On statistically convergent sequences of fuzzy numbers (Q5970182)

Let \(\{x_n\}\) be a sequence of fuzzy numbers. The author proves that the following are equivalent: (a) \(x\) is statistically convergent to \(x_0\). (b) There exist sequences \(y\) and \(z\) of fuzzy numbers such that \(x = y + z\), \(\lim_kd(y_k, x_0) = 0\) and \(\delta\)(supp \(z) = 0\). (c) There exists a subsequence \(k = \{k_n\}\) of \(\mathbb{N}\) such that \(\delta(k) = 1\) and \(\lim_n d(x_{k_n}, x_0) = 0\). He also establishes the equivalence of the following: (a) \(x\) is a statistically Cauchy sequence of fuzzy numbers. (b) There exists a subsequence \(k = \{k_n\}\) of \(\mathbb{N}\) such that \(\delta(k) = 1\) and \(\lim_{m, n}d(x_{k_m}, x_{k_n}) = 0\). (c) There exist sequences \(y\) and \(z\) of fuzzy numbers such that \(x = y + z\), \(\lim_{m, n}d(y_m, y_n) = 0\), and \(\delta(\)supp \(z) = 0\).