Assume $X_i$ are i.i.d with mean $μ$ and the fourth moment of $X_i$ exists. Let $δ < 2$. Show $n^δ Pr(| X_n − μ| >) → 0$ as $n → ∞$.
I know $Pr(| X_n − μ| >)→ 0$ as $n → ∞$ by the WLLN, but I have no idea what to do with $δ$, since that value doesn't converge. I have thought about using Slutsky's Theorem, but again, $n^δ$ doesn't really converge to anything, so I don't know what to do with this.
I think that you have left out an $\epsilon > 0$ in your expression: that is, you want to show that $n^{\delta} \mathbb{P}(|X_n - \mu| > \epsilon) \to 0$ as $n \to \infty$.
Here is a hint: consider using Markov's inequality. Recall that this says that for a non-negative random variable $Z$ with finite mean, we have $\mathbb{P}(Z \geq t) \leq \frac{\mathbb{E}{Z}}{t}$ for all $t \geq 0$. By taking powers inside, note that $\mathbb{P}(Z \geq t) = \mathbb{P}(Z^k \geq t^k)$, where $k \geq 0$.
How might you be able to use this here?