Correction to: On the concept of B-statistical uniform integrability of weighted sums of random variables and the law of large numbers with mean convergence in the statistical sense

1 Correction to: TEST 30, 83–102 (2021) https://doi.org/10.1007/s11749-020-00706-2

Professor Lê Văn Thành (Vinh University, Nghe An, Viet Nam) has so kindly called to the authors’ attention an error in their original article. On line 6 of page 89, the authors incorrectly asserted in the proof of (i) in the necessity half of Theorem 1 that

$$\begin{aligned} \left\{ n \in \mathbb {N}: \sum _{k=1}^{\infty } |a_{nk}| E|X_k|I_{\{|X_{k}|\le a\}} >a \right\} = \varnothing . \end{aligned}$$

The argument for establishing (i) is corrected as follows.

By (2), we can choose M such that

Note that with $a>0$ as in (4),

Combining (*), (4), and (**), we obtain that

$$\begin{aligned} 0&\le \delta _B \left( \left\{ n \in \mathbb {N}: \sum _{k=1}^{\infty } |a_{nk}| E|X_k|> Ma + \frac{\varepsilon }{2} \right\} \right) \\&\le \delta _B \left( \left\{ n \in \mathbb {N}: \sum _{k=1}^{\infty } |a_{nk}|> M \right\} \right) \\&\quad + \delta _B \left( \left\{ n \in \mathbb {N}: \sum _{k=1}^{\infty } |a_{nk}| E|X_k| I_{\{|X_k|> a\}} > \frac{\varepsilon }{2} \right\} \right) \\&= 0 + 0 = 0 \end{aligned}$$

and so

$$\begin{aligned} \delta _B \left( \left\{ n \in \mathbb {N}: \sum _{k=1}^{\infty } |a_{nk}| E|X_k| > Ma + \frac{\varepsilon }{2} \right\} \right) =0. \end{aligned}$$

Thus, the real number $Ma+\frac{\varepsilon }{2}$ is a B-statistical upper bound of the sequence $\left\{ \sum _{k=1}^{\infty } |a_{nk}| E|X_k|: n \in \mathbb {N} \right\}$. Hence (i) holds.