Let $(X_j)_{j\in \mathbb Z}$ be an strictly stationary sequence of random variables with: $$E[X_j]=0, \quad E[X_j^2] < \infty$$ I want to show that for each positive $\varepsilon$: $$ \sum_{n=1}^\infty \mathbb P\left(\max_{1\leq j\leq 2^n}\lvert X_j\rvert>2^{n/2}\varepsilon\right)<\infty. $$ Note that: $$ \mathbb P\left(\max_{1\leq j\leq 2^n}\lvert X_j\rvert>2^{n/2}\varepsilon\right)= \mathbb P\left(\bigcup_{j=1}^{2^n} \left[ |X_j| >2^{n/2}\varepsilon\right]\right)\leq \sum_{j=1}^{2^n} \mathbb P\left( |X_j| >2^{n/2}\varepsilon\right) $$ By stationarity, we have $$\sum_{j=1}^{2^n} \mathbb P\left( |X_j| >2^{n/2}\varepsilon\right)= 2^n \mathbb P\left( |X_1|>2^{n/2}\varepsilon\right)= 2^n \mathbb P\left( |X_1|^2>2^{n}\varepsilon^2\right)$$ My initial idea is to use a comparison test (I accept another solution). However, I am unable to finish.
Could you help me finish?