I have the folllowing succesion of Random Variables defined as:
$$\{X_n\}_{n =1}^\infty \hspace{.3cm} \text{ where } \hspace{.3cm} \mathbb{P} \left(X_n = \sqrt{\ln(n + \alpha)} \right) = \mathbb{P} \left(X_n = -\sqrt{\ln(n + \alpha)} \right) =\frac{1}{2}$$
Does the weak law of large numbers hold for this R.V?, because i think not.
I started with this aproach but get lost in the middle:
$$P \left( \left| \frac{1}{n}\sum_{j=1}^n(X_j-E(X_j) \right| > \varepsilon \right) =\frac{\operatorname{Var}[\frac{1}{n} \sum_{j=1}^n(X_j-E[X_j])]}{\varepsilon^2}$$
Any ideas or suggestions would be appreciated.