Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a
D3161: Random real number such that
| (i) |
$X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection
|
| (ii) |
\begin{equation}
\lim_{x \to \infty} x \mathbb{P}(|X_1| > x)
= 0
\end{equation}
|
| (iii) |
\begin{equation}
\mu_N
: = \mathbb{E}(X_1 I_{\{ |X_1| \leq N \}})
\end{equation}
|
The idea of the proof is to apply
R3260: Weak law of large numbers for random real triangular arrays for a triangular array
\begin{equation}
\begin{split}
& X_1 \\
& X_1 \quad X_2 \\
& X_1 \quad X_2 \quad X_3 \\
& \quad \vdots
\end{split}
\end{equation}
where, in each column, the variables are the one and the same variable and we have $\lambda_N : = N$. To this end, we need to confirm that the assumptions of the theorem in this special case hold. These are
\begin{equation}
\lim_{N \to \infty} \sum_{n = 1}^N \mathbb{P}(|X_n| > N)
= 0
\end{equation}
and
\begin{equation}
\lim_{N \to \infty} \frac{1}{N^2} \sum_{n = 1}^N \mathbb{E}(|X_n|^2 I_{\{ |X_n| \leq N \}})
= \lim_{N \to \infty} \frac{1}{N} \mathbb{E}(|X_1|^2 I_{\{ |X_1| \leq N \}})
= 0
\end{equation}
Since $X_1, X_2, X_3, \ldots$ are identically distributed, hypothesis (ii) implies
\begin{equation}
\sum_{n = 1}^N \mathbb{P}(|X_n| > N)
= \sum_{n = 1}^N \mathbb{P}(|X_1| > N)
= N \mathbb{P}(|X_1| > N)
\to 0
\end{equation}
as $N \to \infty$. This takes care of the first assumption. Denote from now on $Z_n : = X_n I_{\{ |X_n| \leq N \}}$. Result
R4742: Probability calculus expression for tail probability of truncated random unsigned basic real number shows that we have
\begin{equation}
\mathbb{P}(|Z_N| > t)
= \mathbb{P}(|Z_1| > t)
= \left( \mathbb{P}(|X_1| > t) - \mathbb{P}(|X_1| > N) \right) I_{[0, N]} (t)
\end{equation}
Thus, applying result
R4689: Probabilistic Cavalieri principle, we have
\begin{equation}
\begin{split}
\mathbb{E} |Z_N|^2
= \mathbb{E} |Z_1|^2
= \int^{\infty}_0 \mathbb{P}(|Z_1| > t) 2 t \, d t
& = \int^{\infty}_0 \left( \mathbb{P}(|X_1| > t) - \mathbb{P}(|X_1| > N) \right) I_{[0, N]} (t) 2 t \, d t \\
& \leq \int^N_0 2 t \mathbb{P}(|X_1| > t) \, d t \\
\end{split}
\end{equation}
Consider next the auxiliary function $\varphi(t) : = 2 t \mathbb{P}(|X_1| > t)$. Since $\varphi(t) \to 0$ as $t \to \infty$ and since $\varphi(t) \in [0, 2 t]$, then $\sup_{t \in \mathbb{R}} \varphi(t) < \infty$. Using result
R4743: Riemann integral average of vanishing unsigned basic real function vanishes, we thus have
\begin{equation}
\begin{split}
\frac{1}{N} \mathbb{E} |Z_N|^2
\leq \frac{1}{N} \int^N_0 2 t \mathbb{P}(|X_1| > t) \, d t
= \frac{1}{N} \int^N_0 \varphi(t) \, d t
\longrightarrow 0
\end{split}
\end{equation}
as $N \to \infty$. This completes the proof. $\square$
