Looking for the weakest possible conditions to add to $|X_n/n|$ being $L^1$ convergent so that I can deduce that $X_n$ is convergent in probability.

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Basically, I'm working on some probability assumptions and my goal is to prove the convergence in probability of $X_n$, as of now I could succeed in proving the $L^1$ Convergence of $|X_n/n|$, so I was wondering if there were any extra conditions that I could maybe allow myself to additionally assume to deduce the convergence in probability from that.