Example for which the CLT holds but the LLN doesn`t I am currently thinking about the relationship between the law of large numbers and the central limit theorem and I was wondering whether someone can give me an example of a familiy of random variables $(X_i): (\Omega, \mathscr A, P) \to (\Omega_i, \mathscr A_i) $ such that the central limit theorem holds but the law of large numbers does not. 
EDIT: I (believe) I have proved that CLT under these conditions implies WLLN. So i am only interested in the SLLN anymore. 
 A: Assume a sequence of random variables (independent or not) $X_1, \dotsc, X_n, \dotsc$ with $\DeclareMathOperator{\E}{\mathbb{E}}\DeclareMathOperator{\Var}{\mathbb{V}} \E X_i =\mu$, and which satisfies the conditions for some central limit theorem (CLT) such that
$$
    c_n (\bar{X}_n -\mu) \stackrel{\mathbb{D}}{\to} \mathcal{N}(0, 1)
$$
for some sequence of constants $c_n \to \infty$. For the usual IID case we have $c_n = \sqrt{n}/\sigma$. Then we can show convergence in probability for $\bar{X}_n$ to $\mu$. 
So for a counterexample (if it exists) you will have to look for a case where $c_n$ do not grow over any bounds.  
And, such examples exist, even if maybe artificial. Less artificial examples would be interesting. Generalize the situation above by replacing $\bar{X}_n$ by $\bar{X}_{wn}$, some weighted mean of the first $n$ variables in the sequence. Then assume that $X_i \sim \mathcal{N}(0,i^2)$, so the variances is increasing fast. Use the usual optimal weighted mean with weights $w_i=i^{-2}$. Then $\mathbb{V} \bar{X}_{wn}=(\sum_{i=1}^n i^{-2})^{-1}$ so we can choose $c_n=(\sum_{i=1}^n i^{-2})^{1/2}$ which do not grow to infinity. So the law of large numbers do not hold, since 
$$
\sum_{i=1}^n \frac1{i^2}=\pi^2/6.
$$
