses a asymptotic desity, the x = (x ) is statistically coverget. Remark 5.2. I Corollary 5., it ca ot be replaced usual covergece by statistical covergece. Let us cosider the sequece x = (x ) where {, = m x := 2, m N, k, m2. The idex set of upper peak poits of the sequece x = (x ) is H = { k : k m 2, m N }. It is clear that δ(h) = ad x = (x ) is bouded. So, the hypothesis of Corollary 5. is fulfilled. The subsequece (x k ) = ( 2, 3, 5, 6, 7, 8, 0,... ) is covergece to zero. Also, x = (x ) is statistical covergece to zero but it is ot covergece to zero.

11 O statistical covergece ad statistical mootoicity A-geeralizatio of statistical mootoicity Statistical mootoicity ca be geeralized by usig A-desity of a subset K of N for a regular oegative summability matrix A = (a k ). Recall A-desity of a subset K of N, if exists ad is fiite. δ A (K) := lim k K a k = lim k= a k K (k) = = lim (A K)() The sequece x = (x ) is A-statistically coverget to l, if for every ε > 0 the set K ε = {k N : x k l ε} possesses A-desity zero (see for details [3]). Defiitio 6.. A sequece x = (x ) is called A-statistical mootoe, if there exists a subset H of N with δ A (H) = such that the sequece x = (x ) is mootoe o H. A = (a k ) ad B = (b k ) will deote oegative regular matrices. Theorem 6.. If the coditio (6.) lim sup a k b k = 0 k= holds. The, x = (x ) is A-statistical mootoe if ad oly if x = (x ) is B-statistical mootoe. Proof. For a arbitrary H N the iequality 0 (A H )() (B H )() = k H a k k H a k b k k H a k b k, k= b k holds. Uder the coditio (6.) δ A (H) exists if ad oly if δ B (H) exists, ad i this case δ A (H) = δ B (H). Therefore, A-statistical mootoicity of x = (x ) implies B-statistical mootoicity vice versa. Let us cosider strictly icreasig ad oegative sequece {λ } N ad E = {λ } =0. If A = (a k ) is a summability matrix, the A λ := (a λ(),k )

12 268 E. Kaya, M. Kucukasla ad R. Wager is the submatrix of A = (a k ). Thus, the A λ trasformatio of a sequece x = (x ) as (A λ x) = a λ(),k x k. k=0 Sice, A λ is a row submatrix of A, it is regular wheever A is a regular summability matrix (See [6],[0]). The Cesáro submethod s has bee defied by D.H. Armitage ad I.J. Maddox i [], ad later studied by J.A. Osikiewicz i [6]. Theorem 6.2. Let A be a summability matrix ad let E = {λ } ad F = {ρ } be a ifiite subset of N. If F \ E is fiite, the A λ -statistical mootoicity implies A ρ -statistical mootoicity. Proof. Assume that F \E is fiite, ad x = (x ) is A λ -statistical mootoe sequece. From the assumptio there exists a 0 N such that {ρ() : 0 } E. It meas that there is a mootoe icreasig sequece j() such that ρ() := := λ j (). So, the A ρ asymptotic desity of the set K := { N : ρ() = λ j ()} is lim a ρ(),k K (k) = lim a λj(),k K (k) = k= k= =. This give us, x = (x ) is A ρ -statistical mootoe sequece. By the Theorem 6.2 we have followig corollaries: Corollary 6.. A-statistical mootoe sequece is A λ -statistical mootoe. Corollary 6.2. Uder the coditio of Theorem 6.2, if E F is fiite, the the sequece x = (x ), A λ - statistical mootoe if ad oly if A ρ -statistical mootoe. Refereces [] Armitage D.H. ad I.J. Maddox, A ew type of Cesaro mea, Aalysis, 9(-2) (989), [2] Birch, B.J., Multiplicative fuctios with o-decraasig ormal order, J. Lodo Math Soc., 42 (967), 49 5.

13 O statistical covergece ad statistical mootoicity 269 [3] Buck, R.C., Geeralized asymptotic desity, Amer. J. Math., 75 (953), [4] Erdős, P. ad G. Teebaum, Sur les desities de certaies suites détiers, Proc. Lodo Math. Soc., 59 (989), [5] Fast, H., Sur la covergece statistique, Colloq. Math., 2 (95), [6] Fridy, J.A., Submatrices of summability matrices, It. J. Math. Sci., (4) (978), [7] Fridy, J.A., O statistical covergece,, Aalysis, 5 (985), [8] Fridy, J.A. ad M.K. Kha, Tauberia theorems via statistical covergece, J. Math. Aal. Appl., 228 (998), [9] Gadjiev, A.D. ad C. Orha, Some approximatio theorems via statistical covergece, Rocky Moutai J. Math., 32 (2002), [0] Goffma, C. ad G.M. Peterso, Submethods of regular matrix summability methods, Caad. J. Math., 8 (956), [] Halberstam, H. ad K.F. Roth, Sequeces, Claredo Press, Oxford, 966. [2] Idlekofer, K.-H., New approach to probabilistic umber theorycompactificatios ad itegratio, Adv. Studies i Pure Mathematics, 49 (2005), [3] Idlekofer, K.-H., Cesàro meas of additive fuctios, Aalysis, 6 (986), 24. [4] Kátai, I., A remark o umber-theoretical fuctios, Acta Arith., 4 (968), [5] Miller, H.I., A measure theoretical subsequece characterizatio of statistical covergece, Tras. Amer. Math. Soc., 347 (995), [6] Osikiewicz, J.A., Equivalece results for Cesaro submethods, Aalysis, 20() (2000), [7] Steihaus, H., Sur la covergece ordiaire et la covergece asymtotique, Colloq. Math., 2 (95), [8] Zygmud, A., Trigoometric Series, Secod ed., Cambridge Uiv. Press, Cambridge, 979. E. Kaya Tarsus Vocatioal School Uiversity of Mersi Takbaş Köyü Mevkii Mersi, Turkey kayaerdeer@mersi.edu.tr M. Kucukasla Faculty of Sciece, Departmet of Mathematics Uiversity of Mersi Mersi, Turkey mkucukasla@mersi.edu.tr

14 270 E. Kaya, M. Kucukasla ad R. Wager R. Wager Faculty of Computer Sciece, Electrical Egieerig ad Mathematics Uiversity of Paderbor Warburger Straße 00 D Paderbor, Germay