Let $X\_1,...,X\_N$ be IID, mean-zero random variables whose tail is bounded by a subgaussian-tailed variable to the fourth moment, i.e., for some $t \ge t\_0 > 0$
$$
P(|X\_i| > t) \le C\exp\left( -C t^{1/2} \right)
$$
Then what is the best upper bound one can get for the 
$$
P\left( \left|\frac{1}{\sqrt{N}} \sum\_{i=1}^N X\_i \right| > t\right)
$$
?



Concentration inequality for subgaussian^4
For $\gamma\gt 0$ and a random variable $X$, define 
$$\lVert X\rVert\_{\Phi\_\gamma}:=\inf\left\{c\gt 0;\mathbb E\left[\exp\left(\left|X/c\right|^\gamma\right)\right] \leqslant 2\right\}.$$
For $\gamma\geqslant 1$, this defines a norm, and for $0\lt \gamma\lt 1$, this is equivalent to a norm. Obverse that Markov's inequality and the monotone convergence theorem entail 
$$\mathbb P \left\{\left|X\right|\geqslant x\right\}\leqslant 2\exp\left(-\frac{x^\gamma}{\lVert X\rVert^\gamma\_{\Phi\_\gamma}}\right),$$
and rewriting the expectation of the exponential in terms of norm, we derive that if $Y$ satisfies 
$$\mathbb P \left\{\left|Y\right|\geqslant t\right\}\leqslant K\exp\left(-\frac{t^\gamma}{\lambda^\gamma}\right)$$
for each positive $t$, then $\lVert Y\rVert\_{\Phi\_\gamma}\leqslant \left(1+K/2\right)^{1/\gamma}\lambda$. Therefore, the problem reduces to find a good upper bound for $\lVert \sum\_{i=1}^nX\_i\rVert\_{\Phi\_\gamma}$. This is precisely the purpose of Theorems 3 and 4 in [Isoperimetry and integrability of the sum of independent Banach-space valued random variables](https://projecteuclid.org/download/pdf_1/euclid.aop/1176991174) by Michel Talagrand. In particular, this gives that if $X\_1$ satisfies 
$$\mathbb P \left\{\left|X\_1\right|\geqslant x\right\}\leqslant c\_1\exp\left(-c\_2t^\gamma\right)$$
then the quantity $\lVert \sum\_{i=1}^nX\_i\rVert\_{\Phi\_\gamma}/\sqrt n$ can be bounded independently of $n$.


