TITLE: Hoeffding type inequality for bounding deviation of a sequence
QUESTION [5 upvotes]: I'm am trying to work out an Hoeffding type inequality for upper bounding
$$\mathbb{P} \left\{ \bigcap_{n=1}^N \{S_n - \mathbb{E}[S_n] \geq nt\} \right\} $$
where $S_n = \sum_{i=1}^n X_i$, $X_k$ are iid (note that $S_{n+1}$ and $S_n$ are dependent). We can assume that $X_k$ has bounded support for applying Hoeffding inequality but I can't figure out how to do it.
My approach was to consider the complement of the event and to try and use union bound but that's too weak and I end up with a trivial upper bound greater than $1$.

REPLY [4 votes]: To simplify notations, let me assume without loss of generality that all the $X_i$'s are centered and bounded by 1 in absolute value. For sure, one trivial upper bound to your quantity is
$$
 \mathsf P\left(\bigcap_{n\le N} (S_n\ge nt)\right)\le \mathsf P\left(S_N\ge Nt\right)\le e^{-t^2N/2}.
$$
In fact, you cannot hope to do much better than this in general. Indeed, you can always lower bound
$$
\mathsf P\left(\bigcap_{n\le N} (S_n\ge nt)\right)\ge \mathsf P\left(\bigcap_{n\le N} (X_n\ge t)\right)=\mathsf P\left(X_1\ge t\right)^N.
$$
And we can make the right-hand side arbitrarily close to the upper bound, at least up to universal constants (for example, take a standard Gaussian and truncate it to stay in $[-1,1]$).