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.

Thank you in advance


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 $n\to\infty$.


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