Suppose that the following statement is all what we know about the family of bounded random variables $(x_\epsilon)_{\epsilon > 0}$:
"There exists a positive constant $C>0$ and a positive random variable $\epsilon_1$ such that for $\epsilon \leq \epsilon_1$, \begin{equation} \mathbb{E} |x_\epsilon|^p \leq C \mathbb{E} \epsilon^{p}, \end{equation} for all $p > 0$".
How does the above statement on the random variable $x_\epsilon$ relate to the standard modes of convergence in probability theory (almost sure, in probability, etc)?
We can conclude that $x_{\epsilon} \to 0$ in probability. Let $\delta_n$ be positive numbers decreasing to 0 and put $\epsilon = \min \{\epsilon_1,\delta_n \}$ in the given inequality. The right side $\to 0$. It follows that $x_\min \{\epsilon_1,\delta_n\} \to 0$ in $L^{p}$, hence in probability. Now $P\{|x_{\delta_n}| >\eta\} \leq P\{|x_\min \{\epsilon_1,\delta_n\}| >\eta \}+P\{\epsilon_1 <\delta_n\}$. the last term $\to 0$ because the events $\{\epsilon_1 <\delta_n\}$ decrease to empty set.