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ThmDex – An index of mathematical definitions, results, and conjectures.
Weak law of large numbers for random real triangular arrays
Formulation 0
Let X={Xn,m}n1,1mn be a D5163: Random real triangular array such that
(i) Xn,1,,Xn,n is an D2713: Independent random collection for each n1,2,3,
(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}
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Comment 0
Hypothesis (iv) can also be written in the form \begin{equation} \lim_{n \to \infty} \sum_{m = 1}^n \mathbb{P}(X_{n, m} \neq X_{n, m} I_{\{ |X_{n, m}| \leq \lambda_n \}}) = 0 \end{equation}
Formulation 1
Let X = \{ X_{n, m} \}_{n \geq 1, \, 1 \leq m \leq n} be a D5163: Random real triangular array such that
(i) X_{n, 1}, \ldots, X_{n, n} is an D2713: Independent random collection for each n \in 1, 2, 3, \ldots
(ii) \lambda_1, \lambda_2, \lambda_3, \ldots \in (0, \infty) are each a D993: Real number
(iii) \begin{equation} \lim_{n \to \infty} \lambda_n = \infty \end{equation}
(iv) Z_{n, m} : = X_{n, m} I_{\{|X_{n, m}| \leq \lambda_n\}}
(v) \begin{equation} \lim_{n \to \infty} \sum_{m = 1}^n \mathbb{P}(X_{n, m} \neq Z_{n, m}) = 0 \end{equation}
(vi) \begin{equation} \lim_{n \to \infty} \frac{1}{\lambda^2_n} \sum_{m = 1}^n \mathbb{E} Z^2_{n, m} = 0 \end{equation}
Then \begin{equation} \sum_{m = 1}^n \frac{X_{n, m} - \mathbb{E} Z_{n, m}}{\lambda_n} \overset{p}{\longrightarrow} 0 \quad \text{ as } \quad n \to \infty \end{equation}
Also known as
WLLN for random real triangular arrays
Proofs
Proof 0
Let X = \{ X_{n, m} \}_{n \geq 1, \, 1 \leq m \leq n} be a D5163: Random real triangular array such that
(i) X_{n, 1}, \ldots, X_{n, n} is an D2713: Independent random collection for each n \in 1, 2, 3, \ldots
(ii) \lambda_1, \lambda_2, \lambda_3, \ldots \in (0, \infty) are each a D993: Real number
(iii) \begin{equation} \lim_{n \to \infty} \lambda_n = \infty \end{equation}
(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}
Fix \varepsilon > 0 and denote Z_{n, m} : = X_{n, m} I_{\{ |X_{n, m}| \leq \lambda_n \}}, S^X_n : = \sum_{m = 1}^n X_{n, m}, S^Z_n : = \sum_{m = 1}^n Z_{n, m}, and \mu_n : = \sum_{m = 1}^n \mathbb{E} Z_{n, m}. From result R4737: , we have the upper bound \begin{equation} \mathbb{P} \left( \left| \frac{S^X_n - \mu_n}{\lambda_n} \right| > \varepsilon \right) \leq \mathbb{P}(S^X_n \neq S^Z_n) + \mathbb{P} \left( \left| \frac{S^Z_n - \mu_n}{\lambda_n} \right| > \varepsilon \right) \end{equation} Notice that, by construction, we have \{ X_{n, m} \neq Z_{n, m} \} = \{ |X_{n, m}| > \lambda_n \} for all n \in 1, 2, 3, \ldots. Applying results
(i) R4740:
(ii) R4738: Finite subadditivity of probability measure

as well as hypothesis (iv), we can estimate the first term on the right-hand side by \begin{equation} \begin{split} \mathbb{P}(S^X_n \neq S^Z_n) \leq \mathbb{P} \left( \bigcup_{m = 1}^n \{ X_{n, m} \neq Z_{n, m} \} \right) & \leq \mathbb{P} \left( \bigcup_{m = 1}^n \{ |X_{n, m}| > \lambda_n \} \right) \\ & \leq \sum_{m = 1}^n \mathbb{P}(|X_{n, m}| > \lambda_n) \\ & \to 0 \end{split} \end{equation} as n \to \infty. Next, applying results
(i) R4741: Probabilistic Chebyshov's inequality for square function
(ii) R4687: Additivity of variance for a finite number of independent random real numbers
(iii) R4688: Second moment upper bound to real variance

as well as hypothesis (v), we have \begin{equation} \begin{split} \mathbb{P} \left( \left| \frac{S^Z_n - \mu_n}{\lambda_n} \right| > \varepsilon \right) & \leq \frac{1}{\varepsilon^2} \mathbb{E} \left| \frac{S^Z_n - \mu_n}{\lambda_n} \right|^2 \\ & = \frac{1}{\varepsilon^2 \lambda^2_n} \mathsf{Var}(S^Z_n) \\ & = \frac{1}{\varepsilon^2 \lambda^2_n} \sum_{m = 1}^n \mathsf{Var}(Z_{n, m}) \leq \frac{1}{\varepsilon^2 \lambda^2_n} \sum_{m = 1}^n \mathbb{E} Z^2_{n, m} \to 0 \end{split} \end{equation} as n \to \infty. Combining these results, we find that \begin{equation} \begin{split} \mathbb{P} \left( \left| \sum_{m = 1}^n \frac{X_{n, m} - \mathbb{E} (X_{n, m} I_{\{ |X_{n, m}| \leq \lambda_n \}})}{\lambda_n} \right| > \varepsilon \right) & = \mathbb{P} \left( \left| \sum_{m = 1}^n \frac{X_{n, m} - \mathbb{E} Z_{n, m}}{\lambda_n} \right| > \varepsilon \right) \\ & = \mathbb{P} \left( \left| \frac{\sum_{m = 1}^n X_{n, m} - \sum_{m = 1}^n \mathbb{E} Z_{n, m}}{\lambda_n} \right| > \varepsilon \right) \\ & = \mathbb{P} \left( \left| \frac{S^X_n - \mu_n}{\lambda_n} \right| > \varepsilon \right) \\ & \leq \mathbb{P}(S^X_n \neq S^Z_n) + \mathbb{P} \left( \left| \frac{S^Z_n - \mu_n}{\lambda_n} \right| > \varepsilon \right) \\ & \to 0 \end{split} \end{equation} as n \to \infty. \square