Chaos, Solitons and Fractals 42 (2009) 2815–2821 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos Statistical convergence, selection principles and asymptotic analysis G. Di Maio a,1, D. Djurčić b,2, Lj.D.R. Kočinac c,*,2, M.R. Žižović b,2 a b c Dipartimento di Matematica, Seconda Universita di Napoli, Via Vivaldi 43, 81100 Caserta, Italy Technical Faculty, University of Kragujevac, Svetog Save 65, 32000 Čačak, Serbia Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia a r t i c l e i n f o Article history: Accepted 1 April 2009 a b s t r a c t We consider the set S of sequences of positive real numbers in the context of statistical convergence/divergence and show that some subclasses of S have certain nice selection and game-theoretic properties. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Recently, the second, third and fourth authors of this article found out unexpected, interesting relations between two classical mathematical theories: theory of selection principles (related also to game theory and Ramsey theory) and theory of divergent processes in asymptotic analysis (see [5–8] and the survey paper [4]). On the other hand, in [2] the first and the third authors investigated, among others, applications of the idea of statistical convergence in topology, and in particular in selection principles theory. In this paper we continue such an investigation. Among other things we generalize in various directions several results from the mentioned papers [4–8]. On several places we apply without special mention modified techniques used first in [6]. The statistical rapid convergence is introduced and a result involving this concept is shown. We believe that the idea of this convergence and its modifications like statistical fuzzy rapid convergence, can be applied to investigations in various fields, such as intuitionistic fuzzy spaces (see [14], fuzzy dynamical systems and non-linear dynamical systems, control of chaos and quantum (statistical) mechanics (see [20,9]), and so on. Our basic objects will be the set S of sequences of positive real numbers and some its subsets. 2. Basic notions 2.1. Statistical convergence The first idea of statistical convergence appeared (under the name almost convergence) in the first edition (Warsaw, 1935) of the celebrated monograph [26] of Zygmund. However, the notion of statistical convergence of sequences of real numbers was explicitly introduced by Fast in [10] and Steinhaus in [24] (actually, in 1949 on a conference held at the Wroclaw University in Poland) and is based on the notion of asymptotic (or natural) density of a set A  N [13]. Let A  N and n 2 N. Put AðnÞ :¼ fk 2 A : k 6 ng. Then one defines: * Corresponding author. E-mail addresses: giuseppe.dimaio@unina2.it (G. Di Maio), dragandj@tfc.kg.ac.yu (D. Djurčić), lkocinac@ptt.rs (Lj.D.R. Kočinac), zizo@tfc.kg.ac.yu (M.R. Žižović). 1 Supported by MUR. 2 Supported by MN RS. 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.033 2816 G. Di Maio et al. / Chaos, Solitons and Fractals 42 (2009) 2815–2821 jAðnÞj ; n jAðnÞj dðAÞ :¼ lim sup ; n n!1 dðAÞ :¼ lim inf n!1 called the lower and upper asymptotic density of A, respectively. If dðAÞ ¼ dðAÞ, then dðAÞ ¼ lim n!1 jAðnÞj ; n is called the asymptotic (or natural) density of A. All the three densities, if they exist, are in [0, 1]. We recall also that dðN n AÞ ¼ 1  dðAÞ for A  N. A sequence ðxn Þn2N in a topological space X is said to converge statistically (or shortly, s-converge) to x 2 X, if for every neighborhood U of x; dðfn 2 N : xn R UgÞ ¼ 0. In this case we write x ¼ s  lim xn . In [11] (see also [21]) it was shown that for X ¼ R this definition is equivalent to the statement: for every neighborhood U of x there exist a subset A of N with dðAÞ ¼ 1 and n0 ¼ n0 ðUÞ such that n P n0 and n 2 A imply xn 2 U (i.e. limm!1; m2A xm ¼ x. Because we are going to work with sequences of real numbers we shall use this fact, and ‘‘statistical convergence” will mean ‘‘statistical convergence of sequences of real numbers”. We state now some simple, basic facts about statistical convergence. Fact 1. The limit of a statistically convergent sequence is uniquely determined. Fact 2. If a sequence ðxn Þn2N converges to x in the usual sense, then it statistically converges to x (since every finite subset of N has asymptotic density zero), while the converse is not true in general. Take, for example the divergent sequence ðxn Þn2N in R defined by: xn ¼ 1, if n is prime, and xn ¼ 0, otherwise. Since the set of prime natural numbers has asymptotic density 0, this sequence statistically converges to 0. Fact 3. A subsequence of a statistically convergent sequence need not be statistically convergent. (See the previous example.) A subset A of N is statistically dense if dðAÞ ¼ 1. Let us note that the union and intersection of two statistically dense sets in N are also statistically dense. A subsequence ðxnk Þk2N of a sequence ðxn Þn2N is statistically dense in ðxn Þn2N (respectively, has positive asymptotic density) if the set of indices nk is a statistically dense subset of N (respectively, has positive asymptotic density in N). Fact 4. A sequence ðxn Þn2N is statistically convergent if and only if any its statistically dense subsequence is statistically convergent. Statistical convergence has several applications in different fields of mathematics (see [2] and references there in): summability theory, number theory, trigonometric series, probability theory, measure theory, optimization, approximation theory. The statistical convergence of sequences of real numbers was generalized to sequences in metric spaces (see, for instance, [19]) and also to topological and uniform spaces [2]. In [2], several applications to selection principles theory, function spaces and hyperspaces were given. 2.2. Selection principles and games The theory of selection principles, a growing field of mathematics, has old roots going back to the 1920’s and 1930’s to works of Borel (1919), Menger (1924), Hurewicz (1925, 1927), Rothberger (1938), Sierpiński (1926, 1928)) and others. In recent years in many papers it was demonstrated that this theory has deep connections with different areas of mathematics. Because we study the set S and its subsets we shall define selection principles only for subsets of S. For more general definitions and results related to selection principles and their interplay with game theory we refer the interested reader to the survey papers [16,17,22,25]. Let A and B be subsets of S. Then: 1. S1 ðA; BÞ denotes the selection principle: For each sequence ðAn : n 2 NÞ of elements of A there is a sequence ðbn : n 2 NÞ such that for each n; bn 2 An , and fbn : n 2 Ng is an element of B. 2. G1 ðA; BÞ denotes the infinitely long game for two players, ONE and TWO, who play a round for each positive integer. In the n-th round ONE chooses a set An 2 A, and TWO responds by choosing an element bn 2 An . TWO wins a play A1 ; b1 ;    ; An ; bn ;    if fbn : n 2 Ng 2 B; otherwise, ONE wins. Clearly, if ONE does not have a winning strategy in the game G1 ðA; BÞ, then the selection hypothesis S1 ðA; BÞ is true. The converse implication is not always true. A strategy r for the player TWO is a coding strategy if TWO remembers only the most recent move by ONE and by TWO before playing the next move, i.e. the moves of TWO are: b1 ¼ rðA1 ; ;Þ; bn ¼ rðAn ; bn1 Þ; n P 2. G. Di Maio et al. / Chaos, Solitons and Fractals 42 (2009) 2815–2821 2817 3. [18] ai ðA; BÞ, i ¼ 1; 2; 3; 4, denotes the selection hypothesis that for each sequence ðAn : n 2 NÞ of elements of A there is an element B 2 B such that: a1 ðA; BÞ: for each n 2 N the set An n B is finite; a2 ðA; BÞ: for each n 2 N the set An \ B is infinite; a3 ðA; BÞ: for infinitely many n 2 N the set An \ B is infinite; a4 ðA; BÞ: for infinitely many n 2 N the set An \ B is nonempty. 4. [2] s  ai ðA; BÞ; i ¼ 1; 2; 3; 4, denotes the selection hypothesis that for each sequence ðAn ¼ ðan; m Þm2N : n 2 NÞ of elements of A there is an element B 2 B such that: s  a1 ðA; BÞ: for each n 2 N, dðfm 2 N : an; m 2 An n BgÞ ¼ 0; s  a2 ðA; BÞ: for each n 2 N, dðfm 2 N : an; m 2 An \ BgÞ ¼ 1; s  a3 ðA; BÞ: there is a set K  N with dðKÞ ¼ 1 such that dðfm 2 N : ak; m 2 Ak \ BgÞ ¼ 1 for each k 2 K; s  a4 ðA; BÞ: there is a set K  N with dðKÞ ¼ 1 such that for each k 2 K; Ak \ B–;. If in the previous definitions of s  ai properties, i ¼ 2; 3; 4, we replace everywhere asymptotic density 1 by positive asymptotic density, we obtain the definitions of the properties s  a2 ; s  a3 and s  a4 , respectively. Evidently, for arbitrary subclasses A and B of S we have a1 ðA; BÞ ) a2 ðA; BÞ ) a3 ðA; BÞ ) a4 ðA; BÞ; s  a1 ðA; BÞ ) s  a2 ðA; BÞ ) s  a3 ðA; BÞ ) a4 ðA; BÞ; s  a2 ðA; BÞ ) s  a3 ðA; BÞ ) s  a4 ðA; BÞ; S1 ðA; BÞ ) a4 ðA; BÞ; s  ai ðA; BÞ ) s  ai ðA; B ) ai ðA; BÞ; i ¼ 2; 3; 4; a1 ðA; BÞ ) s  a1 ðA; BÞ: 2.3. Rapid variation In his famous, influential 1930 paper [15], Karamata initiated investigation in asymptotic analysis of divergent processes, nowadays known as Karamata theory of regular variation (see [1] and also [23]). In 1970, de Haan [12] defined and investigated rapid variation and so stimulated further development in asymptotic analysis. The book [1] is a nice exposition of Karamata Theory and the theory of rapid variability. A sequence ðxn Þn2N 2 S is rapidly varying (of index of variability 1) if for each k > 1 the following asymptotic condition is satisfied: lim n!1 x½kn ¼ 1: xn If under the same condition the above limit is 0, we say that the sequence ðxn Þn2N is rapidly varying of index of variability 1. R1; s and R1; s denote the classes of rapidly varying sequences of index of variability 1 and 1, respectively. Definition 2.1. A sequence a ¼ ðan Þn2N 2 S is said to belong to the class st  ARVs if for each k > 1 there are a statistically dense set M  N; n0 2 N and c ¼ cðkÞ > 1 such that a½kn P c  an ; for every n P n0 ; n 2 M. Lemma 2.2. Every sequence from the class st  ARVs contains a subsequence divergent to 1. Proof. Let a ¼ ðan Þn2N 2 st  ARVs . Take the statistically dense set T ¼ N n P, where P is the set of prime numbers and consider the subsequence ðam Þm2T of a. For k ¼ 2 there are n0 ¼ n0 ð2Þ, a statistically dense subset M ¼ Mð2Þ of N and c ¼ cð2Þ > 1 such that a½2n P c  an for all n P n0 and n 2 M; without loss of generality one can assume n0 ¼ 1. The set M \ T has asymptotic density 1; let M \ T ¼ fk1 < k2 <   g and let bi ¼ aki ; i 2 N. Finally consider the subsequence ðb2i Þi2N of ðbi Þi2N (hence of a, too). It is easy to see that ðb2i Þi2N diverges to 1. h 3. Results I Let a ¼ ðan Þn2N 2 S and l > 0 and m > 0 with l  m P 1 be fixed. A sequence b ¼ ðbn Þn2N 2 S is said to be ðl; mÞ-weakly asymptotically equivalent with a if there is n0 2 N such that 1=m  bn 6 an 6 l  bn for all n P n0 (see [6]). Denote by fagl;m :¼ fb 2 S : b is ðl; mÞ  weakly asymptotically equivalent to ag: 2818 G. Di Maio et al. / Chaos, Solitons and Fractals 42 (2009) 2815–2821 Theorem 3.1. Let a ¼ ðan Þn2N 2 S and holds. l > 0; m > 0 such that l  m P 1 be fixed. The selection principle s  a2 ðfagl; m ; fagl; m Þ Proof. Let ðxn ¼ ðxn; m Þm2N : n 2 NÞ be a sequence of elements from fagl; m . We are going to construct a sequence in fagl; m having the intersection with each xn in a set of indexes of positive asymptotic density. S Decompose N into pairwise disjoint sets as N ¼ n2N Mn , where Mn ¼ 2n1  ð2N  1Þ :¼ f2n1  ð2m  1Þ : m 2 Ng; n 2 N: It is known [2] that dðMn Þ ¼ 2n . There is m1 2 N such that 1m  x1; m 6 am 6 l  x1; m for all m P m1 . Consider the sequence y1 ¼ ðx1; m ÞmPm1 . Inductively for each n P 2 form a sequence yn as follows. Suppose the sequences y1 ; y2 ;    ; yn1 and numbers m1 ; m2 ;    ; mn1 have been already defined. Let   1 mn ¼ min m 2 N :  xn;m 6 am 6 l  xn;m : m Define mn ¼ ( if mn 6 mn1 ; mn1 ; minfmn1 þ ð2k  1Þ  2n1 : mn1 þ ð2k  1Þ  2n1 P mn g; if mn > mn1 : k2N The sequence yn will be defined in such a way that in the sequence yn1 we replace each element on a position p; p 2 M n ; p > mn , with the corresponding element (of the same index p) from the sequence xn . We proceed with this procedure and obtain a sequence z ¼ ðzm Þm2N , which, by construction, belongs to the class fagl; m and has common elements with each of sequences xn ; n P 1 in a set whose indexes form a subset on N having positive asymptotic density. In fact, for xn ; n > 1, each xn; m from xn with m 2 Mn and m P mn is such a common element. h Let a ¼ ðan Þn2N 2 S and l > 0 be fixed. A sequence b ¼ ðbn Þn2N 2 S l-dominates a if there is n0 2 N such that an 6 l  bn for all n P n0 [6]. Denote by fagl the set of all sequences in S which l-dominate a. The following theorem is shown in a similar way as Theorem 3.1. Theorem 3.2. Let a ¼ ðan Þn2N 2 S and l > 0 be fixed. Then s  a2 ðfagl ; fagl Þ holds. Let A and B be subclasses of S. The symbol GðA; BÞ denotes the infinitely long game for two players, ONE and TWO, who play a round for each positive integer. In the n-th round ONE chooses a sequence sn 2 A, and TWO responds by choosing a S subset T n from sn of positive asymptotic density. TWO wins a play ðs1 ; T 1 ;    ; sn ; T n ;   Þ if n2N T n can be arranged in a sequence from B; otherwise, ONE wins. Evidently, if TWO has a winning strategy in the game GðA; BÞ (or even if ONE does not have a winning strategy in GðA; BÞ), then the selection hypothesis s  a2 ðA; BÞ is true. Let a ¼ ðan Þn2N 2 S be fixed. A sequence b ¼ ðbn Þn2N in S is said to be strongly asymptotically equivalent to a if for every l > 1 the conditions b 2 fagl and a 2 fbgl are satisfied (see [3,6]). This is equivalent to the fact that for every l > 1 there is n0 2 N such that l1  bn 6 an 6 l  bn for all n P n0 , or to the fact limn!1 bann ¼ 1. This relation is an equivalence relation on S. For a fixed a 2 S denote by ½a the set of all sequences from S which are strongly asymptotically equivalent to a. Theorem 3.3. Let a ¼ ðan Þn2N 2 S be given. Then the player TWO has a winning strategy in the game Gð½a; ½aÞ. Proof. Let p1 < p2 <    < pn <    be an increasing sequence of natural numbers and let N¼ [ Mn ; ðM n ¼ 2n1  ð2N  1Þ :¼ f2n1  ð2m  1Þ : m 2 Ng; n 2 NÞ; n2N be a decomposition of N into pairwise disjoint sets (with dðMn Þ ¼ 2n ; n 2 N). The player TWO will use the following strategy. Round 1: Suppose ONE chooses a sequence x1 ¼ ðx1;m Þm2N from ½a. Then TWO picks p1 , finds a position mp1 such that am x1;m  1 6 p1 for m P mp1 , and fix elements x1;m , where m 2 M 1 and m P mp1 (so the set T 1 ¼ fx1;m : m 2 M 1 and 1 m P mp1 ¼ m1 g). Round 2: ONE chooses a sequence x2 ¼ ðx2;m Þm2N from ½a. TWO finds a position mp2 in the sequence x2 such that am x2;m  1 6 p1 for all m P mp2 and puts m2 ¼ maxfm1 ; mp2 g; in the sequence x1 TWO finds now elements x1;m , where 2 m 2 M 2 ; m P m2 and replaces them by elements x2;m with the same index m (so, T 2 ¼ fx2;m : m 2 M 2 and m P m2 g). G. Di Maio et al. / Chaos, Solitons and Fractals 42 (2009) 2815–2821 2819 Round n, n P 3: ONE takes a sequence xn ¼ ðxn;m Þm2N from ½a. TWO first considers a position mpn in the sequence xn such that am xn;m  1 6 p1 for m P mpn and takes mn ¼ maxfmn1 ; mpn g. Now, in the sequence obtained by this procedure in the step n n  1 TWO replaces elements x1;m ; m 2 M n ; m P mn , by elements xn;m of the same indexes m (hence, T n ¼ fxn;m : m 2 M n and m P mn g). This procedure leads to the sequence y ¼ ðym Þm2N , where ym ¼ xn;m , if there are n 2 N with m P mn and m 2 Mn , and ym ¼ x1;m otherwise. The sequence y belongs to S and, by construction, has the intersection in a set of positive asymptotic density with every sequence xn . We prove that y 2 ½a, i.e. that for every e > 0 it holds aym  1 < e for all but finitely many m. Let pk be so large that p1 < e. m k Then for all m > mk we have aym  1 < e. h m Corollary 3.4. Let a 2 S be given. Then s  a2 ð½a; ½aÞ is true. A sequence a ¼ ðan Þn2N 2 S is said to be negligible with respect to a sequence b ¼ ðbn Þn2N from S if for every e > 0, there is n0 ¼ n0 ðeÞ such that an 6 e  bn whenever n P n0 ([6]). Denote by rðaÞ the set of all sequences b in S such that a is negligible with respect to b. The proof of the following theorem is similar to that of Theorem 3.3. Theorem 3.5. Let a ¼ ðan Þn2N 2 S. The player TWO has a winning strategy in the game GðrðaÞ; rðaÞÞ. 4. Results II Definition 4.1. Let A 2 ½0; 1Þ. A sequence b ¼ ðbn Þn2N 2 S statistically converging to A is said to s-converge rapidly to A if the Landau–Hurwicz sequence of b defined by wn ðbÞ ¼ supfjbm  bk j : m P n; k P ng; n 2 N; belongs to de Haan’s class R1;s of rapidly varying sequences of index of variability 1. For A 2 ½0; 1Þ let st  ½AR1;s ¼ fb ¼ ðbn Þn2N 2 S : bs  converges rapidly to Ag: Example 4.2. Let A > 0 be fixed. For each n 2 N define the sequence xn ¼ ðxn; m Þm2N as follows: xn;m ¼  ð1=nÞ  em þ A; if m is not a prime number; 1=n; if m is a prime number: Then each sequence xn belongs to st  ½AR1;s . Theorem 4.3. Let A 2 ½0; 1Þ. Then the player TWO has a winning strategy in the game G1 ðst  ½AR1; s ; st  ½AR1; s Þ. Proof. Assume that in the first round the player ONE chooses a sequence x1 ¼ ðx1; m Þm2N in st  ½AR1; s . w ¼ 0 for every k > 1.] [This means: there is M 1  N with dðM1 Þ ¼ 1 such that limm!1; m2M1 x1; m ¼ A, and limm!1 w½kn n Then TWO takes y1 ¼ x1; m , where m 2 M1 is arbitrary such that x1; m – A. In the second round ONE chooses a sequence x2 ¼ ðx2; m Þm2N in st  ½AR1; s . Let M2 be a statistically dense subset of N witnessing this fact. TWO calculates   1 m2 :¼ min m 2 M2 : jx2;m  Aj 6 jy1  Aj; x2;m – A : 4 and takes y2 ¼ x2; m , where m is (arbitrary) chosen in such a way that m 2 M2 and m P m2 . In the n-th round ONE chooses a sequence xn ¼ ðxn; m Þm2N in st  ½AR1; s . Let M n be a statistically dense subset of N which guaranties this fact. TWO takes yn ¼ xn; m , where m is chosen in such a way that m 2 M n and   1 m P mn :¼ min m 2 Mn : jxn;m  Aj 6 jyn1  Aj; xn;m – A : 4 And so on. In this way during the play x1 ; y1 ; x2 ; y2 ;    ; xn ; yn ;    the player TWO created the sequence y ¼ ðym Þm2N . It is understood that y 2 S; y \ xn ¼ yn for every n 2 N; y (statistically) converges to A and jynþ1  Aj 6 14 jyn  Aj for each n 2 N. h 2820 G. Di Maio et al. / Chaos, Solitons and Fractals 42 (2009) 2815–2821 Claim 1. The sequence ðjyn  AjÞn2N belongs to R1; s . Since (by the construction of y) 1 1 jyn  ynþ1 j 6 jyn  Aj  jyn  Aj 6 wn ðyÞ 6 jyn  Aj þ jyn  Aj; n 2 N; 4 4 we have 3 5 jy  Aj 6 wn ðyÞ 6 jyn  Aj: 4 n 4 This means that for n 2 N it holds wn ðyÞ ¼ hn jyn  Aj; where 34 6 hn 6 54 for n 2 N. Therefore, for each k > 1 we have  ½knn  ðk1Þn1 jy½kn  Aj jy½kn  Aj jy½kn1  Aj jynþ1  Aj 1 1 6 ¼   6 jyn  Aj 4 4 jyn  Aj jy½kn1  Aj jy½kn2  Aj so that limn!1 jy½kn Aj jyn Aj ¼ 0, i.e. ðjyn  AjÞn2N 2 R1;s . Claim 2. ðwn ðyÞÞn2N 2 R1;s . By the above for each k > 1 we have w½kn ðyÞ h½kn jy½kn  Aj 5 jy½kn  Aj  ¼ 6  ; wn ðyÞ 3 jyn  A hn jyn  Aj and thus ðwn ðyÞÞn2N 2 R1;s . So, y 2 st  ½AR1;s , and the theorem is shown. Corollary 4.4. Let A 2 ½0; 1Þ. Then the selection principle S1 ðst  ½AR1; s ; st  ½AR1; s Þ is satisfied. Theorem 4.5. The player TWO has a winning coding strategy in the game G1 ðst  ARVs ; R1; s Þ. Proof. A strategy r for TWO will be defined in the following way. Assume that in the first round ONE plays x1 ¼ ðx1; m Þm2N from st  ARVs . Then TWO responds by choosing rðx1 ; ;Þ ¼ x1; m1  y1 , where x1; m1 is an arbitrary element in x1 . If in the second round ONE has played x2 ¼ ðx2;m Þm2N , then TWO finds x2; m2 2 a2 such that x2; m2 > 2  x1; m1 (which is possible according to Lemma 2.2) and responds by rðx2 ; x1; m1 Þ ¼ x2; m2  y2 . Let in the n-th round ONE play xn ¼ ðxn; m Þm2N ; then TWO chooses xn; mn 2 xn such that xn; mn > 2  xn1; mn1 and plays rðxn ; xn1; mn1 Þ ¼ xn; mn  yn . And so on. We claim that ðyn Þn2N is rapidly varying of index 1. Let k > 1. For n 2 N we have y½kn y½kn y ¼    nþ1 : yn y½kn1 yn Since on the right side of this equality there are ½kn  n factors, and ½kn  n > ðkn  1Þ  n ¼ ðk  1Þn  1, we have y½kn > 2½knn > 2ðk1Þn1 ; n 2 N; yn and therefore lim n!1 y½kn ¼ 1; yn i.e. ðyn Þn2N 2 R1; s . h Corollary 4.6. The selection principle S1 ðst  ARVs ; R1; s Þ is satisfied. In the next theorem we show that, in difference of the previous theorem and its corollaries, for some classes of sequences the player ONE has a winning strategy in the game G1 . Following terminology from [8] we say that a sequence x ¼ ðxn Þn2N 2 S is in the class TrðRVs Þ of translationally regularly varying sequences if for each k 2 R, lim n!1 x½nþk < 1: xn G. Di Maio et al. / Chaos, Solitons and Fractals 42 (2009) 2815–2821 2821 Theorem 4.7. ONE has a winning strategy in the game G1 ðst  ARVs ; TrðRVs ÞÞ. Proof. Let r be a strategy for ONE. Suppose that the first move of ONE is the sequence rð;Þ ¼ x1 ¼ ðx1; m Þm2N from st  ARVs . Let TWO’s response be y1 ¼ x1; m1 2 x1 . Then ONE picks a new sequence x2 ¼ ðx2; m Þm2N 2 st  ARVs such that all its elements are bigger than 2  y1 (again by Lemma 2.2). Let y2 ¼ x2; m2 2 x2 be the TWO’s response. In the n-th round ONE looks at yn1 , TWO’s choice in the ðn  1Þ-th round, and chooses a sequence xn ¼ ðxn; m Þm2N 2 st  ARVs such that all its elements are bigger than 2n  yn1 . And so on. The sequence ðyn Þn2N obtained during the play x1 ; y1 ; x2 ; y2 ;    ; xn ; yn ;    for each k P 1 satisfies lim inf n!1 y½nþk ynþ½k ¼ lim inf ¼ lim inf n!1 n!1 yn yn ynþ½k y    nþ1 ynþ½k1 yn ! P lim inf 2ðnþ1Þ½k ¼ 1; n!1 i.e. for k P 1 we have limn!1 y½nþk yn ¼ 1. Therefore, ðyn Þn2N R TrðRVs Þ. h Corollary 4.8 [8]. ONE has a winning strategy in the game G1 ðARVs ; TrðRVs ÞÞ. 5. Conclusion We apply the idea of statistical convergence to obtain various properties for classes of sequences of positive real numbers and to introduce a kind of convergence, that we call statistical rapid convergence. We hope that further investigation in this direction and further development of the idea of statistical rapid convergence may have nice applications in different fields of applied mathematics and physics: non-linear dynamical systems, control of chaos, quantum mechanics. Acknowledgements The paper has its roots in the third author stay in Caserta during September 2008 as a visiting professor. 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