Tail bound for sum of random variables satisfying subgaussian upper tail bound

Question

So suppose you have a collection of random variables $X_1, \cdots X_n$ that are iid and they all satisfy the tail bound

$$ P(X_i-L>u)\leq \exp(-\frac{u^2}{2\sigma^2})$$

for all $u>0$. Is it true that the sum then satisfies a similarly inequality

$$ P(\sum_{i=1}^n X_i -L>u)\leq \exp(-\frac{u^2}{2C})$$

for some constant $C$ depending only on $\sigma^2$ and possibly $n$. I have searched thoroughly and could not find any information on this anywhere else.

Answer 1

This is not possible. Let $X_i=L$ with probability $1$. Then $P(X_i-L >u)=0$ for all $u >0$ so the hypothesis is satisfied. The conclusion fails when $u =\frac {(n-1)L} 2$.