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A measure theoretical subsequence characterization of statistical convergence. (English) Zbl 0830.40002
Let $$T= (a_{mn})$$ be a triangular matrix with $$a_{mn}>0$$ for $$n\leq m$$, $$\sum_n a_{mn}= 1$$ $$(m\in \mathbb{N})$$ and $$\lim_m a_{mn}= 0$$ $$(n\in \mathbb{N})$$. A sequence of reals $$S= \{s_n\}$$ is said to be statistically $$T$$-summable to $$L$$ (briefly $$s_n\to L$$ (stat $$T$$)) if $$\sum_n [a_{mn}$$: $$|s_n- L|\geq \varepsilon ]\to 0$$ $$(m\to \infty)$$ for every $$\varepsilon>0$$. A subset $$A\subset \mathbb{N}$$ is said to have $$T$$-density zero if $$\lim_m \sum_{n\in A} a_{mn} =0$$.
For a sequence $$S$$ let $$S(x)$$ denote the subsequence of $$S$$ with $$s_n\in S(x) \iff e_n (x) =1$$ where $$\sum_n e_n (x) 2^{-n}$$ is the binary expansion of $$x\in (0, 1]$$ with $$e_n (x)\in \{0, 1\}$$.
The main result of the paper is that $$s_n\to L$$ (stat $$T$$) if and only if there exists a subset $$A= \{k_n\}$$ of $$\mathbb{N}$$ having $$T$$-density zero such that $$m_A (\{x\in (0, 1]$$: $$\lim_n (S(x ))_n= L\})=1$$. Here $$m_A$$ is the unique probability measure defined on the Borel subsets of $$(0,1]$$ having the following property: $m_A (\{x\in (0,1]:\;e_j (x)= 1\})= \begin{cases} {1\over 2} &\text{if } j\not\in A\\ {1\over 2^n} &\text{if } j= k_n \end{cases} ,$ and $$\{e_j (x)\}$$ is a sequence of independent random variables with respect to $$m_A$$.
Reviewer: T.Leiger (Tartu)

##### MSC:
 40A05 Convergence and divergence of series and sequences 40D25 Inclusion and equivalence theorems in summability theory 28A12 Contents, measures, outer measures, capacities 40G99 Special methods of summability
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##### References:
  Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. · Zbl 0184.43301  Patrick Billingsley, Probability and measure, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Wiley Series in Probability and Mathematical Statistics. · Zbl 0411.60001  R. Creighton Buck, Generalized asymptotic density, Amer. J. Math. 75 (1953), 335 – 346. · Zbl 0050.05901 · doi:10.2307/2372456 · doi.org  Yuan Shih Chow and Henry Teicher, Probability theory, Springer-Verlag, New York-Heidelberg, 1978. Independence, interchangeability, martingales. · Zbl 0399.60001  J. S. Connor, The statistical and strong \?-Cesàro convergence of sequences, Analysis 8 (1988), no. 1-2, 47 – 63. · Zbl 0653.40001  Jeff Connor, On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull. 32 (1989), no. 2, 194 – 198. · Zbl 0693.40007 · doi:10.4153/CMB-1989-029-3 · doi.org  Jeff Connor, Two valued measures and summability, Analysis 10 (1990), no. 4, 373 – 385. · Zbl 0726.40009 · doi:10.1524/anly.1990.10.4.373 · doi.org  Richard G. Cooke, Infinite matrices and sequence spaces, Dover Publications, Inc., New York, 1965. · Zbl 0132.28901  H. Fast, Sur la convergence statistique, Colloquium Math. 2 (1951), 241 – 244 (1952) (French). · Zbl 0044.33605  J. J. Sember and A. R. Freedman, On summing sequences of 0’s and 1’s, Rocky Mountain J. Math. 11 (1981), no. 3, 419 – 425. · Zbl 0496.40008 · doi:10.1216/RMJ-1981-11-3-419 · doi.org  A. R. Freedman and J. J. Sember, Densities and summability, Pacific J. Math. 95 (1981), no. 2, 293 – 305. · Zbl 0504.40002  J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301 – 313. · Zbl 0588.40001 · doi:10.1524/anly.1985.5.4.301 · doi.org  J. A. Fridy and H. I. Miller, A matrix characterization of statistical convergence, Analysis 11 (1991), no. 1, 59 – 66. · Zbl 0727.40001 · doi:10.1524/anly.1991.11.1.59 · doi.org  J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993), no. 1, 43 – 51. · Zbl 0794.60012  -, Lacunary statistical summability (to appear). · Zbl 0786.40004  I. J. Schoenberg, The integrability of certain functions and related summability methods., Amer. Math. Monthly 66 (1959), 361 – 375. · Zbl 0089.04002 · doi:10.2307/2308747 · doi.org
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