Let $(X_i)_{i\in \mathbb{N}}$ a sequence of i.i.d. random variables, such that
$$\mathbb{P}(X_i=1)=p=1-\mathbb{P}(X_i=-1)$$
Suppose that $p\geq\frac{3}{4}$ and consider $S_n=\sum_{i=1}^{n}X_i$. I want to show that $\mathbb{P}(S_n<\frac{1}{2}n)$ decays exponentially in $n$. Any suggestion of concentration's inequalities that can help me?
Let $B_n:=(S_n+n)/2$ s.t. $B_n\sim \text{Bin}(n,p)$. Using the bound derived in this paper, one gets
\begin{align} \mathsf{P}(S_n<n/2)&=\mathsf{P}(B_n\le \lceil3n/4-1\rceil) \\ &\le \Phi\!\left(\operatorname{sgn}(a_n-np)\sqrt{2n H(p,a_n/n)}\right), \end{align} where $a_n:=\lceil 3n/4-1\rceil+1$ and $H(p, c):=c\ln(c/p)+(1−c)\ln((1 − c)/(1 − p))$. For $p>3/4$, the sign of $(a_n-np)$ becomes negative for all $n$ large enough so that, eventually, $$ \mathsf{P}(S_n<n/2)\le 1-\Phi\!\left(\sqrt{2n H(p,a_n/n)}\right)\le \frac{\phi(\sqrt{2n H(p,a_n/n))}}{\sqrt{2n H(p,a_n/n)}}. $$