I'm looking for a sequence of identically distributed random variables $(X_n)_n$ with $\text{var} (X_i) > 0$ so that the law of large numbers (either weak or strong) and the central limit theorem do not apply.
I propose: $\mathbb{P} [X_1 = 0] = \mathbb{P} [X_1 = 1] = \frac{1}{2}$; $X_i = X_1$ for $i > 1$. Thus, $(X_n)_n$ are identically distributed (in fact they are identical) but not independent. $\mu = \mathbb{E} [X_1] = \frac{1}{2}$, $\sigma^2 = \text{var} (X_1) = \frac{1}{4}$.
The weak law (and thus the strong law) doesn't hold for $(X_n)_n$ because $\mathbb{P} [|\frac{1}{n} \sum_{i = 1}^n X_i - \mu| > \varepsilon] = \mathbb{P} [|X_1 - \frac{1}{2}| > \varepsilon]$ for $\varepsilon < \frac{1}{2}$. ($\mathbb{P} [|X_1| = 1] = \mathbb{P} [X_1 = 1] = \frac{1}{2}$.)
The central limit theorem doesn't hold because, for $Z_n = \frac{\sum_{i = 1}^n X_i - n \mu}{\sigma \sqrt{n}} = \sqrt{n} (X_1 - 1)$, $\mathbb{P} [Z_n \le 1] = \mathbb{P} [X_1 \le 1 + \frac{1}{\sqrt{n}}] \underset{n \rightarrow \infty}{\longrightarrow} \mathbb{P} [X_1 \le 1] = 1 \ne \Phi (1)$.
I would appreciate any comment on whether this example is right. Is there any more straightforward example?