Strongly \(\lambda \)-statistically and strongly Vallée-Poussin pre-Cauchy sequences in probabilistic metric spaces. (English) Zbl 1509.54009
Let \(\lambda\) be a non-decreasing sequence of positive real numbers going to \(\infty\) with \(\lambda_1=1\) and \(\lambda_{n+1}\le\lambda_n+1\). A sequence \((x_n)\) in a PM space \((P,\mathcal{F},\tau)\) [B. Schweizer and A. Sklar, Probabilistic metric spaces. New York-Amsterdam-Oxford: North-Holland (1983; Zbl 0546.60010)] is said to be strongly \(\lambda\)-statistically convergent to \(l\in P\) if, for every \(t>0\),
\[
\lim_{n\to\infty}\frac{1}{\lambda_n} |\{k\in [n-\lambda_n+1,n]:F_{x_kl}(t)\le 1-t\}|=0\,.
\]
An analogous definition of strongly \(\lambda\)-statistical pre-Cauchy sequence is given.
The sequence \((x_k)\) is said to be de la Vallée-Poussin pre-Cauchy if for every \(\varepsilon>0\) there is \(n_0\) such that for every \(n\ge n_0\), \[ \frac{1}{\lambda_n^2}\,\sum_{j,k\in [n-\lambda_n+1,n]} d_L(F_{x_j,x_k},\varepsilon_0)\le\varepsilon. \] Here \(d_L\) is the modified Lévy distance. The relationships between the various concepts are studied.
(Throughout the paper the names de la Vallée-Poussin and Cesàro are misspelled).
The sequence \((x_k)\) is said to be de la Vallée-Poussin pre-Cauchy if for every \(\varepsilon>0\) there is \(n_0\) such that for every \(n\ge n_0\), \[ \frac{1}{\lambda_n^2}\,\sum_{j,k\in [n-\lambda_n+1,n]} d_L(F_{x_j,x_k},\varepsilon_0)\le\varepsilon. \] Here \(d_L\) is the modified Lévy distance. The relationships between the various concepts are studied.
(Throughout the paper the names de la Vallée-Poussin and Cesàro are misspelled).
Reviewer: Carlo Sempi (Lecce)
Keywords:
probabilistic metric space strongly; \(\lambda \)-statistical convergence; strongly \(\lambda \)-statistical pre-Cauchy; strongly statistical pre-Cauchy; strongly Valle-Poussin pre-CauchyCitations:
Zbl 0546.60010References:
| [1] | J. Connor, J. Fridy and J. Kline, Statistically pre-Cauchy Sequences, Analysis, 14 (1994), 311-317. · Zbl 0810.40001 |
| [2] | J. Connor, J. 1992. R-type summability methods, Cauchy criteria, P-sets and Statistical convergence, · Zbl 0765.40002 |
| [3] | Proc. Amer. Math. Soc., 115 (1992), 319-327. · Zbl 0765.40002 |
| [4] | J. Connor, The statistical and strong P-Cesaro convergence of sequences, Analysis, 8 (1988), 47-63. · Zbl 0653.40001 |
| [5] | H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244. · Zbl 0044.33605 |
| [6] | J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313. · Zbl 0588.40001 |
| [7] | P. Malik and S. Das, On strong λ-statistical convergence of sequences in probabilistic metric (pm) spaces, arXiv preprint arXiv:2007.09173, 2020. |
| [8] | K. Menger, Statistical metrics, Proc. Nat. Acad. Sci., 28 (1942), 535-537. · Zbl 0063.03886 |
| [9] | M. Mursaleen, λ-statistical convergence, Mathematica Slovaca, 50 (2000), no. 1, 111-115. · Zbl 0953.40002 |
| [10] | T. S´alat, On statistically convergent sequences of real numbers, Mathematica Slovaca, 30 (1980), 139-150. · Zbl 0437.40003 |
| [11] | I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375. · Zbl 0089.04002 |
| [12] | C. S¸en¸cimen and S. Pehlivan, Strong statistical convergence in probabilistic metric space, Stoch. Anal. Appl., 26 (2008), 651-664. · Zbl 1147.54015 |
| [13] | B. Schweizer, and A. Sklar, Statistical metric spaces, Pacific J. Math., 10 (1960), 314-334. · Zbl 0091.29801 |
| [14] | B. Schweizer, and A. Sklar and E. Thorp, The metrization of statistical metric spaces, Pacific J. Math., 10 (1960), 673-675. · Zbl 0096.33203 |
| [15] | B. Schweizer, and A. Sklar, Probabilistic Metric Spaces, North Holland: New York, Amsterdam, Oxford, 1983. · Zbl 0546.60010 |
| [16] | R. M. Tardiff, Topologies for Probabilistic Metric spaces, Pacific J. Math., 65 (1976), 233-251. · Zbl 0337.54004 |
| [17] | E. Thorp, Generalized topologies for statistical metric spaces, Fundamenta Mathematicae, 51 (1962), 9-21. · Zbl 0105.11501 |
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