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I'm reading this book on concentration inequalities and I'm trying to solve all of the exercises in the book. The following problem is from the book which I couldn't manage to solve. I have also posted it on MSE a while ago, but I received no response, yet. I'd appreciate it if anyone can give me a hint for solving this problem.

Probelm : Let $X_1,X_2,\ldots ,X_n$ be iid copies of a real random variable $X$ that for some $p\geq 1$ obeys

\begin{align*} \mathbb{P}\left(\left\vert X\right\vert > u\right)& \leq \exp\left(-u^p\right), \end{align*}

for all $u>0$. For any $s\in \mathbb{R}^n$ and with $q$ denoting the conjugate of $p$ (i.e., $1/p+1/q=1$) prove that

\begin{align*} \mathbb{P}\left(Z:=\sum _{i=1}^n s_i X_i > t \right)&\le L \exp\left(-\frac{1}{L}\min \left(\frac{t^2}{\left\Vert s\right\Vert_2^2},\frac{t^p}{\left\Vert s\right\Vert_q^p}\right)\right), \end{align*} where $L>0$ is a constant that only depends on $p$, but not $n$.

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Here's some work. I don't think this counts as a complete answer, though. Sorry.

First, $$ \left|\sum_i s_iX_i\right| \le \sum_i|s_i X_i| \le \left(\sum_i |s_i|^q\right)^{1/q}\left(\sum_i |X_i|^p \right)^{1/p}, $$ by the triangle inequality and Holder's inequality.

Taking this, \begin{align*} P(|Z|>t) &\le P\left( \sum_i |X_i|^p > t^p\left(\sum_i |s_i|^q\right)^{-p/q}\right) \\ &= P\left( \sum_i |X_i|^p > \frac{t^p}{ \left\Vert s\right\Vert_q^p }\right) . \end{align*}

I don't know how to sharpen it with the minumum, though. I tried using the fact that $P(\max_i |X_i| \le t) = P(|X_i|\le t)^n$ and the complement rule to get: $$ P\left( \sum_i |X_i|^p > \frac{t^p}{ \left\Vert s\right\Vert_q^p }\right) \le P\left( \max_i |x_i|^q > \frac{t^p}{ n \left\Vert s\right\Vert_q^p }\right) \le 1 - \left[1 - \exp\left(-\left\{\frac{t^p}{ n \left\Vert s\right\Vert_q^p }\right\}^p \right) \right]^n. $$ Not sure though.

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