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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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[1] Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. · Zbl 0184.43301
[2] Patrick Billingsley, Probability and measure, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Wiley Series in Probability and Mathematical Statistics. · Zbl 0411.60001
[3] R. Creighton Buck, Generalized asymptotic density, Amer. J. Math. 75 (1953), 335 – 346. · Zbl 0050.05901 · doi:10.2307/2372456 · doi.org
[4] Yuan Shih Chow and Henry Teicher, Probability theory, Springer-Verlag, New York-Heidelberg, 1978. Independence, interchangeability, martingales. · Zbl 0399.60001
[5] J. S. Connor, The statistical and strong \?-CesĂ ro convergence of sequences, Analysis 8 (1988), no. 1-2, 47 – 63. · Zbl 0653.40001
[6] 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
[7] 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
[8] Richard G. Cooke, Infinite matrices and sequence spaces, Dover Publications, Inc., New York, 1965. · Zbl 0132.28901
[9] H. Fast, Sur la convergence statistique, Colloquium Math. 2 (1951), 241 – 244 (1952) (French). · Zbl 0044.33605
[10] 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
[11] A. R. Freedman and J. J. Sember, Densities and summability, Pacific J. Math. 95 (1981), no. 2, 293 – 305. · Zbl 0504.40002
[12] 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
[13] 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
[14] J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993), no. 1, 43 – 51. · Zbl 0794.60012
[15] -, Lacunary statistical summability (to appear). · Zbl 0786.40004
[16] 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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