# Show that a sequence satisfies the weak law of large numbers

Let $$\{X_n\}_{n \geq 1}$$ be a sequence of independent random variables such that $$\mathbb P( X_n = \pm n^a)=1/2$$.

Show that if $$a<1/2$$, then the sequence satisfies the weak law of large numbers.

Since $$\mathbb E(X_i)=0 \; \;\forall \; \; i \geq 1$$, we need to show that $$\forall \; \; ε>0 \; \; \lim_{n \rightarrow \infty} \mathbb P\Biggl(\dfrac{1}{n} \Bigg|\sum_{i=1}^n X_i\Bigg| \geq ε\Biggr) = 0$$ but I don't know how to proceed.

Use Chebyshev's inequality $$\mathbb P\Biggl(\frac{1}{n}\Bigg|\sum_{i=1}^n X_i\Bigg| \geq \varepsilon\Biggr) \leq \frac{\text{Var}\left(\frac{1}{n}\sum_{i=1}^n X_i\right)}{\varepsilon^2} = \frac{\sum_{i=1}^n\text{Var}(X_i)}{n^2\varepsilon^2}. \tag{1}\label{1}$$ Here $$\text{Var}(X_i)=i^{2\alpha}$$ and $$\sum_{i=1}^n\text{Var}(X_i)=\sum_{i=1}^n i^{2\alpha}\sim n^{2\alpha+1}$$ Since $$2\alpha+1<2$$, $$n^{2\alpha+1}=o(n^2)$$. Therefore r.h.s. in \eqref{1} tends to zero as $$n\to\infty$$.