Let $a_1,a_2,\dots$ be random variables taking values in $\mathbb{R}^n$. Suppose they are i.i.d with mean zero. Given an arbitrary norm on $\mathbb{R}^n$, I want to know how fast $\mathbb{E}[\|\frac{1}{n}\sum\limits_{i=1}^{n}a_i\|^{2}]$ tends to zero.
For example, if $\|\cdot\|$ is the Euclidean norm, then it is an elementary result from statistics that $ \mathbb{E}[\|\frac{1}{n}\sum\limits_{i=1}^{n}a_i\|_2^{2}]\leq \frac{1}{n}\mathbb{E}[\|a_1\|_2^{2}]$.
If we have another norm, then because norms on $\mathbb{R}^n$ are equivalent, I can say that there is a $C$ depending on the particular norm such that $\mathbb{E}[\|\frac{1}{n}\sum\limits_{i=1}^{n}a_i\|^{2}] \leq \frac{C}{n}\mathbb{E}[\|a_1\|^{2}]$.
However, in my application I need to find such a $C$ that works simultaneously for many (infinitely many) norms. Is this possible?