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Lacunary statistical convergence and inclusion properties between lacunary methods. (English) Zbl 0952.40001

The integer sequence θ={k r } is called a lacunary sequence if it is increasing and lim r (k r -k r-1 )=. A complex number sequence x={x k } is said to be s θ -convergent to L if for each ε>0 one has

lim r 1 k r -k r-1 #{k:k r-1 <kk r and|x k -L|ε}=0·

Let S θ be the family of all sequences x which are s θ -convergent to some L. In this paper, which continues the work of J. A. Fridy and C. Orhan [Pac. J. Math. 160, No. 1, 43-51 (1993; Zbl 0794.60012)], the author studies inclusion properties between S θ and S β , where θ and β are two arbitrary lacunary sequences.

40A05Convergence and divergence of series and sequences
40D05General summability theorems
40C05Matrix methods in summability
11B05Topology etc. of sets of numbers