In the discrete time case, we have the following result:
Let $\gamma>0$ and $p_{0}\in \mathbb{N}$. Moreover let $(Z_{n})_{n\in\mathbb{N}}$ be a sequence of random variables. If for every $p\geq p_{0}$ there exists a constant $c_{p}>0$ such that for all $n\in \mathbb{N}$, \begin{eqnarray*} \left(\mathbb{E}\left|Z_{n}\right|^{p}\right)^{1/p}&\leq&c_{p}\cdot n^{-\gamma}\,, \end{eqnarray*} then for all $\varepsilon>0$ there exists a random variable $\eta_{\varepsilon}$ such that \begin{eqnarray*} \left|Z_{n}\right|&\leq&\eta_{\varepsilon}\cdot n^{-\gamma+\varepsilon}\,, \end{eqnarray*} almost surely, for all $n\in \mathbb{N}$. Moreover, $\mathbb{E}\left|\eta_{\varepsilon}\right|^{p}<\infty$ for all $p\geq 1$.
Now, my question is that: Is the following result is correct in contentious time ?
Let $\gamma>0$ and $p_{0}\in \mathbb{N}$. Moreover let $(Z_{T})$ be a random variables. If for every $p\geq p_{0}$ there exists a constant $c_{p}>0$ such that for $T \to \infty$, \begin{eqnarray*} \left(\mathbb{E}\left|Z_{T}\right|^{p}\right)^{1/p}&\leq&c_{p}\cdot T^{-\gamma}\,, \end{eqnarray*} then for all $\varepsilon>0$ there exists a random variable $\eta_{\varepsilon}$ such that \begin{eqnarray*} \left|Z_{T}\right|&\leq&\eta_{\varepsilon}\cdot T^{-\gamma+\varepsilon}\,, \end{eqnarray*} almost surely.
Thanks for all your kindly help!