Sufficient $\varepsilon$-$\delta$-criterion for polynomial tail decay

While reading a proof of the Bobkov-Houdre conjecture, I stumbled upon the following claim (Step 1 on page 1 of this paper):

If for a random variable $X$ and positive $\varepsilon$, $\delta$ there exists positive $t_0$ such that \begin{equation*} P \left( \left| X \right| \geq \left( \sqrt{2} + \varepsilon \right) t \right) \leq \frac{1}{2-\delta} P \left( \left| X \right| \geq t\right) \end{equation*} for $t \geq t_0$, then \begin{equation*} P \left( \left| X \right| \geq t \right) \leq \frac{C_p}{t^p} \end{equation*} for all $p \in (0,2)$.

As this claim is not proved it must be well-known, but unfortunately I was not able to fill in the details. Could somebody please help me out?

Solution 1:

Fix $p\in(0,2)$ and pick $\varepsilon$, $\delta$ small enough such that $$ \frac{\log(2-\delta)}{\log(\sqrt{2}+\varepsilon)}\ge p. $$ Now we can apply the assumption $n$ times to get $$ P(|X|\ge(\sqrt{2}+\varepsilon)^n t_0)\le \frac{1}{(2-\delta)^n}P(|X|>t_0)\le \frac{1}{(2-\delta)^n}\le \frac{1}{(\sqrt{2}+\varepsilon)^{np}}, $$ so that the property is readily verified for $t$ of the form $(\sqrt{2}+\varepsilon)^n t_0$, with $n$ integer. You can then extend the result for all $t$'s using the fact that $P(|X|\ge t)$ is non-increasing.