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Statistically pre-Cauchy sequences and Orlicz functions. (English) Zbl 1150.40006

Summary: Let \(x= (x_i)\) be a sequence and let \(M\) be a bounded Orlicz function. We prove that \(x\) is statistically pre-Cauchy if and only if \[ \lim_n\frac 1{n^2}\sum_{j, i\leq n} M ( \frac{|x_i-x_j|}{\rho})=0. \] This implies a theorem due to J. Connor, J. Fridy and J. Kline [Analysis 14, No.4, 311–317 (1994; Zbl 0810.40001)].

MSC:

40A30 Convergence and divergence of series and sequences of functions

Citations:

Zbl 0810.40001
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