I’m trying to construct a sequence of independent random variables $X_1, X_2, \ldots$ with $\mathbb E[|X_n|] = \infty$ for every $n$, and for which we have $S_n^* := \frac{X_1 + \cdots + X_n}{\sqrt n}$ converges in distribution to the standard normal distribution with density $e^{-x^2/2}/\sqrt{2\pi}$.
The proof of the Central Limit Theorem with which I’m most familiar involves taking the characteristic functions $\varphi_{S_n^* }(t)$ of $S_n^* $, showing they converge pointwise to $e^{-t^2/2}$ as $n \to \infty$, and using Lévy’s Continuity Theorem to show that the distributions of $S_n^* $ converge weakly to the standard normal distribution $\mathcal N_{0,1}$. The only way I know to do this involves using the Taylor expansion of $\varphi_{S_n^* }$, which requires that $\mathbb E[|X_n|] <\infty$ for all $n$. (More specifically, the Taylor expansion requires that $\varphi_{S_n^* }(t)$ be at least twice differentiable, which is equivalent to $\mathbb E[|S_n^* |^2]<\infty$, which we don’t have if $\mathbb E[|X_n|] = \infty$ for all $n$).
The other ideas I’ve tried have involved either (a) taking a distribution of the form $\mathbb P_{X_n} = \sum_{k \in \mathbb Z} p_{n,k} \delta_k^* $, where $p_{n,k} = \mathbb P[X_n = k]$ and $\delta_k$ is the Dirac mass at $k \in \mathbb Z$, or (b) taking $X_n$ with a continuous density $f_n$ with respect to Lebesgue measure. In both cases, we want $\mathbb E[|X_n|] = \infty$. We know $\varphi_{S_n^* }(t) = \prod_{i = 1}^n \varphi_{X_i}(t/\sqrt{n})$ by independence. In the discrete case (a),
$$
\varphi_{X_i}(t) = \sum_{k \in \mathbb Z} p_{i,k} \cos(kt),
$$
and in the continuous case (b),
$$
\varphi_{X_i}(t) = \int_{\mathbb R} \cos(xt) f_i(x) dx.
$$
In both cases I’m not sure how to find a nice expression for $\prod_i\varphi_{X_i}(t/\sqrt{n})$, and using a Taylor approximation for $\varphi_{X_i}$ is essentially a non-starter if $\mathbb E[|X_i|] = \infty$ for all $i$.
Any suggestions for how I might proceed?
Best Answer
The following is an example of sequence $ \{X_n,n\ge 1 \} $ of independent random variables with $ \mathsf{E}[|X_n|]=\infty $ and $ S_n^\ast := \frac{X_1+\cdots+X_n}{\sqrt{n}} \overset{d}{\to} N(0,1) $.
Suppose $ \{\xi_n,n\ge1\} $ is a sequence of i.i.d. random variables and $ \xi_n\overset{d}{=} U(-1,1) $. Denote \begin{align*} X_n&=\sqrt{3}\xi_n + \frac{1_{\{|\xi_n|\le n^{-2}\}}}{|\xi_n|}. \\ &=\sqrt{3}\xi_n +Z_n. \tag{1} \end{align*} Then \begin{gather*} \mathsf{E}[|Z_n|]=\infty, \quad \mathsf{E}[|X_n|]=\infty,\\ \frac{1}{\sqrt{n}}\sum_{j=1}^n\sqrt{3}\xi_j\overset{d}{\to} N(0,1). \tag{2} \end{gather*} Due to $ \sum\limits_{n\ge 1}\mathsf{P}(Z_n\ne0)<\infty $, \begin{gather*} \sum_{n=1}^{\infty}Z_n <\infty,\qquad \text{a.s.}.\\ \frac{1}{\sqrt{n}}\sum_{j=1}^n Z_j\to 0, \qquad \text{a.s.}.\tag{3} \end{gather*} From (1)-(3) get \begin{equation*} \frac{1}{\sqrt{n}}\sum_{j=1}^n X_j \overset{d}{\to} N(0,1). \end{equation*}