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Keywords:
compact sets; natural density; statistically bounded sequence; statistical cluster point
Summary:
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.
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