Previous |  Up |  Next


compact sets; natural density; statistically bounded sequence; statistical cluster point
In this paper we study the set of statistical cluster points of sequences in $m$-dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in $m$-dimensional spaces too. We also define a notion of $\Gamma $-statistical convergence. A sequence $x$ is $\Gamma $-statistically convergent to a set $C$ if $C$ is a minimal closed set such that for every $\epsilon > 0 $ the set $ \lbrace k\:\rho (C, x_k ) \ge \epsilon \rbrace $ has density zero. It is shown that every statistically bounded sequence is $\Gamma $-statistically convergent. Moreover if a sequence is $\Gamma $-statistically convergent then the limit set is a set of statistical cluster points.
[1] J. S. Connor: The statistical and strong $p$-Cesàro convergence of sequences. Analysis 8 (1988), 47–63. MR 0954458 | Zbl 0653.40001
[2] H.  Fast: Sur la convergence statistique. Collog. Math. 2 (1951), 241–244. MR 0048548 | Zbl 0044.33605
[3] J. A. Fridy: On statistical convergence. Analysis 5 (1985), 301–313. MR 0816582 | Zbl 0588.40001
[4] J. A. Fridy: Statistical limit points. Proc. Amer. Math. Soc. 118 (1993), 1187–1192. DOI 10.1090/S0002-9939-1993-1181163-6 | MR 1181163 | Zbl 0776.40001
[5] J. A. Fridy and C.  Orhan: Statistical limit superior and limit inferior. Proc. Amer. Math. Soc. 125 (1997), 3625–3631. DOI 10.1090/S0002-9939-97-04000-8 | MR 1416085
[6] E.  Kolk: The statistical convergence in Banach spaces. Acta Comm. Univ. Tartuensis 928 (1991), 41–52. MR 1150232
[7] S.  Pehlivan and M. A. Mamedov: Statistical cluster points and turnpike. Optimization 48 (2000), 93–106. DOI 10.1080/02331930008844495 | MR 1772096
[8] M. A.  Mamedov and S. Pehlivan: Statistical cluster points and Turnpike theorem in nonconvex problems. J.  Math. Anal. Appl. 256 (2001), 686–693. DOI 10.1006/jmaa.2000.7061 | MR 1821765
[9] T. Šalát: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139–150. MR 0587239
Partner of
EuDML logo