Let X={Xn,m}n≥1,1≤m≤n be a D5163: Random real triangular array such that
(i) | Xn,1,…,Xn,n is an D2713: Independent random collection for each n∈1,2,3,… |
(ii) | λ1,λ2,λ3,…∈(0,∞) are each a D993: Real number |
(iii) | lim |
(iv) | \begin{equation} \lim_{n \to \infty} \sum_{m = 1}^n \mathbb{P}(|X_{n, m}| > \lambda_n) = 0 \end{equation} |
(v) | \begin{equation} \lim_{n \to \infty} \frac{1}{\lambda^2_n} \sum_{m = 1}^n \mathbb{E} (|X_{n, m}|^2 I_{\{ |X_{n, m}| \leq \lambda_n \}}) = 0 \end{equation} |
Then
\begin{equation}
\sum_{m = 1}^n \frac{X_{n, m} - \mathbb{E} (X_{n, m} I_{\{ |X_{n, m}| \leq \lambda_n \}})}{\lambda_n}
\overset{p}{\longrightarrow} 0
\quad \text{ as } \quad
n \to \infty
\end{equation}