I am trying to compute a tight bound on a tail event.
Let $(X_n, n \geq 1), (Y_n, n \geq 1)$ be two independent sequences of i.i.d.$\sim\mathcal{N}(0,1)$ random variables, and let $Z_n =\sum_{j=1}^n X_jY_j$ for $n \geq 1$. Let also $t > 0$ be fixed. Show that there exists $c > 0$ such that $P(\{Z_n > nt\}) \leq \exp(−cn)$ for all $n \geq 1$.
I think I can not use Hoeffding's inequality as the $X_iY_i$ are not constrained to any interval. So I tried the following approach: I computed that for every $s\geq 0, E(\exp(sX_iY_i))=\frac{1}{\sqrt{1-s^2}}$,for $X\sim\mathcal{N}(0,1)$, which is well-defined if $|s|<1$, well-defined but infinite for $s=1$ or $s=-1$, and ill-defined otherwise $s=1/2$. Given this, I can make the following computation, for $s$ in the interval above:
\begin{align} P(\{Z_n > nt\}) &\leq E(\exp(sZ_n))\cdot \exp(-snt)\\ &= (\frac{1}{\sqrt{1-s^2}})^n \cdot \exp(-snt) \\ &= \exp(-\frac12n\log(1-s^2))\cdot \exp(-snt) \\ &= \exp(-n (\frac12\log(1-s^2)+st)) \end{align} Then I minimized over all the possible $s$ in that interval, computing the partial derivative of $(\frac12\log(1-s^2)+st)$ wrt to $\partial s$ and I got stuck because it is equal to $t-\frac{s}{1-s^2}$ and I am not able to conclude which $s$ gives me the minimum. Do you have some hints?