I was trying to add some details to a result concerning the strong law of large numbers for pairwise independent random variables : here's what I've done :
Let $\epsilon > 0,1<p<2, \, (X_i)_{i\geq1}$ be a sequence of identically distributed and pairwise (not mutually!) independent random variables :
\begin{multline*} P\left \{\left|\sum_{k=1}^n (X_k-EX_k) \right|>\varepsilon n\right\} = P\left \{\left|\sum_{k=1}^n (X_k-EX_k) \right|>\varepsilon n, \, \exists k \, : |X_k| >n \right \} \\ + P\left \{\left|\sum_{k=1}^n (X_k-EX_k) \right|>\varepsilon n, \, \forall k \, : |X_k|\leq n \right\} \\ \leq P\left \{ \exists k \, : |X_k| >n \right \} + P\left \{\left|\sum_{k=1}^n (X_k-EX_k)I(|X_k| \leq n) \right|>\varepsilon n, \, \forall k \, : |X_k|\leq n \right\} \\ \leq P\left \{ \bigcup_{k = 1}^{n}|X_k| >n \right \} + P\left \{\left|\sum_{k=1}^n (X_k-EX_k)I(|X_k| \leq n) \right|>\varepsilon n \right\} \\ \leq \sum_{k = 1} ^{n} P\left \{|X_k| >n \right \} + P\left \{\left|\sum_{k=1}^n (X_k-EX_k)I(|X_k| \leq n) \right|>\varepsilon n / 2 \right\} \end{multline*}
and so by the above inequality, Markov's inequality, the triangular inequality and using the fact that the $X_k's$ are pairwise independent we get :
\begin{multline*} \sum_{n = 1}^{\infty}n^{-1} P \left \{\left|\sum_{k=1}^n (X_k-EX_k) \right|>\varepsilon n\right\} \leq \sum_{n = 1}^{\infty}n^{-1} \sum_{k = 1} ^{n} P\left \{|X_k| >n \right \} \\ + \sum_{n = 1}^{\infty}n^{-1} P\left \{\left|\sum_{k=1}^n (X_k-EX_k)I(|X_k| \leq n) \right|>\varepsilon n / 2 \right\} \\ \leq \sum_{n = 1}^{\infty}n^{-1} \sum_{k = 1} ^{n} n^{-p} E|X_k|^p + \sum_{n = 1}^{\infty}n^{-1} \frac{4n^{-2}}{\varepsilon^2}( E \sum_{k = 1} ^{n} \left|(X_k-EX_k)I(|X_k| \leq n) \right|^2 + 0) \\ \leq \sum_{n = 1}^{\infty}n^{-1-p} \sum_{k = 1} ^{n} E|X_k|^p + c \sum_{n = 1}^{\infty}n^{-3} \sum_{k = 1} ^{n} E \left|X_k \right|^2 I(|X_k| \leq n) \\ \end{multline*}
now all I need to reach the conclusion in the paper is to show that : $\sum_{n = 1}^{\infty}n^{-3} \sum_{k = 1} ^{n} E \left|X_k \right|^2 I(|X_k| \leq n) \leq c \sum_{k = 1} ^{n} E \left|X_k \right|^p \sum_{n =k}^{\infty}n^{-1-p} $