Chernoff-Cramer bound

The Chernoff-Cramèr inequality is a very general and powerful way of bounding random variables. Compared with the famous Chebyshev bound, which implies inverse polynomial decay inequalities, the Chernoff-Cramèr method yields exponential decay inequalities, at the cost of needing a few morehypotheses on random variables’ .

Theorem: (Chernoff-Cramèr inequality)

Let ${\{X_i\}}_{i=1}^n$ be a collection of independent randomvariables such that  $\forall i$ in a right neighborhood of $t=0$, i.e. for any $t\in (0,c)$ (Cramèr condition).

Then a zero-valued for $x=0$, positive, strictly increasing, strictly convex function $\mathrm{\Psi }(x):[0,\mathrm{\infty })\mapsto R^{+}$ existssuch that:

$$\mathrm{Pr}\left\{\sum _{i=1}^n\left(X_i-E[X_i]\right)>\epsilon \right\}\le \exp \left(-\mathrm{\Psi }(\epsilon )\right)\forall \epsilon \ge 0$$

Namely, one has:

$\mathrm{\Psi }(x)$$=$

$\psi (t)$$=$${\displaystyle \sum _{i=1}^n}\ln E\left[e^{t\left(X_i-E{X}_i\right)}\right]={\displaystyle \sum _{i=1}^n}\left(\ln E\left[e^{t{X}_i}\right]-tE\left[X_i\right]\right)$

that is, $\mathrm{\Psi }(x)$ is the Legendre of the cumulant generating function of the ${\sum }_{i=1}^n\left(X_i-E[X_i]\right)$ random variable.

Remarks:

1) Besides its importance for theoretical questions, the Chernoff-Cramér bound is also the starting point to derive many deviationor concentration inequalities, among which Bernstein (http://planetmath.org/BernsteinInequality), Kolmogorov, Prohorov (http://planetmath.org/ProhorovInequality), Bennett (http://planetmath.org/BennettInequality),Hoeffding and Chernoff ones are worth mentioning. All of theseinequalities are obtained imposing various further conditions on $X_i$ random variables, which turn out to affect the general form of the cumulant generating function $\psi (t)$.
2) Sometimes, instead of bounding the sum of n independent random variables, one needs to estimate they ” i.e. the quantity $\frac{1}{n}{\sum }_{i=1}^n{X}_i$; in to reuse Chernoff-Cramér bound, it’s enough to note that

$$\mathrm{Pr}\left\{\frac{1}{n}\sum _{i=1}^n\left(X_i-E[X_i]\right)>{\epsilon }^{\prime }\right\}=\mathrm{Pr}\left\{\sum _{i=1}^n\left(X_i-E[X_i]\right)>n{\epsilon }^{\prime }\right\}$$

so that one has only to replace, in the above stated inequality, $\epsilon$ with $n{\epsilon }^{\prime }$.
3) It turns out that the Chernoff-Cramer bound is asymptotically sharp, as Cramèr theorem shows.