Is anything wrong in the following reasoning?
Suppose that $X_1$ is a real-valued random variable such that $E[|X_1|^t] \leq \gamma$ for some $\gamma \in (0, \infty)$ and some $t > 0$. From Markov's inequality, for any $x \geq 0$, $P[X_1 > x] \leq 1 \wedge \gamma x^{-t}$. Let $X_2$ be a random variable with cumulative distribution function $F_2$.
For any $x \in \mathbb{R}$, let $F_2(x) = 1_{[\gamma^{-1/t}, \infty)}(x)(1 - \gamma x^{-t})$. Then, for any $x \in \mathbb{R}$, if $x \leq 0$, $$P[X_1 > x] \leq 1 = P[X_2 > x],$$ and if $x > 0$, $$P[X_1 > x] \leq 1 \wedge \gamma x^{-t} = P[X_2 > x].$$
Therefore, this shows that any random variable $X_1$ with $E[|X_1|^t] \leq \gamma$ is stochastically dominated by $X_2$.