Let $X_1,X_2,\ldots,X_n$ be an i.i.d. sequence of $n$ positive random variables with mean $E[X_1]=\mu_X<\infty$ and the second moment $E[X_1^2]=\infty$.

I am interested in upper-bounding $P\left(|\frac{1}{n}\sum_{i=1}^nX_i-\mu_X|\geq x\right)$ (though a lower bound on $P\left(\frac{1}{n}\sum_{i=1}^nX_i-\mu_X \geq x\right)$ would be great as well). My go-to method for this is Chebyshev's inequality, however, the second moment for $X_1$'s (and, hence, the variance) is infinite so that's of no use.

However, I do have an expression for the tail probability of individual member of this i.i.d. sequence, $P(X_1>x)$ in a (more-or-less) nice form. Can I use it to get the desired bound(s) on the sequence average?